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the sensitivities of second generation gravitational wave detectors such as advanced ligo and advanced virgo are expected to be limited by the thermal noise of the test masses within a significant range of signal frequencies around 100 hz  @xcite . to reach even better sensitivities ,
it has been proposed to use laser beams with an intensity pattern other than that of the fundamental gaussian beam to reduce the effects of this thermal noise  @xcite .
a beam whose intensity is distributed more homogeneously over the mirror surface , for the same clipping losses , benefits from a more effective averaging over the mirror surface distortions caused by thermal effects  @xcite .
the specific advantage of using higher order lg modes , as opposed to mesa  @xcite and conical  @xcite beams , is that they are compatible with spherical mirrors as currently used in gw detectors and other high precision optical setups .
research into the potential of the lg@xmath0 mode in gravitational wave detectors has been carried out using numerical simulations and table - top experiments  @xcite .
the sensing and control signals for an lg@xmath0 beam were found to perform as well as for the fundamental mode in all aspects examined and the lg@xmath0 behaved as expected in short linear and triangular optical cavities .
however , an optical cavity resonant for a higher  order gaussian mode is degenerate so that a number of modes can resonate at the same time .
this is a fundamental difference to a well designed cavity for the fundamental gaussian mode , in which any resonant enhancement of other modes can be suppressed .
this degeneracy can potentially cause additional optical losses .
simulations have shown that the use of the lg@xmath0 beam , compared to the fundamental mode , lg@xmath1 , could result in a significant contrast defect at the dark fringe  @xcite .
it is the aim of this paper to investigate how mirror surface distortions affect the purity of an lg@xmath0 beam in high finesse cavities , by analytical calculation and numerical simulation .
we will focus on the direct coupling from a distorted mirror and how this affects the mode content in a linear cavity , with our final aim to produce specifications for the mirror surfaces .
ideally the mirrors in gravitational wave detectors should be perfectly smooth with a radius of curvature matching that of the incident beam . however , real mirrors deviate from a perfect surface , altering the beams which interact with them
. if a beam @xmath2 is incident on a distorted surface described by @xmath3 and uniform reflectivity @xmath4 , then the reflected beam is given by : @xmath5 fig .  [
fig : coupling_diagram ] illustrates this effect .    .
the surface @xmath3 is defined as a height field across a plane perpendicular to the optical axis along @xmath6 . ]    in order to investigate the effects of surface distortions measured mirror surface maps can be used .
the term _ mirror map _ refers to an array of data detailing the optical properties of a mirror , often its surface height in nanometers .
this data can be used to represent realistic mirrors in numerical simulations of gravitational wave detectors .
mirror maps have been produced from uncoated advanced ligo mirror substrates which represent the best mirror surfaces of this kind currently available . in the following
we have made use of one such map , the surface map of the substrate etm08  @xcite . the deviation of this map surface from a perfectly spherical surface with radius of curvature 2249.28 m is shown in fig .
[ fig : etm08 ] .
this substrate shows an rms surface figure error of 0.523 nm .
zernike polynomials are well suited for the purposes of describing mirror surface distortions .
zernike polynomials can be used to describe classical distortions such as tilts and curvatures  @xcite .
they are a complete set of functions which are orthogonal over the unit disc and defined by radial index , @xmath7 , and azimuthal index , @xmath8 , with @xmath9 . for any index @xmath8
we have one odd and one even polynomial  @xcite : @xmath10 where @xmath11 is the normalised radial coordinate , @xmath12 is the azimuthal angle , @xmath13 is the amplitude and @xmath14 is the radial function .
the radial function is given by the following sum : @xmath15 for @xmath16 even and 0 otherwise .
this gives @xmath17 non - zero zernike polynomials for each value of @xmath7 ( for @xmath18 the odd polynomial is zero ) .
[ fig : zern ] shows the surfaces described by the zernike polynomials corresponding to orders ( n ) 0 to 4 .
the lower order polynomials represent some common optical distortions , some of which are summarised in table  [ table : zernike ] .
.summary of some common names for the lower order zernike polynomials  @xcite . [ cols="^,^,^",options="header " , ]     these amplitude limits were applied to the etm08 mirror map , resulting in coupling of 19ppm into modes other than lg@xmath0 and 0.043ppm into the other order 9 modes .
the cavity defined in sec .  [ sec : ligo_cavs ] was simulated with this limited map , resulting in 815ppm impurity in the circulating beam .
this is a very good improvement on the original impurity of 0.114 , illustrating that a high beam purity is achievable with these mirror requirements . to achieve an even higher beam purity
will involve reducing the amplitudes of these polynomials further , as well as additional zernike requirements .
we have investigated the coupling which occurs when laguerre - gauss modes are incident on a mirror with surface distortions .
taking an analytical approach we used zernike polynomials to represent mirror surface distortions and derived an approximate equation for the significant coupling when an lg mode is reflected from a surface defined by a particular zernike polynomial .
this derivation resulted in a condition for significant coupling , @xmath19 , where @xmath8 is the azimuthal index of the zernike polynomial and @xmath20 and @xmath21 are the azimuthal indices of the incident and coupled modes respectively .
this is a significant result as it allows us to predict which order 9 modes will be largely coupled by particular zernike polynomials and suggest which modes will have large amplitudes in the arm cavities    we investigated the performance of lg@xmath0 in high finesse cavities by simulation with advanced ligo mirror maps .
this illustrated the degraded purity of the circulating beam in realistic cavities due to higher order mode degeneracy .
the results were then analysed by looking at the zernike polynomials representing our example mirror map .
the analysis and results were consistent with the predictions made from eq .
[ eq : k_eq ] .
this suggested that astigmatism was causing a significant amount of coupling , particularly into the lg@xmath22 and lg@xmath23 modes .
this was confirmed when the cavity was simulated again with the astigmatism removed from the mirror map and we observed a dramatic increase in the lg@xmath0 mode purity .
the analytical description enabled us to identify the specific zernike polynomials which cause large couplings as well as the lg modes which would dominate as a result . using this
we were able to derive certain requirements for our example mirror map , etm08 , in terms of limits on the amplitudes of the zernike polynomials z@xmath24 , z@xmath25 and z@xmath26 ( table  [ table : limits ] ) . using this map
the resulting circulating beam impurity was found to be 815ppm , a significant reduction from the original impurity of 0.114 .
this investigation has demonstrated that a high beam purity is achievable using an lg@xmath0 beam when modifications are made to the low order zernike polynomials in advanced ligo mirrors . implementing the lg@xmath0 beam in gravitational wave detectors
will be challenging as we require very small amplitudes on these lower order polynomials .
we should also consider that the example mirror surfaces considered here refer to uncoated substrates .
the coating process is likely to add to the lower order features in the mirror surfaces .
however , using this analytical approach we can derive specific requirements for the mirror surfaces leading to designs for suitable mirrors for these higher order beams .
we would like to thank garilynn billingsley for providing the advanced ligo mirror surface maps and for advice and support on using them .
we would also like to thank david shoemaker and stefan hild for useful discussions .
this work has been supported by the science and technology facilities council and the european commission ( fp7 grant agreement 211743 ) .
this document has been assigned the ligo laboratory document number ligo  p1100081 .
the product of two laguerre - gauss modes is : @xmath27 the following derivation follows from eq .
[ eq : m_condition ] .
currently we are concerned with the magnitude of the coupling coefficients , so we ignore any constant phase shifts and integrate with respect to @xmath12 : @xmath28 in order to further simplify the equation the following variable substitution is made : @xmath29 and a new limit to the integral : @xmath30 where @xmath31 is the zernike radius .
this gives the integral : @xmath32 substituting in the sums representing the laguerre polynomials and the radial zernike function as a function of @xmath33 gives : @xmath34 this type of integral results in the lower incomplete gamma function : @xmath35 when @xmath36 is equal to @xmath7 , an integer , the function is given by the following sum : @xmath37 therefore , the final equation for this approximation of the magnitude of the coupling coefficients is given by : @xmath38
clipping loss is given by : @xmath39 where the integral represents the normalised power reflected by a perfect mirror .
@xmath40 defines an infinite plane perpendicular to the beam axis .
the magnitude squared of an lg mode is : @xmath41 where @xmath42 is the beam radius , @xmath43 and @xmath20 are the mode indices and @xmath4 is the radial position . integrating over the surface and taking a similar approach as for the coupling coefficients
we get : @xmath44 where @xmath45 , @xmath31 is the radius of the mirror and @xmath46 is the lower incomplete gamma function .
for certain zernike polynomials ( those with non - zero @xmath8 ) their amplitudes in a surface depend on the orientation of that surface with respect to the zernike surface .
for example , consider the two polynomials responsible for astigmatism , z@xmath47 .
the two polynomials actually describe the same shape , with one just rotated by 90 degrees with respect to the other . therefore , rotating a surface , such as the one described by mirror map etm08 , will change the amplitudes of these two polynomials within the surface . fig .
[ fig : etm08_zs_rotated ] illustrates this effect .
the plots show the amplitudes of the order 2 zernike polynomials present in the etm08 mirror surface as it is rotated .
as expected the amplitude of the z@xmath48 polynomial remains constant as it has no angular dependence .
the amplitudes of the z@xmath47 polynomials oscillate and , at a certain rotation ( around 120@xmath49 ) the astigmatism of the surface is completely described by z@xmath50 , and 90@xmath49 later completely described by z@xmath51 .
the root mean squared amplitude ( @xmath52 ) of the polynomials remains constant .
we have used zernike polynomials to describe mirror surface distortions and analyse the coupling that occurs from lg@xmath0 into other order 9 modes .
this approach appears to be very suitable as we have been able to identify specific polynomials which extract significant amounts of power from the input mode .
however , there is an alternative method used to investigate mirror surface distortions , which involves looking at the spatial frequencies present in real mirrors . in this section
we compare these two methods .
to look at the spatial frequencies present in realistic mirrors we perform a 2d fourier transform of the surface height data .
the resulting spectra is then analysed and synthetic maps are created with the same spatial frequencies .
this method focuses on identifying particular spatial wavelengths which cause a large degree of coupling from lg@xmath0 .
many synthetic maps are created and used in simulations of gravitational wave detectors . a statistical approach
is then taken to determine the extent of the coupling when specific spatial frequencies are present in the mirror .    in the zernike approach
we look at the different polynomials present in mirror surface distortions .
this can be thought of as equivalent to looking at the spectra of the mirror surfaces as the different polynomials represent different spatial frequencies .
the plot in fig .
[ fig : map_spectra ] illustrates this .
the spectrum of the etm08 advanced ligo mirror map is shown , along with the spectra of maps made up from the zernike polynomials present in the etm08 mirror .
each of the zernike maps recreates the ligo map with polynomials up to a certain order .
the plots show that as the order of zernike polynomials present increases the higher spatial frequencies are represented in the mirror map .
this is because these higher order polynomials represent the higher order spatial frequencies .
looking at the spatial frequencies present in the zernike polynomials we found that the frequencies depended on the order , @xmath7 .
a consequence of this is that if we just consider the spatial frequencies present in the mirror maps we will not be able to distinguish between polynomials with different @xmath8 . as we have seen , the azimuthal index is very significant as it determines which modes are largely coupled from lg@xmath0 . therefore looking at the spatial frequencies does nt identify the important shapes in the mirror surfaces .
the zernike approach would seem to be the most suitable as this allows us to identify the interesting polynomials and modes .                                `
zernike polynomials and their use in describing the wavefront aberrations of the human eye ' ( stanford university , 2003 ) http://scien.stanford.edu/pages/labsite/2003/psych221/projects/03/pmaeda/index.html
higher order laguerre  
 gauss ( lg ) beams have been proposed for use in future gravitational wave detectors , such as upgrades to the advanced ligo detectors and the einstein telescope , for their potential to reduce the effects of the thermal noise of the test masses . 
 this paper details the theoretical analysis and simulation work carried out to investigate the behaviour of lg beams in realistic optical setups , in particular the coupling between different lg modes in a linear cavity . 
 we present a new analytical approximation to compute the coupling between modes , using zernike polynomials to describe mirror surface distortions . 
 we apply this method in a study of the behaviour of the lg@xmath0 mode within realistic arm cavities , using measured mirror surface maps from the advanced ligo project . 
 we show mode distortions that can be expected to arise due to the degeneracy of higher order spatial modes within such cavities and relate this to the theoretical analysis . 
 finally we identify the mirror distortions which cause significant coupling from the lg@xmath0 mode into other order 9 modes and derive requirements for the mirror surfaces .