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optical vortices are light beams that carry orbital angular momentum and are characterised by a phase that increases azimuthally about a singularity at the centre of the beam@xcite .
these beams can be generated directly in a cylindrically symmetric laser cavity .
it is , however , simpler in practice to approximate them by illuminating diffractive optics with more readily available laser modes@xcite .
a variety of possible applications for optical vortices@xcite have driven research into their production .
these applications include optical trapping and manipulation of colloidal particles@xcite and cold atoms@xcite , the generation of bose - einstein condensate vortices@xcite , phase - contrast microscopy@xcite , image filtering for extra - solar planet detection@xcite , stimulated emission depletion ( sted ) microscopy@xcite and the encoding of optical quantum information@xcite .
the considerable utility of optical vortices has driven the development of several methods for their production .
most commonly , a phase shift of the form @xmath0 is applied to an incident laser beam where @xmath1 is the topological charge and @xmath2 is the azimuthal coordinate about the beam centre in the transverse plane .
one method to apply the required phase shift is to use a computer - controlled spatial - light modulator ( slm)@xcite .
this approach has the considerable advantages of being flexible and allowing the phase profile to be dynamically varied on 10 ms timescales .
slm s , however , suffer from high cost and limited spatial and phase resolution .
conversion efficiencies can be quite high for low - order optical vortices but begin to suffer from resolution limits as the order is increased .
another approach is to use lithographic processes to construct a phase mask@xcite .
these phase masks have restrictions in terms of phase resolution , spatial resolution , maximum phase shift and useable area , although advanced lithographic techniques can relax many of these restrictions .    here
, we report on the production by direct machining of spiral phase mirrors ( spms ) that are suitable for generating optical vortices .
we demonstrate that the resultant phase mirrors can deterministically produce high - quality optical vortices over a large range of topological charges from 1 to 1000 .
these mirrors can be produced economically and the size of the optics is restricted only by the geometry of the machining enclosure .
the spms are machined from aluminium using a single - crystal diamond tool .
the machining is performed by an ultra - precision lathe , produced by moore nanotechnology . the tool position relative to the part surface
is adjusted while the part is rotated at a constant rate on the spindle .
high - resolution position encoders ensure that the tool is positioned with nanometric precision for each point on the part surface .
the cutting speed is limited by the acceleration of the lathe s linear axes , although we can relax this restriction by breaking the spm into segments .
each segment is a linear ramp with a total height corresponding to a @xmath3 phase shift .
figure [ sem ] shows an sem image of a charge 100 spm composed of 25 segments , each of which imparts an @xmath4 phase ramp . for the charge 100 spm , we estimate that the central defect accounts for 0.006% of the total mirror surface , while the sharp jumps between segments collectively account for approximately 3% of the total area .    for our analysis , we produced 25 mm diameter optics , which required cutting times ranging from 30 minutes to two hours for charges of 1 to 100 , respectively .
the size of the optic can be scaled without losing quality or increasing cutting time simply by selecting a diamond tool of an appropriate radius .
to verify the performance of our phase plate , we made transverse beam profile measurements that can be compared to theoretical models .
the illumination of an spm with a tem@xmath5 mode results in a beam profile that can be expressed as a hypergeometric gaussian mode ( hygg)@xcite . these modes have a cylindrically symmetric amplitude and propagate from a gaussian profile at the spm into a ring - shaped profile in the far - field .
figure [ propagation ] a ) shows the calculated radial amplitude of a propagating charge 15 hygg mode immediately after it is created .
experimental intensity profiles are obtained by illuminating machined spms with a 532 nm laser beam that has been transmitted through a mode - cleaning cavity locked to the tem@xmath5 mode .
the resulting intensity patterns are recorded on a ccd camera .
the radial intensity distribution is then obtained by fitting an interpolating function to the raw pixel data and integrating the resultant function azimuthally in polar co - ordinates .
the experimental data is normalised such that the area under the curve is equal to that of the normalised hygg integrated over the same region .    .
b ) , c ) measured ( solid red trace ) and calculated ( dashed blue trace ) radial intensity profiles after propagation and at the location of the spm , respectively .
d ) measured intensity profile for a charge 15 optical vortex imaged in the far - field .
e ) measured phase of the same vortex imaged in the near - field . ]    figures [ propagation ] b ) and c ) show the calculated ( dashed blue ) and measured ( solid red ) radial intensity profiles after propagation through 0.096 rayleigh lengths and at the location of the spm , respectively .
the beam waist of the input gaussian mode is 450 @xmath6 and the propagation distance has been corrected from a measured 9.5 cm to 11.5 cm , a discrepancy which we attribute to the spm not being located exactly at the beam waist .
the excellent agreement of the model with our data indicates that the spm is performing as designed .
figure [ propagation ] d ) shows the intensity of a charge 15 optical vortex that has been imaged in the far - field .
the phase of the vortex in the near - field , figure [ propagation ] e ) , has been obtained by interfering the vortex with a collimated gaussian beam and scanning the relative phase of the two beams .
ccd images of the interference patterns are captured and the phase profile is obtained by fitting a sine function to the intensity recorded at each pixel .
we verify that the machined spms produce high - quality vortices over a range of topological charges by examining the intensity and phase profiles of charge 1 and charge 100 vortices .
figure [ charge1and100 ] ( a - c ) shows the results for the charge 1 vortex . due to the slow evolution of the charge 1 vortex , imaging optics were used to record the intensity pattern and the propagation distance which was used for the calculated intensity was selected to fit the measured profile .
figure [ charge1and100 ] ( d - f ) shows the results for a charge 100 vortex . there is some deviation from the calculated intensity in the peripheral fringes ; however , the primary feature is in good agreement with the calculated intensity distribution . as with the charge 15 analysis
, the propagation distance was corrected to account for the spm not being located at the beam waist .
the phase profile shows three distinct regions .
the central region has zero charge and originates from the undiffracted portion of the beam .
an intermediate region shows a charge of 25 , which we believe is an artefact of the 25 step design used to create the spm .
the inner region is , however , of little consequence because the outer region , beyond a radius of 0.25 @xmath7 , contains 98% of the optical power and shows the intended charge 100 optical vortex .    the intensity distribution of the same charge 100 vortex , imaged in a plane that features the smallest observed ring thickness , is shown in figures [ highcharge ] a ) and b ) .
the minimal undiffracted order , even at high charges , is notable when compared to slm - produced vortices@xcite .    in an attempt to find the limits of the direct machining process , we manufactured a charge 1000 spm ; however , our ability to properly characterise the vortices degraded after a charge of roughly 100 due to limited ccd resolution .
figure [ highcharge ] c ) shows the measured intensity profile of the charge 1000 optical vortex .     scale .
c ) intensity profile of a charge 1000 optical vortex .
d ) the vortex produced by a charge 15 spm with a @xmath8 period modulation of the phase ramp . ]
spiral phase mirrors that can be manufactured by this technique are not limited to linear azimuthal phase ramps .
modulated phase profiles@xcite can be produced provided that the modulation is slow compared to the loop bandwidth of the positional feedback on the lathe . to demonstrate this flexibility , we cut a charge 15 optical vortex with a sinusoindally modulated phase ramp , shown in figure [ highcharge ] d ) .
the modulation has a period of @xmath9 and a depth of 0.1 .
it is also possible to cut the spms on a curved surface . in this way
, spms could be created on spherical or even aspherical mirrors to provide integrated focusing optics .
we have demonstrated that the direct machining of spiral phase mirrors using a diamond tool can produce high - quality , high - charge optical vortices .
in addition to low cost , technical simplicity and flexibility , this fabrication method has the advantage of being applicable to a variety of metallic and soft crystalline optical materials , which may allow the production of transmissive as well as reflective vortex producing optics .
we thank a. luiten , a. g. white , m. hosseini and n. devlin for fruitful discussion .
we gratefully acknowledge the technical support provided by t. stewart and e. alarie from moore nanotechnology .
this research is supported by the australian research council ( project numbers ce110001027 and dp1092891 ) .
we report on the generation of high - order optical vortices by spiral phase mirrors . 
 the phase mirrors are produced by direct machining with a diamond tool and are shown to produce high - quality optical vortices with topological charges ranging from 1 to 1000 at a wavelength of 532 nm . 
 the direct machining technique is flexible and offers the promise of high - precision , large - diameter spiral phase mirrors that are compatible with high optical powers .