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Sunday, September 27, 2020

The market for Covid-19 vaccines and the tipping point to herd immunity

by Ajay Shah.

Many firms are developing Covid-19 vaccines. Enormous resources have to be deployed, up front, to develop a vaccine and to build manufacturing capacity. It is likely that many vaccines will get through to approval in mature regulatory regimes. Not all vaccines will work identically for all situations, e.g. some vaccines may work better for an elderly person than others.

It is commonly assumed that the global market size for a Covid-19 vaccine is about 6 billion people. In this article, we argue that this might not be the case. Let's think about the situation in the market once one or more vaccine reaches the market.

The buyers perspective before vaccine sales have commenced

The private gain for an individual from buying a vaccine are shaped by the probability of getting sick when leading an unconstrained life. This is shaped by the extent to which Covid-19 has burned through the communities that the person plans to engage with. As an example, in the slums of Bombay or Delhi, herd immunity has set in. A person living there knows that few people in her circles are now getting sick, and she feels relatively safe. Well known factors such as age and co-morbidities will also shape the threat perception of each person. Therefore, for her, the gains from a vaccine are relatively modest, and the willingness to pay is small.

In each city of the world, there is a different numerical value for the attack rate (the fraction of people who are infectious) and the extent of immunity. The state of the epidemic in Pune is different from that in Bombay. As time passes, each city is inching towards herd immunity, and the passage of time thus diminishes interest in paying for a vaccine. Vaccine IP and manufacturing facilities are wasting assets.

It it were possible to develop a combination of tests that add up to an immunity passport', then the price of this test and the odds of coming out positive would shape the demand function for the vaccine.

Progress on immunisation and herd immunity

Into this world, let us imagine that the sale of multiple vaccines commences. At first, there would be a rush of demand and high prices. As immunisation progresses, the attack rate would go down and the gains from buying the vaccine would further go down. In places like Bombay and Delhi, where a considerable proportion of the population has already been exposed to the disease, when a modest fraction of the population is vaccinated, this could tip the population over into herd immunity, and the disease could die down.

In such a world, vaccine makers face the prospect of a short hot market. At first, vaccine demand will be high and the factories will not be able to keep pace. Competition will come about and that will exert pressure on prices. In a city like Bombay, with about 20 million people, after (say) 5 million persons buy the vaccine, this may significantly change the threat perception in the eyes of the average individual. Vaccine demand would then decline.

Under such numerical values, the market potential in Bombay is not roughly \$50$\times$20 million people or \$1 billion, but perhaps more like \$25$\times$5 million people or about \$125 million.

All of this reduced revenue potential will go to the first few firms that get 5 million doses into the Bombay market. Competition would exert downward pressure on the price, demand would tail off as herd immunity sets in, and there would be a price crash. The late comers would flood the market with output but would obtain low revenues in return.

The vaccine demand collapse in a simple model and in the real world

We have always known that a vaccine is not just a private good; there is a positive externality. The novel idea of this article is about tipping points.

Consider a simple model in which herd immunity is achieved at 60%. Suppose 50% of the population is already immune and knows it. The first 10% that gets the vaccine tip the system over to $R_0<1$ and then the fires start dying out. Once the fires start dying out, the attack rate goes down, the threat perception changes, and the incentive for private people to buy the vaccine drops a lot. Under these conditions, the positive externality imposed by vaccine purchase by the early vaccine buyers, upon the overall system, is particularly large.

A key factor that drives behaviour in this model is that when a person is immune, she knows it and then has no incentive to buy a vaccine. In the real world, people don't know whether they are immune, and would be more inclined to buy a vaccine just to be safe. In the limit, the veil of ignorance is complete, nobody is able to assess the threat, and everyone wants to buy a vaccine.

In the real world, the veil of ignorance is not complete. At every place, people do have a personal judgement about the threat level based on the extent to which their friends and family are getting sick (or not) per month. Age and co-morbidities will also shape vaccine demand. As a general principle, it is always wise to think that humans are sentient optimising creatures. Individuals have a noisy estimator of the threat that they face and this will shape their willingness to pay for a vaccine.

Wall street tells Main street what to do

These problems feed into the thought process of private firms and shape the commitments of capital to the problems of vaccine development and manufacturing when faced with a novel epidemic.

Numerous vaccines are under development. The process of vaccine approval is necessarily slow. At present, we generally think that over time, one by one, many of these vaccines will get through to the market. By the reasoning of this article, the first few will get through, within a few months the market will collapse, and all funding will be yanked for other projects. This will be a bit reminiscent of how funding for vaccines against Sars-Cov-1 was abruptly yanked when the funders realised that Sars-Cov-1 had reached $R_0<1$.

The numerical values used here (e.g. 60% for herd immunity, 5 million immunised in Bombay to tip over into herd immunity, $50, etc.) are of course purely illustrative. To translate these ideas into practical calculations requires data on the extent to which immunity has come about. In many places worldwide, there are good estimates of the persons who have antibodies, but there is more to immunity than measured antibodies. In India, the information available about the state of the disease in (say) Bombay is rather poor. If we take this dynamics of the vaccine market seriously, vaccine makers have an incentive to create such datasets. Alongside the construction of such datasets, there is a need for derivatives trading on underlyings such as the fraction of Bombay residents who have antibodies. The argument of this article is a special case of the long-standing problems of incentives for vaccine development. An effective pathway for state intervention, and philanthropic capital, lies in offering contracts for R&D and manufacturing which change the incentives of private persons to engage in these activities. Implications To the extent that this reasoning is correct, individuals will at first face a vaccine market with high prices and shortages. For many individuals, particularly for low-risk persons, there is a tradeoff between paying more to get the vaccine early versus paying less to get it late or even to not get vaccinated if the pandemic has subsided. For firms with a vaccine under development, this article paints a winner-takes-all scenario, where the first few vendors who get output on scale will capture all the revenue. To the extent that this reasoning is correct, plodding along to the finish line late will induce low revenues. For policy makers and philanthropic capital, it is important to avoid a coronavirus winter', a collapse in coronavirus research of the kind which happened after the SARS epidemic achieved$R_0<1\$. There is enormous knowledge, and capable teams, which has been created by the early gold rush of building vaccines against SARS-Cov-2. This knowledge should not be lost. As an example, it would be nice if research groups will publish research papers and release code before they put out the lights. We need to think of the sustainable frameworks, where we achieve a new normal of high R&D into pathogens that can trigger pandemics.

Tuesday, September 22, 2020

Introduction

The Covid-19 pandemic has forced all of us to live, work, learn, and communicate online. This has led to an increase in the demand for reliable, efficient, and speedy internet access during the pandemic. However, those who are already disadvantaged are suffering greater digital exclusion during this time, in the form of inadequate internet connectivity. Thus, countries, especially developing countries have been presented with an opportunity to deploy different regulatory and policy tools to improve internet access and provide meaningful internet connectivity to their citizens.

In my recently published paper for the Digital Pathways at Oxford Paper Series, I try and understand how Covid-19 has served as a catalyst for positive regulation in improving internet connectivity through the discussion of initiatives taken by four governments - Panama, South Africa, Kenya, and the State of Kerala (in India). Specifically, I ask two questions:

• What regulatory and policy steps were taken by governments and regulators to meet the increased demand for access to the internet during the Covid-19 pandemic?
• What changes in regulation are necessary to nudge mobile network operators (MNOs) to work with governments to ensure continued and affordable access to the internet?

The paper examines the different approaches that have met with some success in the four countries and provides various policy options for governments to maintain and improve internet connectivity during the pandemic. In this blog post, I will be summarising the various policy options that are available to governments.

Temporary allocation of spectrum by regulators

Low spectrum allocation adversely impacts network infrastructure and performance; reduces the reliability and quality of mobile broadband services; and can affect the future deployment of mobile broadband technologies. Thus, one option available with governments to maintain efficient and reliable internet connectivity during a crisis is to temporarily allocate unassigned spectrum to MNOs in a fair and non-discriminatory manner. Through the four cases discussed, the paper found that spectrum allocation is a viable option during an emergency only in cases where the total (permanent) allocation of spectrum to MNOs has been inefficient. Thus, while this option was expressly considered/offered by all four countries, there was uptake only in Panama and South Africa. Despite the opportunity cost to allocating spectrum free of charge (in terms of foregone revenue from auction proceeds), both these countries pursued this method in view of the insufficient existing spectrum allocation and by attaching certain conditions to the allocation of spectrum.

The regulators in Panama and South Africa temporarily allocated spectrum to MNOs through the passage of emergency resolutions and regulations, which set out:

• The frequency of spectrum that was open for temporary allocation;
• The duration of the temporary allocation;
• The application procedure and the requirements that needed to be met by applicants, such as demonstrating network performance (as in South Africa);
• Whether the temporary allocation was free of charge or not - in both Panama and South Africa the allocation was free of charge; and
• The conditions or expectations that were tied to the allocation of spectrum, in the form of reduced data costs or network expansion that could benefit the users.

While allocating spectrum during an emergency, countries should take care to ensure that such spectrum allocation does not become permanent; affect the long-tem spectrum allocation plan of the regulator; and does not reduce the overall competitiveness of the sector, by entrenching the dominance of a few players. The paper also briefly discusses alternative innovative approaches to spectrum management that could have been employed by the regulators.

Temporary freeze on internet and mobile payments

The COVID-19 pandemic has forced people to stay at home. This has meant that people are even more reliant on the internet to work from home, to study, to have any medical consultation, to stay in touch with friends and family, or to consume online entertainment. At the same time, the impact of the pandemic has been the hardest on the marginalised sections of society, who may find it difficult to keep up payments on their internet or mobile bills.

In such a situation, another policy option that can be considered by governments is to put in a place a temporary freeze on internet and mobile payments for a certain period of time, as was done in Panama. Drawing from the Panamanian example, any government considering such a temporary freeze can use a legal instrument that clearly defines:

• The duration of the temporary freeze on payments, i.e. the time period for which payment of any internet or mobile bill can be suspended and deferred;
• The criteria for intended beneficiaries, depending on whether the benefit is expected to be universally applied or restricted to a smaller identified class (as in Panama);
• The method of repayment, specifying the time period over which the pending bills have to be paid, the number of instalments, and whether the repayment is interest free;
• Whether there is any impact on the credit history of an individual if they avail of this measure - In Panama, the government clarified that there would be no impact on credit history or the quality of services offered by MNOs.

Having a clear narrowly tailored legal instrument that lays down the obligations of the MNOs, avoids a situation as in India, where the industry body, the Cellular Operators Association of India and the regulator, TRAI were at loggerheads about the nature of the obligation placed on MNOs. TRAI had wanted the MNOs to ensure continued service to all prepaid SIM card owners during the period of the lockdown, whereas the MNOs wanted to restrict it only to the "underprivileged and needy customers", so as to avoid an unjustified subsidy for a larger customer basis.

Prohibition on price increase

Similar to the temporary freeze on internet payments, where MNOs are required to continue service for a limited duration, even in cases of non-payment by users (albeit with an obligation to pay back), this regulatory measure prohibits a price increase in the data plans during the period of the emergency/pandemic. Such a measure was put in place in South Africa through the Electronic Communications, Postal, and Broadcasting (ECPB)Directions in March, although the ECPB Directions were amended in May 2020 to remove such restrictions.

Implementation of tax measures

In order to encourage MNOs to pass on certain benefits to consumers, in the form of reduced data costs, governments can implement various tax measures, such as reducing Value Added Tax, as in Kenya, where the VAT was reduced across the board from 16% to 14% with effect from 01 April 2020. In Panama, the government introduced a "Solidarity Plan" or "Plan Solidario" as a temporary support program to mitigate the economic impact of COVID-19. As part of this Plan, the government offered MNOs income tax deduction on any contributions in cash or kind or any other services towards the government's crisis efforts. Partially in response to this, all the MNOs in Panama came together to announce a "Solidarity Mobile Plan", which was a free basic package for accessing the internet.

Support to telecom infrastructure service providers

Another option available with governments, while not directly regulatory in nature, involves coordination and cooperation with the telecom infrastructure service providers to ensure uninterrupted internet service during a lockdown. This is because practical problems such as inadequate/interrupted power supply, or the necessary municipal officials not being available may affect the service providers.

The State of Kerala in India was unique amongst the cases discussed in the paper in that the Kerala State IT Department worked with the Kerala State Electricity Board in identifying the mobile towers that were exclusively reliant on Diesel Generator (DG) sets, and monitoring the regular supply of diesel to these towers during the national lockdown. This was done to prevent major power outages that could disrupt cell service. In South Africa, to support the legal obligation for continued service by ISPs and MNOs, the telecom infrastructure providers were given regulatory support, in the form of prioritised infrastructure approvals, postponement on license fee renewals, and temporary deferment of wayleaves.

Utilisation of the un-utilised money in the Universal Access Fund

Countries such as Kenya and India have a form of a universal access fund, which comprises of mandatory contributions by MNOs. For instance, in India, a universal access levy is statutorily levied on MNOs to contribute towards the "Universal Service Obligation Fund", which is aimed at ensuring widespread, non-discriminatory, and universal access to ICT services in India. A large amount of money is lying un-utilised in the funds in both these countries (estimated to be INR 51,500 crore in India) and this money can potentially be re-directed by the government towards specific connectivity measures during an emergency, such as reducing data costs or improving network resilience. Alternatively, as in Colombia, governments can temporarily suspend the payment obligations into these universal access funds for MNOs, so that the savings can be passed on to the users.

Provision of zero-rated access to specific websites

In some countries, such as in South Africa, data costs are fairly high, leading to real concerns about affordability and accessibility to the internet during a pandemic. In such a situation, providing zero-rated access, i.e. free access, to certain important health and education websites, may help keep citizens up to date about the latest medical information and research about the virus; as also help students access online educational resources. Consequently, the South African government passed a law requiring MNOs to provide zero-rated access to certain government and local educational websites. Currently, over 1000 health and education websites are offering zero-rated (i.e. free) access to their content.

However, it is worth bearing in mind that zero-rating is a complex issue, especially as the debate in India demonstrates. Hence, any government adopting such a policy should consider the following factors:

• Zero-rating is often technically complex to implement, since ISPs may not have the technical architecture to distinguish amongst the websites visited by a particular user; and hence, will be unable to determine whether the user is accessing a COVID-19 zero-rated website or a regular website;
• A zero-rating pandemic policy may limit the zero-rated websites to health and educational websites. However, as a matter of practice, with many websites, notably YouTube, it is difficult to distinguish between the educational and entertainment value of the website;
• Any requirement, as imposed in South Africa, that the zero-rated website must be a "local" educational content website, may run into the problem that even "local" websites host third party non-local content in the form of embedded videos and text or store the content on a foreign cloud server. Theoretically, access to these websites will not be "local", and hence, they will fall outside the intended benefit of the government's zero-rating policy;
• Finally, and perhaps most importantly, zero-rating inherently involves privileging certain websites and content over the other, whether the decision is being taken by the government or MNOs or both. In the long run, this can threaten and potentially undermine net neutrality.

Regular monitoring of network capacity

Maintaining reliable and uninterrupted access to the internet also involves ensuring that the back end of the entire telecom infrastructure service system works. Thus, governments can coordinate with MNOs to regularly monitor network performance to assess whether there is sufficient network capacity to meet the increased demand for internet access during a pandemic. Collecting the requisite data will help the government form empirically sound policy. For instance, in Kerala, much before the national lockdown was announced, the State IT Secretary held a meeting with the Telecommunication Department and and all the MNOs in the Kerala Circle to understand internet consumption pattern, bandwidth utilisation, and network capacity in the state. The MNOs had agreed to increase network capacity by 30-40% if required. However, as it turned out, based on the periodic reports that were submitted by MNOs, the government and the MNOs realised that there was no need to increase server capacity or allocate additional spectrum. Similarly, in April, the Kerala State IT Department issued a government order approving the upgradation of networks from 3G to 4G by MNOs in specific tower locations, which had otherwise been delayed.

Conclusion

By examining the regulatory response in Panama, South Africa, Kenya, and India (specifically Kerala), the paper presents various policy options that can be used by a government to improve maintain and improve internet connectivity during a pandemic. Although the paper is situated within the COVID-19 pandemic response, the policy options can be used in any emergency situation that creates additional stress on the existing digital divide and infrastructure. It is worth noting that the paper only focuses on temporary regulatory measures that are intended to maintain and improve internet connectivity during the period of an emergency, although these may have medium to long-term benefits as well.

However, any regulatory measure undertaken by a government to improve internet connectivity during an emergency should be capable of having a tangible impact in the short-term, apart from/ in addition to any medium or long-term benefits. This is because in an emergency such as the COVID-19 pandemic, any increase in the demand for the internet or reduced capacity to afford continued internet services requires immediate and urgent policy intervention. For instance, in Kenya, the pandemic expedited the approval of the innovative Loon Project, that is aimed at using the high altitude internet balloons to bring 4G coverage to underserved and remote areas of Kenya. While this is certainly an innovative example of positive regulation, it may not have the desired short-term effects given the complexity and scale of the project. The Kenyan government also constituted a COVID-19 ICT Advisory Committee that was commissioned to submit a report on the methods of improving "universally affordable connectivity" within six months. However, there is an opportunity cost of establishing a Committee in the middle of a pandemic, in that the government's time and money could have been better spent in pursuing other positive regulations.

References

AA4I (2020): Alliance for Affordable Internet, Meaningful connectivity: A new standard to raise the bar for internet access.

Bhandari(2020): Vrinda Bhandari, Improving internet connectivity during COVID-19, Digital Pathways at Oxford Paper Series No. 4, Oxford, United Kingdom.

European Commission (2017): European Commission, Zero-rating practices in broadband markets.

GSMA (2020): GSMA, Keeping the world connected: Development challenges in times of COVID-19.

Hadzik (2019): Senka Hadzic, A global south perspective on alternative spectrum policy, Research ICT Africa Policy Brief 1: December 2019.

ISPA (2020): ISPA, COVID-19: Most frequently asked questions for ISPs.

ITU (2020): International Telecommunications Union, Pandemic in the internet age: Communications, industry responses.

Vrinda Bhandari is a practicing advocate in Delhi.

Friday, September 11, 2020

Introduction

A critical element of organisational planning is the ability to anticipate and manage risks. Many organisations worry about the risk of disruptions to their operations. They spend time and resources to anticipate such events, and put in place mechanisms to mitigate their impact. This is known as business continuity planning (BCP). It can range from keeping standby suppliers for raw materials, finding alternate delivery channels to reach customers, building redundancies along networks and so on. The objective of BCP is to ensure that business operations do not get disrupted.

Courts are an important part of the institutional eco-system of business. In times of general distress, like the one posed by the pandemic, courts are an essential service. As the state uses all the powers at its disposal to deal with the pandemic, judicial checks and balances are likely to be most needed. As businesses try to minimise their losses, disputes between economic actors are likely to increase. The role of courts to adjudicate disputes becomes more important in these times. An important question in this context is: are key institutions such as courts able to ensure business continuity?

In this article, we analyse this question in the context of the functioning of one of India's largest commercial tribunals, the National Company Law Tribunal (NCLT), in the post-Covid world. The NCLT's jurisdiction extends to matters under the Companies Act, 2013 (CA2013), the Insolvency and Bankruptcy Code, 2016 (IBC), and the Limited Liability and Partnership Act, 2008 (LLP Act). It hears a wide range of matters such as disputes between shareholders, enforcement actions against companies and their management, schemes of corporate restructuring, and insolvency proceedings. It would not be unfair to say that the NCLT's functioning likely affects the functioning of firms in the country.

India implemented one of the most stringent lockdowns in response to the pandemic. A near complete closure of all activities that were deemed non-essential commenced on 25th March and lasted till nearly the end of May. From June onwards, a phase-wise unlocking process commenced. In order to study the impact of the lockdown on the functioning of the NCLT, we constructed a novel data-set, drawing upon the daily cause-lists published by the NCLT.

Data and methodology

Data

We use a novel data-set derived from the daily cause-lists published by the NCLT. A causelist is a list of cases that are scheduled to be heard in a courtroom. While causelists suffer from lack from standardisation, both in the template and in the manner in which data fields are populated, they are a rich source of information about court functioning. Table 1 provides the list of the fields that we used in our analysis. Some of these are original, that is verbatim from the causelist, while others are derived by cleaning up and organising the original information available.

Table 1: Causelist data

Field name Description

Date Causelist date
Bench/Court Bench name and court room number
CP/CA Unique case identifier
Case purpose Purpose of hearing
Remarks Post hearing remarks
Case purpose category Categorisation of hearing purpose under broad heads
Remarks category Categorisation of post hearing remarks under broad heads

We use causelists for the period from 1st February to 30th June for our analysis. In India, the lockdown started from 25th March and was subsequently extended till the end of May. However, from 20th April, a range of conditional relaxations began to be introduced. On 30th May, 2020, the Central Government effectively allowed a phasewise opening of economic activity outside containment zones with effect from 1st June, 2020. In line with this timeline, we divide our analysis period into three phases: pre-lockdown, lockdown and unlock (Table 2). The pre-lockdown phase allows us to observe the regular functioning of the NCLT. The lockdown and the unlock phases allow us to observe court functioning in the post-Covid world.

Table 2: Study period

Phase Dates Days of data

Pre-lockdown 1st February to 24th March 34
Lockdown 25th March to 31st May 31
Unlock 1st June to 30th June 22

The NCLT Registry issued several circulars and practice directions with regard to its functioning during the lockdown period. The first of such circulars, issued on 23rd March, 2020, suspended the functioning of the NCLT with effect from 23rd March, 2020 until 31st March, 2020. The suspension was subsequently extended until the end of the lockdown. During this time, all the benches of the NCLT were directed to schedule hearings for 'urgent matters' through video-conferencing on designated days of the week.

For our analysis period, from the NCLT website, we get data for 22 bench-court combinations. We use 18 of these, namely 6 courtrooms of the NCLT bench in New Delhi (including the Principal Bench), 5 courtrooms of the NCLT bench in Mumbai, 2 courtrooms for the bench in Kolkata, and one each for the benches in Bengaluru, Chandigarh, Cuttack, Guwahati and Jaipur. We exclude 4 bench-court combinations, 2 for Chennai, and one each for Allahabad and Kochi due to sparse causelist availability. Ahmedabad bench is excluded as no data is available.

Methodology

We use the input-output approach to analyse court functioning, where scheduled hearings are treated as inputs and outcomes of hearings as outputs. Matters that come to the NCLT often go through a cycle of multiple hearings before they are finally completed. The input-output approach assumes that the NCLT's objective is to hold hearings on substantive questions that arise before it in respect of a matter and dispose them of. In doing so, the court seeks to move the matter forward towards a timely completion. A hearing that results in disposal as an outcome enables this. Hearings that result in a next date being given, for whatever reason, extend the completion timeline. Regy and Roy 2017 have previously used the idea of 'failed hearings'(adjourned hearings) to estimate judicial delays in debt recovery tribunals. We apply this concept to estimate the productivity of the court in the aftermath of the pandemic.

The input-output approach allows us to measure effective court capacity as:

$Effective court capacity = Hearings scheduled x Disposal rate$

A limitation of this approach is that it does not take into account the quality of the order passed by the NCLT. Our analysis is restricted to the volume of hearings and the number of disposals.

Input: volume of hearings scheduled

We find a sharp drop in the average number of daily hearings conducted by the NCLT during the lockdown period (Figure 1). In the pre-lockdown world, across all benches in our study, an average of 588 hearings were scheduled per day. This declined to 30 hearings per day during the lockdown period, a 95% decline. In the unlock period, the total hearings per day marginally increased from 30 to 41 per day. However, even this was a 93% decline from the pre-lockdown levels.

Figure 2 shows the location wise variation in the number of hearings scheduled. For each location graph, the red dotted line indicates the start of the lockdown period and the green dotted line indicates the start of the unlock period. The numbers on the top indicate the average daily hearings scheduled in each period.

Several interesting findings emerge. First, the bulk of the hearings during the lockdown period were conducted by benches in three locations, namely Mumbai, New Delhi and Chandigarh. Second, there is time variation in when courts started functioning during the lockdown. We observe that while courtrooms in Mumbai started scheduling hearings from 22nd April, the courtrooms in New Delhi started from 5th May onwards. Third, Mumbai, New Delhi and Chandigarh benches are also the ones that managed to ramp up capacity during the unlock period. Other benches have not, even till 30th June, built up a steady pattern of scheduling hearings.

The reason for the variation in the volume of hearings across locations remains a puzzle. We note that while the e-filing facility was available across some of the benches, with time, the litigants before the other benches were instructed to file their proceedings through an e-mail to the Registrar. The availibility of the e-filing facility does not explain the volume of hearings post lockdown. For instance, the Chandigarh bench, which was hearing cases all through this period, implemented e-filing only towards the later half of June. With Mumbai and New Delhi being the worst affected by the pandemic, the variation in volumes also cannot be attributed to the severity of the pandemic at a location. It is unclear then as to what has caused this location level variation and why some benches were able to resume functioning as early as mid-April while others could not do so even towards the end of June.

Output: disposal rates

Table 3 gives us an overview of the outcome of hearings during the pre-lockdown, lockdown and unlock period. In the pre-lockdown period, while a large number of hearings were getting scheduled, nearly 82% of these resulted in a next hearing date being given. During the lockdown period, this changed. The disposal rate improved significantly, from 17.9% to 54.5%.

Several factors might explain the improvement in disposal rates in the post lockdown period. One possibility is that during the lockdown, since the NCLT was hearing urgent matters only, they had to be disposed of. The second is that the pre-lockdown scheduling of nearly 40-50 cases per courtroom per day, was unrealistic. It resulted in a few matters getting actually heard and a next date being given in the remaining. Since the number of hearings getting scheduled during the lockdown period were low, these matters were actually getting the attention of the court which resulted in an improved disposal rate. Finally, it is also possible that the manner in which courts have dealt with hearings in the lockdown period changed. They were less amenable to allowing re-scheduling.

The pattern of a higher disposal rate during the lockdown period continued in the unlock period. However, there was some decline in the disposal rates compared to the lockdown period (from 54.5% to 48.4%).

Table 3: Outcome of hearings (as % of period totals)

Next date For order Disposed Total

Pre-lockdown
Number of hearings 15,813 968 2,489 19,270
% of hearings 82.1 5.0 12.9 100.0

Lockdown
Number of hearings 373 26 420 819
% of hearings 45.5 3.2 51.3 100.0

Unlock
Number of hearings 432 52 354 838
% of hearings 51.6 6.2 42.2 100.0

* Disposal rate = percentage of hearings with outcome "disposed, dismissed, admitted or allowed" + percentage of hearings with outcome "For order"

Estimating effective court capacity in the lockdown period

Table 4 brings together the input (hearings scheduled) and output (disposal rate) to give us a sense of the effective court capacity in the three phases. It shows that court capacity even in the unlock period is around 19% of pre-lockdown capacity. Table 4 suggests that the NCLT can adopt a very different mix of hearings and disposal from its pre-lockdown period to increase its overall output. For instance, at a disposal rate of 50%, even scheduling half of the pre-lockdown hearings will result in a higher effective capacity. However, this will require courts to analyse the process learnings from the post-lockdown period which resulted in higher disposal rates and apply them on an ongoing basis.

Table 4: Effective court capacity across the three periods

Phase Hearings (daily avg.) Disposal (%) Output (daily avg.)
($A$) ($B$) ($A*B\right)$

Pre-lockdown 588 17.9 105
Lockdown 30 54.5 16
Unlock 41 48.4 20

Conclusion

The lockdown has reduced the output of an already overburdened justice delivery system. Our analysis of the NCLT output is one case-study that demonstrates this. The functioning of courts during the pandemic is in contrast with the functioning of the overall economy during the same time. By June, most sectors of the economy had resumed operations to a large extent. The manufacturing sector IIP had returned to 80% of its February levels. Railway freight traffic, cargo traffic at ports and air cargo traffic had come back to 88%, 86% and 61% of their February levels respectively. Even a hard hit sector, like airlines, had resumed aircraft traffic to 26% of February levels. By June, the employment rate had come back to 95% of its February levels.

Courts have been reasonably quick in transitioning to virtual hearings. The relatively higher disposal rate at the NCLT demonstrates that a combination of the electronic filing system and virtual hearings, is workable. Despite this, we find that the NCLT has not reverted to even 20% of its output in the pre-lockdown period. The output is much lower for a majority of the benches. This is likely to substantially increase the pendency at the tribunal.

An extended disruption in court functioning can adversely affect the enforcement of civil liberties, property rights and contracts. This can have a debilitating effect on the rule of law. While most of the discourse on court capacity in India focuses on the inadequacy of judges, significant gains can be made by process improvements at the NCLT. Several scholars and policymakers have highlighted the need for a deeper focus on the management and business process planning for courts and tribunals in India (for example, see Datta 2016; Datta and Shah 2015). The courts too have recognised this need time and again (example). A BCP is an integral part of the business proceess engineering of courts.

A silver lining to the devastation caused by the pandemic is that it has accelerated some important reforms. Renewed focus on process management at courts is likely to give us maximum bang for the buck.

References

Regy, Prasanth, and Shubho Roy (2017), Understanding Judicial Delays in Debt Tribunals, NIPFP Working Paper Series, National Institute of Public Finance and Policy, New Delhi, India.

Datta, Pratik, and Ajay Shah. "How to Make Courts Work?" The LEAP Blog, 22 Feb. 2015.

Datta, Pratik, (2016), Towards a Tribunal Services Agency, Indira Gandhi Institute of Development Research, Mumbai Working Papers, Indira Gandhi Institute of Development Research, Mumbai, India.

The authors are researchers with the Finance Research Group. They would like to thank Ajay Shah for discussions and inputs on this article and Rahul Somani for developing the code for constructing the data-set.

Monday, September 07, 2020

If school fee regulation had to be done, how can it be done better?

by Bhuvana Anand and Shubho Roy.

In the Indian policy discourse, there is a strand of thought which sees schools as aggressors and parents as victims in a fight over school fees. The National Education Policy 2020 for example is concerned with “...the commercialization and economic exploitation of parents by many for-profit private schools...”. The COVID-19 pandemic has increased disputes between parents and schools over fees. Some parents are demanding a moratorium on fees. Some state governments are insisting that schools continue paying teachers. The managers of schools are struggling to balance these demands.

It is useful to take two steps back and ask: Is there a role for state power in controlling prices? In a market economy, market prices coordinate supply and demand, resulting in optimum resource allocation. Any state manipulation of the market price, such as a price cap, distorts this process and hampers mutually beneficial exchanges (Coyne and Coyne 2015). Price caps in general lead to poor outcomes. We see reduced investments and innovation (Ross 1983), collusion amongst suppliers (Grayson 1974, Knittel and Stango 2003), and regulatory capture. In the school education section in India, there is some evidence that implementing fee caps results in perverse outcomes (Agarwal et al. 2019).

Government control of prices has gone out of fashion in some areas (cement, steel) but this continues to be mainstream thinking in fields like education policy in India. Nine states and union territories have legislation that establishes fee regulation for private schools. Other states, such as Delhi, use executive power to regulate school fees. States with fee regulations impose absolute caps on fees or limit annual hikes.

In a previous article (Anand and Roy 2020), we discussed regulatory capture by teachers in fee setting. In this article, we discuss the hold-out problems that fee regulation creates. The current approach to fee regulation gives parents the power to hold up fee hikes. The exception to this is Maharashtra Educational Institutions (Regulation of Fee) Act, 2011. The Maharashtra law reduces the chances of a single parent holding up a fee hike. If there was a desire to have price controls in elementary education, the Maharashtra law gives us a better method for implementation as compared with what has been done elsewhere.

Fee regulation and the power of parents

Multiple states in India allow parents to challenge the decision of school management to hike school fees. This is, in and of itself, an awkward arrangement: Customers will always be biased in favour of lower prices. For example, if customers could hold up price hikes by security guards, they would always do so. Similarly, if exporters are given power on government control of the exchange rate, they will always favour depreciation.

Under the conventional arrangements, if parents find that schools have hiked fees 'unreasonably', they can complain to district/divisional fee regulatory committees. These committees are composed of parents, school representatives and government officials. Most laws characterise such action as raising a “grievance” or a “complaint” and require quasi-judicial proceedings to determine fees.

In Chandigarh, students, parents or guardians can register “complaints” with the regulatory body against schools for charging 'excessive' fees. Any hike beyond 8% of previous year’s fees, or attempting to profit (Section 7) is considered excessive.

In Bihar, schools may increase fees up to 7% over the previous year’s fee. Any increase beyond the 7% cap, requires the approval of a fee-regulatory committee. Any parent can file “objections” to the committee in case a school charges over the fee cap (Section 5).

Uttar Pradesh allows schools to hike fees in proportion to teacher salaries. However fees cannot be increased by more than CPI + 5% of previous year’s fees. Parents may complain to the district fee regulatory committee in cases where the school does not address their “objections” (Section 8).

In each of these states, an individual parent can start legal proceedings against the school management on the question of fee hikes. In the legal proceedings, the district fee regulatory committee has the power to question the fee increase. The school management has to justify why the fee increase is legitimate. If the district fee regulatory committee is not satisfied, it may impose fines on the school. In Bihar and Uttar Pradesh, if a school fails to pay fines on time or repeats offences, the committee may recommend revoking recognition which will lead to the school shutting down. Managers of schools are potentially subject to this legal process, the corresponding legal risk, and the cost of time and resources required to acquire political capital ahead of time to navigate these situations at the district fee regulatory committee as and when they might arise in the future.

In most states with fee regulation, one parent is capable of holding the entire school at ransom for raising fees. A single parent can engage in what is called a hold-out (Epstein 1993). Schools may need to hike fees to meet operational expenses (teacher salaries), comply with regulations (setting up CCTV cameras) and for infrastructure development (ed-tech). A bulk of parents may be willing to raise fees to meet these costs and improve the quality of education for their children. But the law allows a single parent to make it costly for the school to raise prices.

Maharashtra

Maharashtra regulates fee dispute with an added nuance: it imposes a minimum threshold that parents have to meet before raising a complaint against schools.

Schools in Maharashtra can hike fees only after approval from a school-level committee. This committee is composed of parents, teachers, and school management. Where parents disagree with the decision of the school-level committee, they may appeal to the divisional fee regulatory committee (Section 6). However, the Act does not give the power to an individual parent to raise a legal dispute. The Act creates a category called an ‘aggrieved parents group’ thus:

"aggrieved parents group" means the group of parents of the children, not be less than 25 per cent of the total parents of the children of affected standard or school, as the case may be, who are aggrieved by any decision under the Act (Section 2).

In Maharashtra, parents can challenge the decision of the school-level committee only if 25% (or more) of the parents affected by the hike come together to register a dispute. By raising the bar to 25% of affected parents, the Maharashtra law makes it difficult for one parent to stop the school from ensuring financial viability or improving services.

In the modern age of mobile phones, it may not be hard to solve coordination problems and muster a quarter of the parents as a coalition. The basic problem remains: should customers be given access to a non-market mechanism to influence prices? But this appears better than giving each parent the power to initiate a legal process.

The way forward

The literature shows that school education can be sticky; moving schools frequently can have social, emotional and learning consequences (see here, and here). Parents need to plan household finances for a 5-15 year horizon. Schools, on the other hand, want autonomy to run, expand and modify their offering, and to be able to respond to environmental demands, unforeseen circumstances, and competition from other schools. Parents want to remove consumption volatility in their purchase of vegetables, this does not mean we use state power to prevent price changes in vegetables. If parents wanted to eliminate price uncertainty enough, the market economy might find solutions on its own: e.g. perhaps schools could offer price lock-ins for a group of years at a time.

The present configuration of state power in fee regulation is biased against the school management. The law ignores the possibility that parents have no interest in the long term financial viability of a school. Once their children graduate from a school, they have no interest in whether the school survives. On the other hand, the management, which may run the school for decades (more than a century in some cases), has to consider the long term. By allowing one parent to hold up fee hikes, such laws hamper the long-term financial viability of schools, and create a bias in favour of school promoters who are politically well connected.

The rational response to a disagreement over fees would be parents shifting out when the school raises fees above what they are willing to pay. The Indian state, however, has created artificial constraints on the supply of schools through multiple laws (Singh and Sudhakar 2020). Parents have limited options to choose from for their child’s education.

The foundational mistake in Indian education policy is imposing artificial restrictions on the entry of new schools. The Indian state attempts to respond to the consequences of this bad idea by introducing other bad ideas like fee regulation and involving parents and teachers in school fee setting. This does not solve fee hike disputes and instead introduces new problems. We need to address the core problem head-on and not entangle ourselves into more complicated laws and administrative mechanisms to address them.

References

Controls are not the answer, Jackson C. Grayson, International Economic Review Vol. 21, Issue No. 2, 1974.

How are private school fees regulated?, Ritika Agarwal, Atreyi Bhaumik, Adit Shankar and Anindya Tomar, in Anatomy of K-12 Governance in India, Centre for Civil Society, October 2019.

Holdouts, Externalities, and the Single Owner: One More Salute to Ronald Coase, Richard A. Epstein, The Journal of Law & Economics, John M. Olin Centennial Conference in Law and Economics at the University of Chicago, April 1993.

Moving Matters: The Causal Effect of Moving Schools on Student Performance, Amy Ellen Schwartz, Leanna Stiefel and Sarah A. Cordes, Association for Education Finance and Policy, 2017.

Price Ceilings as Focal Points for Tacit Collusion: Evidence from Credit Cards, Christopher R. Knittel and Victor Stango, The American Economic Review Vol. 93, Issue No. 5, December 2003.

Pricing education: An example from Uttar Pradesh, Bhuvana Anand and Shubho Roy, The Leap Blog, July 2020.

Restrictions on for-profit education in India, Akash Pratap Singh and Tarini Sudhakar, Latest Analysis, Centre for Civil Society, May 2020.

Switching Schools: Reconsidering the Relationship Between School Mobility and High School Dropout, Joseph Gasper, Stefanie DeLuca and Angela Estacion, American Educational Research Journal, 2012.

The economics of price controls, Christopher J Coyne and Rachel L. Coyne, Chap. 2 in Flaws and Ceilings: Price Controls and the Damage They Cause, Institute of Economic Affairs, 2015.

The Private Schooling Phenomenon in India: A Review, Geeta Gandhi Kingdon, The Journal of Development Studies, 2020.

The Price of Education, Ernest Ross, Foundation for Economic Education, 1983.

Bhuvana Anand is a researcher at Centre for Civil Society and Shubho Roy is a researcher at the University of Chicago. The authors thank Tarini Sudhakar at Centre for Civil Society for research support.

Thursday, September 03, 2020

Thomas Laubach

by Ila Patnaik and Ajay Shah.

Our dear friend, Thomas Laubach, died in the US, yesterday.

Thomas reached out to us in 2011. He wanted to spend time in a research group in India that worked in macroeconomics. Thomas, his lovely wife Tahniyat and their three children were in India for 3 months, then. Apart from NIPFP, they visited Amravati where Tahniyat’s family came from.

Thomas fell in love with Delhi in those sunny winter months. At NIPFP, his happy smile encouraged even the youngest members of the staff to go and talk to him. He tried to communicate his excitement and passion about what had been achieved in the last century.

We were engaged, in those years, in trying to figure out a first principles understanding of Indian macroeconomics. Thomas, a former Ph.D. student of Ben Bernanke, was a great influence for this work. He brought immense knowledge of the edifice of modern macroeconomics of advanced economies, and at the same time was sympathetic and supportive of the idea that we had to figure out what makes sense under Indian conditions from scratch. We spent endless hours with him, groping in the facts, looking for the conceptual machinery that made sense and was consistent with the big facts. He cared about understanding the world, and being useful in the world, over and beyond the normal academic incentives.

At the time, Thomas was an academic at Goethe University in Frankfurt. Shortly after he finished at NIPFP, he joined the US Federal Reserve Board. Whenever one of us visited DC, Thomas would be happy to meet. He would always find time for a quick coffee in work hours. Tahniyat would be warm and welcoming and cook a lovely Indian meal at home.

Through the following years, we met Thomas regularly, as he would visit India regularly, and we would coordinate days with him in Delhi and Bombay. He loved walking in Bombay, and his answer to `what would you like to do today' was always "I want to see Bombay through your eyes". He loved stepping out to restaurants in Delhi for the local food. When he could not get leave to come, he  missed India, and would write to us remembering his lovely winter days of Delhi, calling his feelings the IWS (India Withdrawal Symptoms). His last visit to India was on a short trip in February 2019.

 In Bandra (West), a favourite walking region

He was very curious about how India worked. He knew that the emerging markets were going to become more important, and that what made them tick was fundamentally different from what was well understood elsewhere. Sometimes it felt like he was an anthropologist talking with us about what we were doing, and he would go under the hood, seeking the thick description.

We always looked upon his time at the US Fed as a busy interlude, and then he would get back to doing something closer to research, with more control of his time. We had planned that he would join a Sahyadris trek on a post-Fed winter visit to India. It is so cruel, that life worked out like this.

Tuesday, September 01, 2020

Positions for Legal Researchers

The Finance Research Group, Mumbai is an inter-disciplinary group of researchers working in the fields of financial markets, household and firm finance, bankruptcy law, land markets and public finance management and public procurement. In these fields, the group engages in academic and policy oriented research and advocacy.

The Finance Research Group is looking for two full-time legal researchers to work in the following fields:

1. public finance management and public procurement; and
2. commercial courts.

The researchers will be assigned to a project in these two fields respectively. The researchers will be expected to support research in law, economics and public policy in these fields, assist in organising advocacy and dissemination events and generate independent academic outputs and commentary. The positions will involve working with people from varying backgrounds including economics, management, law and public policy.

An indicative list of the research work done by researchers during their time at the Finance Research Group is below:

Researchers will be trained and encouraged to pursue research and produce outputs of the kind described above.

The office functions on free and open source software systems like Linux, Latex, R and others. If appointed, the candidate will be required to learn and use these software systems. The candidate must be willing to adapt to technology, work long hours and deliver quality products within defined timelines.

Previous work experience of atleast one year is required. A background in law is required, and one in economics/ public policy/statistics/ data science knowledge, is preferred. Candidates must be curious and passionate about research. The compensation will be competitive for the role and responsibilities for which the researcher is recruited.

Interested candidates must email their resume with the subject line: Application for research position at the Finance Research Group, to Ms. Jyoti Manke at careersatFRG@gmail.com


text:
J01020100−7122208: an accreted evolved blue straggler that was not ejected from a supermassive black hole
ABSTRACT

J01020100−7122208 is a star whose origin and nature still challenges us. It was first believed to be a yellow supergiant ejected from the Small Magellanic Cloud, but it was more recently claimed to be a red giant accelerated by the Milky Way’s central black hole. In order to unveil its nature, we analysed photometric, astrometric, and high-resolution spectroscopic observations to estimate the orbit, age, and 16 elemental abundances. Our results show that this star has a retrograde and highly-eccentric orbit, $e=0.914_{-0.020}^{+0.016}$. Correspondingly, it likely crossed the Galactic disc at 550 pc from the Galactic Centre. We obtained a spectroscopic mass and age of $1.09\pm 0.10\, {\rm M}_\odot$ and 4.51 ± 1.44 Gyr, respectively. Its chemical composition is similar to the abundance of other retrograde halo stars. We found that the star is enriched in europium, having [Eu/Fe] = 0.93 ± 0.24, and is more metal-poor than reported in the literature, with [Fe/H]  = −1.30 ± 0.10. This information was used to conclude that J01020100−7122208 is likely not a star ejected from the central black of the Milky Way or from the Small Magellanic Cloud. Instead, we propose that it is simply a halo star that was likely accreted by the Milky Way in the distant past, but its more »

Authors:
; ; ; ; ;
Award ID(s):
Publication Date:
NSF-PAR ID:
10381816
Journal Name:
Monthly Notices of the Royal Astronomical Society
Volume:
509
Issue:
3
Page Range or eLocation-ID:
p. 4637-4652
ISSN:
0035-8711
Publisher:
Oxford University Press
National Science Foundation
##### More Like this
1. Abstract

This work presents a new detection of H2absorption arising in a high-velocity cloud associated with either the Milky Way or the Large Magellanic Cloud (LMC). The absorber was found in an archival Far Ultraviolet Spectroscopic Explorer spectrum of the LMC star Sk-70°32. This is the fifth well-characterized H2absorber to be found in the Milky Way’s halo and the second such absorber outside the Magellanic Stream and Bridge. The absorber has a local standard of rest central velocity of +140 km s−1and a H2column density of 1017.5cm−2. It is most likely part of a cool and relatively dense inclusion (T≈ 75 K,nH∼ 100 cm−3) in a warmer and more diffuse halo cloud. This halo cloud may be part of a still-rising Milky Way Galactic fountain flow or an outflow from the Large Magellanic Cloud.

2. ABSTRACT

We present a high-resolution (R ∼ 35 000), high signal-to-noise (S/N = 350) Magellan/MIKE spectrum of the bright extremely metal-poor star 2MASS J1808−5104. We find [Fe/H] = −4.01 (spectroscopic LTE stellar parameters), [Fe/H] = −3.8 (photometric stellar parameters), and [Fe/H] = −3.7 (spectroscopic NLTE stellar parameters). We measured a carbon-to-iron ratio of [C/Fe] = 0.38 from the CH G-band. J1808−5104 is thus not carbon-enhanced, contrary to many other stars with similarly low-iron abundances. We also determine, for the first time, a barium abundance ([Ba/Fe] = −0.78), and obtain a significantly reduced upper limit for the nitrogen abundance ([N/Fe] < −0.2). For its [Ba/Fe] abundance, J1808−5104 has a lower [Sr/Ba] ratio compared to other stars, consistent with behaviour of stars in ultra-faint dwarf galaxies. We also fit the abundance pattern of J1808−5104 with nucleosynthesis yields from a grid of Population III supernova models. There is a good fit to the abundance pattern that suggests J1808−5104 originated from gas enriched by a single massive supernova with a high explosion energy of E = 10 × 1051 erg and a progenitor stellar mass of M = 29.5 M⊙. Interestingly, J1808−5104 is a member of the Galactic thin disc, as confirmed by our detailed kinematic analysis and calculated stellar actions and velocities. Finally, we alsomore »

3. ABSTRACT

We report the spectroscopic analysis of 20 halo ab-type RR Lyrae stars with heliocentric distances between 15 and 165 kpc, conducted using medium-resolution spectra from the Magellan Inamori Kyocera Echelle (MIKE) spectrograph. We obtain the systemic line-of-sight velocities of our targets with typical uncertainties of 5–10 km s−1 and compute orbital parameters for a subsample out to 50 kpc from the Galactic centre, including proper motion data from Gaia DR3. The orientation of our stars’ orbits, determined for an isolated Milky Way and for a model perturbed by the Large Magellanic Cloud, appears to suggest an accreted origin for at least half of the sample. In addition, we derive atmospheric parameters and chemical abundance ratios for seven stars beyond 20 kpc. The derived α-abundances of five of these stars follow a Milky Way halo-like trend, while the other two display an underabundance of α-elements for their [Fe/H], indicating an association with accretion events. Furthermore, based on the [Sr/Ba] ratio, we can speculate about the conditions for the formation of a potential chemically peculiar carbon-enhanced metal-poor (CEMP) RR Lyrae star. By analysing the stars’ orbital parameters and abundance ratios, we find hints of association of two of our stars with two massive satellites, namely themore »

4. ABSTRACT

The star formation and gas content of satellite galaxies around the Milky Way (MW) and Andromeda (M31) are depleted relative to more isolated galaxies in the Local Group (LG) at fixed stellar mass. We explore the environmental regulation of gas content and quenching of star formation in z = 0 galaxies at $M_{*}=10^{5\!-\!10}\, \rm {M}_{\odot }$ around 14 MW-mass hosts from the Feedback In Realistic Environments 2 (FIRE-2) simulations. Lower mass satellites ($M_{*}\lesssim 10^7\, \rm {M}_{\odot }$) are mostly quiescent and higher mass satellites ($M_{*}\gtrsim 10^8\, \rm {M}_{\odot }$) are mostly star forming, with intermediate-mass satellites ($M_{*}\approx 10^{7\!-\!8}\, \rm {M}_{\odot }$) split roughly equally between quiescent and star forming. Hosts with more gas in their circumgalactic medium have a higher quiescent fraction of massive satellites ($M_{*}=10^{8\!-\!9}\, \rm {M}_{\odot }$). We find no significant dependence on isolated versus paired (LG-like) host environments, and the quiescent fractions of satellites around MW-mass and Large Magellanic Cloud (LMC)-mass hosts from the FIRE-2 simulations are remarkably similar. Environmental effects that lead to quenching can also occur as pre-processing in low-mass groups prior to MW infall. Lower mass satellites typically quenched before MW infall as central galaxies or rapidly during infall into a low-mass group ormore »

5. Abstract

Our view of the variety of stellar structures pervading the local Milky Way has been transformed by the application of clustering algorithms to the Gaia catalog. In particular, several stellar streams have been recently discovered that are comprised of hundreds to thousands of stars and span several hundred parsecs. We analyze one such structure, Theia 456, a low-density stellar stream extending nearly 200 pc and 20° across the sky. By supplementing Gaia astrometric data with spectroscopic metallicities from Large Sky Area Multi-Object Fiber Spectroscopic Telescope and photometric rotation periods from the Zwicky Transient Facility and the Transiting Exoplanet Survey Satellite, we establish Theia 456's radial velocity coherence, and we find strong evidence that members of Theia 456 have a common age (≃175 Myr), common dynamical origin, and formed from chemically homogeneous prestellar material ([Fe/H] = −0.07 dex). Unlike well-known stellar streams in the Milky Way, which are in its halo, Theia 456 is firmly part of the thin disk. If our conclusions about Theia 456 can be applied to even a small fraction of the remaining ≃8300 independent structures in the Theia catalog, such low-density stellar streams may be ubiquitous. We comment on the implications this has for themore »

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# Universal, strong and long-ranged trapping by optical conveyors

David B. Ruffner and David G. Grier
Department of Physics and Center for Soft Matter Research, New York University, New York, NY 10003, USA
###### Abstract

Optical conveyors are active tractor beams that selectively transport illuminated objects either upstream or downstream along their axes. Formed by the coherent superposition of coaxial Bessel beams, an optical conveyor features an axial array of equally spaced intensity maxima that act as optical traps for small objects. We demonstrate through measurements on colloidal spheres and numerical calculations based on the generalized Lorenz-Mie theory that optical conveyors’ interferometric structure endows them with trapping characteristics far superior to those of conventional optical tweezers. Optical conveyors form substantially stiffer traps and can transport a wider variety of materials over a much longer axial range.

# 1 Introduction: optical conveyors

Long a staple of science fiction [1], tractor beams are traveling waves that transport illuminated objects back to their sources, opposite to the direction of energy flow. Such retrograde transport has been demonstrated in optical trapping experiments using collimated solenoid beams [2], interfering plane waves [3], and optical conveyors created with coaxial Bessel beams [4, 5, 6]. Among the more surprising proposals arising from these initial studies is that optical conveyors might act as universal tractor beams, transporting small objects at uniform speed regardless of their composition [4]. This contrasts with the performance of conventional optical tweezers, which can only trap bright-seeking objects over a limited domain of size, shape and composition [7, 8, 9]. Another surprising suggestion is that optical conveyors might exceed the trapping stiffness even of diffraction-limited optical tweezers because their intensity actually vanishes at regular intervals along the optical axis [4].

Here, we report the results of a head-to-head comparison between the trapping characteristics of optical tweezers and optical conveyors involving both experimental measurements on micrometer-scale spheres and also numerical evaluations of optical forces using the generalized Lorenz-Mie theory. In confirming the superior performance of optical conveyors, these studies also enable us to establish the axial range over which optical conveyors can usefully transport material, and provide guidance for developing long-ranged tractor beams.

The electric field of a monochromatic optical conveyor of angular frequency $\omega$, linearly polarized along $\hat{x}$ and propagating along $\hat{z}$ through a medium of refractive index $n_{m}$, is the superposition of two Bessel beams [5, 6, 4],

 $\mathbf{E}(\mathbf{r},t)=\frac{1}{2}\,E_{0}e^{-i\omega t}\,\left[\mathbf{b}_{1% }(k\mathbf{r})+e^{i\varphi(t)}\mathbf{b}_{2}(k\mathbf{r})\right],$ (1)

each of which may be described as a conical superposition of plane waves [6],

 $\mathbf{b}_{j}(k\mathbf{r})=\int_{0}^{2\pi}\hat{\epsilon}(\theta_{j},\phi)\,e^% {i\mathbf{k}(\theta_{j},\phi)\cdot\mathbf{r}}\,d\phi,$ (2a) where $\hat{\epsilon}(\theta,\phi)=\cos\phi\,\hat{\theta}+\sin\phi\,\hat{\phi}$ (2b) is the polarization of a plane wave incident on the optical axis at polar angle $\theta$ and azimuthal angle $\phi$, and where $\mathbf{k}(\theta,\phi)=k(\sin\theta\cos\phi\,\hat{x}+\sin\theta\sin\phi\,\hat% {y}+\cos\theta\,\hat{z})$ (2c)

is the corresponding wave vector. In the paraxial approximation, which is appropriate for long-range tractor beams, Eq. (2) reduces to

 $\mathbf{b}_{j}(k\mathbf{r})\approx J_{0}\!\left(\sqrt{1-\eta_{j}^{2}}kr\right)% e^{i\eta_{j}kz}\,\hat{x},$ (3)

where $\eta_{j}$ is related to the Bessel beam’s cone angle by $\eta_{j}=\cos\theta_{j}$. The approximate expression in Eq. (3) differs from the exact expression in Eq. (2) by terms involving higher-order Bessel functions[20, 23], which vanish on the optical axis. The numerical results developed in Sec. 2.1 are based on Eq. (2). Analytical results developed in Sec. 2.2 are more readily obtained from the approximation in Eq. (3).

The two Bessel beams comprising an optical conveyor share the same amplitude $E_{0}$, frequency $\omega$ and polarization along $\hat{x}$, but differ in their relative phase, $\varphi(t)$, and also in their axial wave numbers, $\eta_{j}k$, which are reduced from the plane wave value, $k=n_{m}\omega/c$, by the dimensionless factor $\eta_{j}\in(0,1]$. Here, $c$ is the speed of light in vacuum. The superposition is usefully characterized by the mean convergence factor $\eta=(\eta_{1}+\eta_{2})/2$ and the difference $\Delta\eta=\left|\eta_{1}-\eta_{2}\right|$. The upper limit of an optical conveyor’s range, $R$, is set by the non-diffracting range of the most strongly converging Bessel beam [10],

 $R\leq A\cot\theta_{1}=A\frac{\eta_{1}}{\sqrt{1-\eta_{1}^{2}}}$ (4)

where $A$ is the radius of the beam’s aperture. Longer ranges can be achieved with values of $\eta$ approaching 1.

Figures 1(a) and 1(b) show the measured [11] three-dimensional intensity distribution of optical conveyors with equal values of $\eta$ and differing values of $\Delta\eta$ that were projected with the holographic optical trapping technique [12, 4]. These beams were powered by a linearly polarized diode-pumped solid-state laser (Coherent Verdi) operating at a vacuum wavelength of $\lambda=532~{}\text{nm}$ that was shaped by a liquid crystal spatial light modulator (Holoeye Pluto) before being projected with a microscope objective lens (Nikon Plan-Apo, $100\times$, numerical aperture 1.4, oil immersion). The mode projected by this method is not the simple superposition of Bessel beams, but rather incorporates contributions from small range of axial wave numbers around $\eta_{j}k$ [4]. Equation (1) therefore should be considered an idealized model for the actual beam.

The intensity distribution, $I(\mathbf{r},t)=\frac{1}{2}n_{m}\epsilon_{0}c\left|\mathbf{E}(\mathbf{r},t)% \right|^{2}$, has maxima at axial positions

 $z_{n}(t)=\frac{2\pi n+\varphi(t)}{\Delta\eta\,k},$ (5)

each of which can act as an optical trap for a small object. Here, $\epsilon_{0}$ is the permittivity of space. Varying $\varphi(t)$ as a function of time moves these extrema, and thus conveys trapped objects along the beam. Increasing $\Delta\eta$ reduces the spacing between maxima, as shown in Figs. 1(a) and 1(b), and so provides control over the optical force profile.

# 2 Trap stiffness

We previously have proposed [4] that optical conveyors should make better traps than conventional optical tweezers because their intensity vanishes altogether between maxima. The data in Figs. 1(c) and 1(d) demonstrate this to be true. Figure 1(c) shows the measured trajectory of a colloidal silica sphere diffusing through water in one of the potential energy wells of the static optical conveyor from Fig. 1(b). The optical conveyor has a peak intensity of $79~{}\text{mW}/\mu\text{m}^{2}$, as measured with imaging photometry. Holographic characterization [13, 14, 15] reveals the sphere’s radius to be $a_{p}=0.730\pm 0.005~{}\mu\text{m}$, and its refractive index to be $n_{p}=1.424\pm 0.005$. Holographic tracking [13, 16] yields the sphere’s position with 1 nm precision in-plane and 3 nm resolution axially [17, 15] at 16.7 ms intervals. Figure 1(d) shows the same sphere diffusing in a conventional optical tweezer with the same peak intensity projected by the same instrument.

In both cases, the trapped particle explores the optical force landscape under the influence of random thermal forces. The optical conveyor restricts the particle’s axial excursions to less than half the range of the optical tweezer, resulting in a nearly isotropic trajectory. This suggests that an optical conveyor makes a substantially stiffer trap, even though the 2.8:1 aspect ratio of the optical tweezer’s trajectory approaches the theoretical limit for a diffraction-limited Gaussian trap [18].

Modeling the traps as cylindrically symmetric harmonic potential energy wells,

 $U(\mathbf{r})=\frac{1}{2}k_{r}r^{2}+\frac{1}{2}k_{z}z^{2},$ (6)

we may estimate the transverse and axial trap stiffness, $k_{r}$ and $k_{z}$ from the particle’s trajectory using thermal fluctuation analysis [19, 21]. These measurements are performed for fixed values of the relative phase, $\varphi(t)$, so that the optical traps do not move during the measurement. For the trajectory in Fig. 1(c), we obtain $k_{r}=2.2\pm 0.1~{}\text{pN}/\mu\text{m}$ and $k_{z}=0.89\pm 0.03~{}\text{pN}/\mu\text{m}$, and an anisotropy of $\sqrt{k_{r}/k_{z}}=1.57\pm 0.04$. The equivalent results for the optical tweezer are $k_{r}=2.8\pm 0.4~{}\text{pN}/\mu\text{m}$, $k_{z}=0.41\pm 0.08~{}\text{pN}/\mu\text{m}$ and $\sqrt{k_{r}/k_{z}}=2.6\pm 0.4$. The optical conveyor performs as well as the optical tweezer in the transverse direction and is nearly twice as stiff along the axis.

The performance of single beam optical traps typically is limited by their axial trapping ability. Figure 1(e) shows how the optical conveyor’s stiffness varies with $\Delta\eta$ for fixed $\eta$, and compares this with the performance of a diffraction-limited optical tweezer. Highlighted points correspond to the data from Figs. 1(c) and 1(d). The optical conveyor’s axial stiffness exceeds that of a diffraction-limited optical tweezer by as much as a factor of two. This advantage is emphasized by the ratio between axial and transverse stiffness plotted in Fig. 1(f). Dashed horizontal dashed lines represent the measured and theoretical maximum performance of a Gaussian optical tweezer. Optical conveyors exceed this performance for $\Delta\eta>0.06$, which corresponds to an axial period less than 6.7 $\mu\text{m}$.

# 2.1 Generalized Lorenz-Mie theory

Optical conveyors’ superior trapping performance is consistent with predictions of generalized Lorenz-Mie theory [22, 23, 6, 24, 25], which are plotted as continuous curves in Figs. 1(e) and 1(f). For these calculations, each Bessel beam is expanded as a series,

 $\mathbf{b}_{j}(\mathbf{r})=\sum_{n=1}^{\infty}\sum_{m=-n}^{n}\left[a_{mn}(% \theta_{j})\,\mathbf{M}_{nm}^{(1)}(k\mathbf{r})+b_{mn}(\theta_{j})\,\mathbf{N}% _{nm}^{(1)}(k\mathbf{r})\right],$ (7)

in the vector spherical harmonics, $\mathbf{M}_{nm}^{(1)}(k\mathbf{r})$ and $\mathbf{N}_{nm}^{(1)}(k\mathbf{r})$, that constitute the natural basis for transverse electric (TE) and transverse magnetic (TM) waves, respectively [26]. The expansion coefficients, $a_{mn}(\theta_{j})$ and $b_{mn}(\theta_{j})$, have been reported previously for individual Bessel beams [22]; they can be superimposed according to Eq. (1) to create an optical conveyor with specified values of $\eta$, $\Delta\eta$ and relative phase $\varphi(t)$.

The associated expansion of the scattered field,

 $\mathbf{E}_{s}(\mathbf{r},t)=E_{0}\,e^{-i\omega t}\,\sum_{n=1}^{\infty}\sum_{m% =-n}^{n}\left\{\left[r_{mn}(\theta_{1})+e^{i\varphi(t)}r_{mn}(\theta_{2})% \right]\,\mathbf{M}_{nm}^{(3)}(k\mathbf{r})+\left[s_{mn}(\theta_{1})+e^{i% \varphi(t)}s_{mn}(\theta_{2})\right]\,\mathbf{N}_{nm}^{(3)}(k\mathbf{r})\right\},$ (8a) is obtained by matching boundary conditions for the electric and magnetic fields at the sphere’s surface [23, 24, 25, 27, 28, 29]. The expansion coefficients, $\displaystyle r_{mn}(\theta_{j})$ $\displaystyle=-a_{n}\,a_{mn}(\theta_{j})\quad\text{and}$ (8b) $\displaystyle s_{mn}(\theta_{j})$ $\displaystyle=-b_{n}\,b_{mn}(\theta_{j}),$ (8c)

are related to the expansion coefficients of the incident field by the standard Mie coefficients, $a_{n}$ and $b_{n}$, [27], which depend on the radius and refractive index of the sphere.

The superposed incident and scattered fields, $\mathbf{E}(\mathbf{r},t)+\mathbf{E}_{s}(\mathbf{r},t)$, contribute to the Maxwell stress tensor, $\mathbf{T}(\mathbf{r},t)$, whose integral over a closed surface provides an estimate for the optically-induced force on a sphere centered at $\mathbf{r}$:

 $\mathbf{F}(\mathbf{r},t)=\oint_{S}\hat{n}\cdot\mathbf{T}(\mathbf{r}^{\prime},t% )\,d\mathbf{r}^{\prime},$ (9)

where $\hat{n}$ is the unit normal to the surface $S$ enclosing the sphere. In practice, the integral is computed directly from the combined expansion coefficients using established techniques [7, 8, 9, 30]. This calculation must also be repeated for each value of the relative phase $\varphi(t)$.

A trap’s position, $\mathbf{r}_{0}(t)$, for a given value of $\varphi(t)$ is calculated numerically as a solution of $\mathbf{F}(\mathbf{r}_{0}(t),t)=0$. The traps’s effective stiffness along $\hat{r}_{\nu}$ then is calculated as

 $k_{\nu}=-\left.\partial_{\nu}F_{\nu}(\mathbf{r},t)\right|_{\mathbf{r}=\mathbf{% r}_{0}(t)}.$ (10)

Results for the predicted transverse and axial stiffness are plotted in Fig. 1(e) for conveyor beams with $\eta=0.8$ and $\Delta\eta$ ranging up to 0.13, and for values of the sphere’s radius and refractive index obtained with holographic microscopy. The curves are scaled to a maximum intensity of $41~{}\text{mW}/\mu\text{m}^{2}$, which is less than the measured value presumably because holographically projected Bessel beams are not uniformly bright. This discrepancy does not affect the computed ratio of stiffnesses in Fig. 1(f), which also agree well with experimental results, with no adjustable parameters.

# 2.2 Dipole approximation

Generalized Lorenz-Mie theory is useful for computing the forces on specific particles in particular traps. To assess trends in optical traps’ capabilities, we invoke the dipole approximation in which the time-averaged force [31, 32],

 $\mathbf{F}(\mathbf{r},t)=\frac{1}{2}\,\Re\left\{\alpha_{e}\,\sum_{\nu=1}^{3}E_% {\nu}(\mathbf{r},t)\nabla E_{\nu}^{\ast}(\mathbf{r},t)\right\},$ (11)

is proportional to the object’s polarizability $\alpha_{e}=\alpha_{e}^{\prime}+i\alpha_{e}^{\prime\prime}$. The polarizability of a dielectric sphere is related to its size and refractive index by the Clausius-Mossotti-Draine relation [33]

 $\alpha_{e}=\frac{4\pi\epsilon_{0}n_{m}^{2}K\,a_{p}^{3}}{1-i\frac{2}{3}Kk^{3}a_% {p}^{3}},$ (12)

where $K=(n_{p}^{2}-n_{m}^{2})/(n_{p}^{2}+2n_{m}^{2})$ is the Lorentz-Lorenz factor. Absorptivity increases the imaginary part of $\alpha_{e}$. Conductivity contributes an imaginary part to $K$. The dipole approximation typically applies in the Rayleigh limit, for particles much smaller than the wavelength of light.

An object near the axis of an optical conveyor experiences an axial force

 $\frac{F_{z}(\mathbf{r},t)}{E_{0}^{2}}\approx-\frac{1}{4}\alpha_{e}^{\prime}% \Delta\eta\,\sin\bigl(\Phi(z,t)\bigr)+\alpha_{e}^{\prime\prime}\,\eta\,\cos^{2% }\!\left(\frac{1}{2}\Phi(z,t)\right)$ (13)

and a transverse force

 $\frac{F_{r}(\mathbf{r},t)}{E_{0}^{2}}\approx\alpha_{e}^{\prime}\,\frac{kr}{2}% \left(1-\eta^{2}-\frac{1}{4}\Delta\eta^{2}\right)\cos^{2}\!\left(\frac{1}{2}% \Phi(z,t)\right)+\frac{1}{4}\alpha_{e}^{\prime\prime}\,\eta\,\Delta\eta\,kr% \sin\bigl(\Phi(z,t)\bigr),$ (14)

where $\Phi(z,t)=\Delta\eta\,kz-\varphi(t)$, and where we have assumed $kr\ll 1$. These results are obtained by averaging over times long compared with the optical cycle but short compared with variations in $\varphi(t)$. The particle is trapped where the force vanishes, which occurs at axial positions $Z_{n}(t)$ that are displaced from the intensity maxima by an amount that depends on the particle’s light-scattering properties,

 $Z_{n}(t)-z_{n}(t)=\frac{2}{\Delta\eta k}\,\tan^{-1}\!\left(\frac{\alpha_{e}^{% \prime\prime}}{\alpha_{e}^{\prime}}\frac{2\eta}{\Delta\eta}\right).$ (15)

Equations (10) and (16) then yield the traps’ axial stiffness,

 $k_{z}=\frac{1}{4}\left|\alpha_{e}^{\prime}\right|kE_{0}^{2}\,\Delta\eta^{2},$ (16)

which is strictly positive. An optical conveyor thus can trap and transport any dipolar particle, regardless of its light-scattering characteristics. This universal, material-independent trapping capability does not require feedback [34] or fine tuning of the beam’s properties [35]. Decreasing the inter-trap separation by increasing $\Delta\eta$ enhances intensity gradients and thus increases trap stiffness.

Picking the largest possible value of $\Delta\eta$ to optimize axial trapping is not necessarily the best strategy. An optical conveyor’s transverse stiffness,

 $k_{r}=k_{z}\frac{{\alpha_{e}^{\prime}}^{2}\left(1-\eta^{2}-\frac{1}{4}\Delta% \eta^{2}\right)-2{\alpha_{e}^{\prime\prime}}^{2}\eta^{2}}{\frac{1}{2}{\alpha_{% e}^{\prime}}^{2}\Delta\eta^{2}+2{\alpha_{e}^{\prime\prime}}^{2}\eta^{2}},$ (17)

can vanish or even change sign as $\Delta\eta$ increases.

The dependence of trap stiffness on particle size predicted by Eqs. (12), (16) and (17) is plotted as dashed curves in Fig. 2 for silica spheres in an optical conveyor with $\eta=0.8$ and $\Delta\eta=0.086$. Results from the dipole approximation agree well with the Lorenz-Mie results that are plotted as solid curves in Fig. 2, at least for $ka_{p}\leq 1$. The Lorenz-Mie predictions, in turn, agree quantitatively with results from Fig. 1, which are plotted as discrete points.

The dipole approximation severely underestimates the optical conveyor’s stiffness for $ka_{p}>1$, and suggests that the particle studied in Fig. 1 would not have been stably trapped in the transverse direction. Rather than displaying a single crossover to instability, the generalized Lorenz-Mie result displays limited domains of instability for particles in particular size ranges. Results from the dipole approximation therefore are useful for establishing strict lower bounds on the performance of optical conveyors.

For example, requiring stable transverse trapping ($k_{r}>0$) according to Eq. (17) establishes an upper bound on $\Delta\eta$ and, through Eq. (4), a lower bound on the range over which an optical conveyor can transport small objects:

 $R_{\perp}\leq\frac{A}{2}\left(\frac{\alpha_{e}^{\prime}}{\alpha_{e}^{\prime% \prime}}-\frac{\alpha_{e}^{\prime\prime}}{\alpha_{e}^{\prime}}\right) (18)

The practically accessible range of transport may be substantially greater than $R_{\perp}$ for particles larger than the wavelength of light.

# 2.3 Comparison with optical tweezers

Conventional optical tweezers do not share optical conveyors’ universal trapping ability. To show this, we model an optical tweezer as a focused Gaussian beam whose axial electric field profile is [36, 37]

 $\mathbf{E}_{G}(z,t)=E_{0}\frac{z_{R}}{\sqrt{z^{2}+z_{R}^{2}}}e^{ikz}e^{i\zeta(% z)}e^{-i\omega t}\,\hat{\epsilon},$ (19)

where $z_{R}\approx 2R^{2}/(kA^{2})$ is the Rayleigh range of a Gaussian beam converging at distance $R$ from an aperture of radius $A$, and where $\zeta(z)=\tan^{-1}(z/z_{R})$ is the Gouy phase. Because we are interested in long-ranged axial transport, we assume $R>A$. The axial component of the associated force,

 $\frac{F_{G}(z)}{E_{0}^{2}}=-\frac{1}{2}z_{R}^{2}\,\frac{z\alpha_{e}^{\prime}-k% \left(z^{2}-z_{R}^{2}\right)\alpha_{e}^{\prime\prime}}{\left(z^{2}+z_{R}^{2}% \right)^{2}},$ (20)

forms a trap only for particles satisfying

 $\left(\frac{\alpha_{e}^{\prime}}{\alpha_{e}^{\prime\prime}}\right)^{2}>4kz_{R}% (kz_{R}-1).$ (21)

This condition is most easily satisfied in strongly converging beams for which $z_{R}$ is small. The condition on $\alpha_{e}$ is qualitatively consistent with numerical studies [7, 8, 9] which describe the difficulty of trapping high index or absorbing particles with laser tweezers.

# 3 Transport range

The ultimate rate of an optical conveyor is limited by the non-diffracting range of the constituent Bessel beams. Equation (21), by contrast, establishes an inherent upper limit on an optical tweezer’s range

 $R_{G}=\frac{A}{2}\,\sqrt{1+\sqrt{1+\left(\frac{\alpha_{e}^{\prime}}{\alpha_{e}% ^{\prime\prime}}\right)^{2}}}.$ (22)

At ranges $R$ beyond $R_{G}$, radiation pressure overwhelms trapping forces due to axial intensity gradients and ejects the particle. The upper bound of an optical tweezer’s range therefore is smaller than the lower bound of an optical conveyor’s. This means that optical conveyors can transport objects over substantially longer axial ranges, and at much lower numerical apertures. The price for long-ranged transport is paid in the strength of an optical conveyor’s traps.

Figure 3(a) is a volumetric reconstruction of an optical conveyor with $\eta=0.96$ and $\Delta\eta=0.04$ that was projected with a $60\times$ objective lens (Nikon Plan-Apo, numerical aperture 1.4 oil immersion). Holographic projection limits the conveyor’s range to $R=70~{}\mu\text{m}$, and thus suggests an effective numerical aperture of 0.43 given the convergence angle of $\theta=19^{\circ}$. The holographically measured trajectory plotted in Fig. 3(b) shows this optical conveyor trapping and transporting a 1.5 $\mu\text{m}$-diameter silica sphere over 66 $\mu\text{m}$. The hologram at the bottom of this figure was recorded when the sphere was located at the position indicated by the sphere in Fig. 3(b). This measurement demonstrates that an optical conveyor can transport objects along its entire length, even at low numerical aperture.

Figure 3(c) shows how the traps’ stiffness falls off with range. The solid and dashed curves are Lorenz-Mie calculations of the axial trap stiffness of optical conveyors and optical tweezers, respectively, and are scaled for a peak intensity of $41~{}\text{mW}/\mu\text{m}^{2}$. One set of curves is calculated for a 1.5 $\mu\text{m}$silica sphere, and agrees reasonably well with experimental results for the optical tweezer in Fig. 1(c), the optical conveyor in Fig. 1(d) and the optical conveyor in Fig. 3(a). The other set is calculated in the Rayleigh regime at $ka_{p}=0.5$, and agrees quantitatively with the dipole result from Eq. (20).

Despite the dipole prediction from Eq. (13), the force experienced by a large particle in an optical conveyor can be purely repulsive. Lorenz-Mie calculations reveal that additional radiation pressure due to off-axis scattering can overwhelm trapping forces due to axial intensity gradients for particles larger than the wavelength of light. Although a conveyor can be projected with a range exceeding this limit, it will not be able to transport large particles in the retrograde direction, and so will not act as a tractor beam.

The dependence of the trapping force on range for small particles shows no such crossover from stable trapping to repulsion. While the optical tweezer is inherently limited to $R/A<9$ the optical conveyor extends indefinitely, albeit with a stiffness that falls off as $R^{-4}$. The maximum trapping force in this regime falls off as $R^{-2}$. Both the force and the stiffness scale with the intensity of the beam, and therefore with the laser power. These considerations demonstrate that optical conveyors are viable candidates for long-ranged tractor beams, particularly for objects that are smaller than the wavelength of light.

# 4 Conclusions

Interference endows optical conveyors with trapping characteristics surpassing those of conventional single-beam optical traps. Most notably, optical conveyors are universal traps that can hold and transport small objects regardless of their light scattering properties. Under conditions where optical tweezers also are effective, moreover, optical conveyors make stiffer traps, particularly in the axial direction.

Optical conveyors act as universal traps because their intensity vanishes at points along the optical axis [5, 6, 4]. This ensures that retrograde forces arising from intensity gradients can counteract radiation pressure to hold illuminated objects in place. These traps can be positioned any placed along the propagation-invariant range of the conveyor by varying the relative phase of the constituent Bessel beams. As a consequence, optical conveyors can transport objects over substantially larger axial distances than conventional optical traps [5, 6, 4], which are inherently limited by radiation pressure.

Further improvements in performance almost certainly can be realized by appropriately structuring the intensity and phase gradients [38, 39, 40] along the projected beam. The range and trapping strength also should improve in optical conveyors created with radially polarized light [41, 42, 43]. These insights apply also to other interferometrically structured beams of light, such as solenoidal waves [2]. Both as tractor beams, and also as ordinary optical traps, interferometrically structured beams of light offer clear benefits for optical micromanipulation.

# Acknowledgments

This work was supported principally by the National Science Foundation through grant number DMR-1305875, and in part by a grant from NASA.

# References

• 1 E. E. Smith, “Spacehounds of IPC,” Amazing Stories, July (1931).
• 2 S.-H. Lee, Y. Roichman, and D. G. Grier, “Optical solenoid beams,” Opt. Express 18(7), 6988–6993 (2010).
• 3 O. Brzobohatý, V. Karásek, M. Šiler, L. Chvátal, T. Čižmár, and P. Zemánek, “Experimental demonstration of optical transport, sorting and self-arrangement using a ’tractor beam’,” Nature Photonics 7(2), 123–127 (2013).
• 4 D. B. Ruffner and D. G. Grier, “Optical conveyors: a class of active tractor beams,” Phys. Rev. Lett. 109(16), 163903 (2012).
• 5 T. Čižmar, V. Garcés-Chávez, K. Dhokalia, and P. Zemánek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86(17), 174101 (2005).
• 6 T. Čižmar, V. Kollárová, Z. Bouchal, and P. Zemánek, ‘‘Sub-micron particle organization by self-imaging of non-diffracting beams,” New J. Phys. 8, 43 (2006).
• 7 T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knoner, A. M. Branczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A 9(8), S196–S203 (2007).
• 8 A. B. Stilgoe, T. A. Nieminen, G. Knöener, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “The effect of Mie resonances on trapping in optical tweezers,” Opt. Express 16(19), 15,039–15,051 (2008).
• 9 B. Sun, Y. Roichman, and D. G. Grier, “Theory of holographic optical trapping,” Opt. Express 16(20), 15,765–15,776 (2008).
• 10 J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
• 11 Y. Roichman, I. Cholis, and D. G. Grier, “Volumetric imaging of holographic optical traps,” Opt. Express 14(22), 10,907–10,912 (2006).
• 12 D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
• 13 S.-H. Lee, Y. Roichman, G.-R. Yi, S.-H. Kim, S.-M. Yang, A. van Blaaderen, P. van Oostrum, and D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Opt. Express 15(26), 18,275–18,282 (2007).
• 14 F. C. Cheong, K. Xiao, D. J. Pine, and D. G. Grier, “Holographic characterization of individual colloidal spheres’ porosities,” Soft Matter 7(15), 6816–6819 (2011).
• 15 B. J. Krishnatreya, A. Colen-Landy, P. Hasebe, B. A. Bell, J. R. Jones, A. Sunda-Meya, and D. G. Grier, “Measuring Boltzmann’s constant through holographic video microscopy of a single sphere,” Am. J. Phys. 82(1), 23–31 (2014).
• 16 F. C. Cheong, B. J. Krishnatreya, and D. G. Grier, “Strategies for three-dimensional particle tracking with holographic video microscopy,” Opt. Express 18(13), 13,563–13,573 (2010).
• 17 H. Moyses, B. J. Krishnatreya, and D. G. Grier, “Robustness of holographic video microscopy against defects in illumination,” Opt. Express 21(5), 5968–5973 (2013).
• 18 A. Rohrbach, “Stiffness of optical traps: quantitative agreement between experiments and electromagnetic theory,” Phys. Rev. Lett. 95(16), 168102 (2005).
• 19 E. L. Florin, A. Pralle, E. H. K. Stelzer, and J. K. H. Horber, “Photonic force microscope calibration by thermal noise analysis,” Appl. Phys. A 66, S75–S78 (1998).
• 20 Z. Bouchal and M. OlivÃk, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42(8), 1555-1566 (1995).
• 21 M. Polin, K. Ladavac, S.-H. Lee, Y. Roichman, and D. G. Grier, “Optimized holographic optical traps,” Opt. Express 13(15), 5831–5845 (2005).
• 22 J. M. Taylor and G. D. Love, “Multipole expansion of Bessel and Gaussian beams for Mie scattering calculations,” J. Opt. Soc. Am. A 26(2), 278–282 (2009).
• 23 J. Chen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nature Photonics 5(9), 531–534 (2011).
• 24 M. Šiler and P. Zemánek, “Optical forces in a non-diffracting vortex beam,” J. Quant. Spectr. Rad. Trans. 126, 78–83 (2013).
• 25 N. Wang, J. Chen, S. Liu, and Z. Lin, “Dynamical and phase-diagram study on stable optical pulling force in Bessel beams,” Phys. Rev. A 87(6), 063812 (2013).
• 26 G. Gouesbet, “T-matrix formulation and generalized Lorenz-Mie theories in spherical coordinates,” Opt. Comm. 283(4), 517–521 (2010).
• 27 C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley Interscience, 1987).
• 28 G. Gouesbet and G. Gréhan, Generalized Lorenz-Mie Theories (Springer-Verlag, 2011).
• 29 J. P. Barton, “Internal and near-surface electromagnetic fields for a spheroidal particle with arbitrary illumination,” Appl. Opt. 34(24), 5542–5551 (1995).
• 30 J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66(10), 4594–4602 (1989).
• 31 J. P. Gordon, “Radiation forces and momenta in dielectric media,” Phys. Rev. A 8(1), 14–21 (1973).
• 32 P. C. Chaumet and M. Nieto-Vesperinas, “Time-averaged total force on a dipolar sphere in an electromagnetic field,” Opt. Lett. 25(15), 1065–1067 (2000).
• 33 B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333(2), 848–872 (1988).
• 34 S. Sukhov and A. Dogariu, “Negative nonconservative forces: optical “tractor beams” for arbitrary objects,” Phys. Rev. Lett. 107(20), 203602 (2011).
• 35 A. Novitsky, C.-W. Qiu, and A. Lavrinenko, “Material-independent and size-independent tractor beams for dipole objects,” Phys. Rev. Lett. 109(2), 023902 (2012).
• 36 A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11(5), 288–290 (1986).
• 37 K. Harada, O. Kamimura, H. Kasai, T. Matsuda, A. Tonomura, and V. V. Moshchalkov, “Direct observation of vortex dynamics in superconducting films with regular arrays of defects,” Science 274(5290), 1167–1170 (1996).
• 38 Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100(1), 013602 (2008).
• 39 T. Čižmár and K. Dholakia, “Tunable Bessel light modes: engineering the axial propagation,” Opt. Express 17(18), 15,558–15,570 (2009).
• 40 M. Lei, Z. Li, S. Yan, B. Yao, D. Dan, Y. Qi, J. Qian, Y. Yang, P. Gao, and T. Ye, “Long-distance axial trapping with focused annular laser beams,” PLoS ONE 8(3), e57984 (2013).
• 41 H. Kawauchi, K. Yonezawa, Y. Kozawa, and S. Sato, “Calculation of optical trapping forces on a dielectric sphere in the ray optics regime produced by a radially polarized beam,” Opt. Lett. 32(13), 1839–1841 (2007).
• 42 M. Michihata, T. Hayashi, and Y. Takaya, “Measurement of axial and transverse trapping stiffness of optical tweezers in air using a radially polarized beam,” Appl. Opt. 48(32), 6143–6151 (2009).
• 43 L. Carretero, P. Acebal, and S. Blaya, “Three-dimensional analysis of optical forces generated by an active tractor beam using radial polarization,” Opt. Express 22(3), 3284–3295 (2014).


text:
# Also in the Article

Recursive equations with uncontrolled mating
This protocol is extracted from research article:
A theoretical derivation of response to selection with and without controlled mating in honeybees
Genet Sel Evol, Feb 17, 2021;

Procedure

First, we consider breeding colonies with uncontrolled mating (Fig. 1). A worker group W in year t receives half of its genetic material from its queen Q, which in turn is an offspring from a selected colony in year $t-2$, and thus on average, has a breeding value equal to $Bt-2+S1,t-2$. The other half of its genetic material comes from the drones which Q mated with. The drone producing queens are unselected and from year $t-2$. With probability $pt$, they are breeding queens of that year and thus have an average breeding value equal to $Bt-4+S1,t-4$. With probability $1-pt$, they are passive queens of year $t-2$. In this case, a further distinction is necessary, since the colony from which a passive queen originates may be an unselected breeding colony (probability $qt-2$) with an average breeding value equal to $Bt-4$ or a passive colony (probability $1-qt-2$) with an average breeding value equal to $Pt-4$. Combining all paths of inheritance, we arrive at the following recursive equation for the average true breeding values in the breeding population:

If we assume that $pt=p$, $qt=q$, and $S1,t=S1$ are constant over years, grouping terms for breeding and passive populations yields:

In the passive population, a worker group W in year t receives half of its genetic material from its queen Q, which in turn comes either from an unselected breeding colony (probability $qt$) with average true breeding value $Bt-2$, or from a passive colony (probability $1-qt$) with average breeding value $Pt-2$ (Fig. 1). Therefore, Q has an average breeding value that is equal to $qtBt-2+(1-qt)Pt-2$. For the paternally inherited genetic material, the same considerations as with uncontrolled mating for the breeding population apply. This yields:

which in analogy to Eq. 2, again under the assumption of constant $pt=p$, $qt=q$, and $S1,t=S1$, can be rearranged to

Note: The content above has been extracted from a research article, so it may not display correctly.

Q&A


text:
How do you solve and graph 3x + 8 ≥ 4x ?

Jul 24, 2018

$x \le 8$

Explanation:

$3 x + 8 \ge 4 x$

Subtract $\textcolor{b l u e}{3 x}$ from both sides:
$3 x + 8 \quad \textcolor{b l u e}{- \quad 3 x} \ge 4 x \quad \textcolor{b l u e}{- \quad 3 x}$

$8 \ge x$

Therefore,
$x \le 8$

Here's a graph of it on a number line:

(mathwarehouse.com)

The closed circle on $8$ means that $8$ is one of the solutions (and anything less than it).

Hope this helps!