Optics and Lasers in Engineering 50 (2012) 1501–1507

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Optics and Lasers in Engineering

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Effect of aberrations in vortex spatial filtering

Manoj Kumar Sharma n, Joby Joseph, P. Senthilkumaran

Department of Physics, Indian Institute of Technology Delhi, New Delhi 110016, India

a r t i c l e i n f o

Article history:

Received 30 April 2012

Received in revised form

18 June 2012

Accepted 20 June 2012Available online 12 July 2012

Keywords:

Hilbert transform

Edge enhancement

Aberrations and anisotropic vortex

66/$ - see front matter & 2012 Elsevier Ltd. A

x.doi.org/10.1016/j.optlaseng.2012.06.012

esponding author. Tel.: þ91 98 6840 2488.

ail address: imaging.mks@gmail.com (M.K. Sh

a b s t r a c t

Edge enhancement is a very important operation in image processing and a spiral phase plate can be

used as a radial Hilbert mask for isotropic edge enhancement. In this paper we analyze the effect of

various Seidel aberrations on the performance of radial Hilbert mask or the vortex phase mask. The

aberrated vortex phase mask is implemented optically with the help of a high resolution, spatial light

modulator (SLM). It has also been shown that out of various aberrations astigmatism can introduce

anisotropy in the Hilbert mask which causes selective edge enhancement.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

In image processing, edges of an image are very important tounderstand the characteristics of images. Therefore, the edges areenhanced to obtain better contrast. A spiral phase plate (SPP) oftopological charge m, with transmittance function by expðimyÞ canbe implemented as a spatial filter to obtain edge enhancement [1].If the phase profile of the vortex mask is analyzed, it is obviousthat there is a phase difference of mp at any two symmetricpositions in any radial line with respect to the vortex core. Similarcharacteristics can be seen in the 1D-Hilbert transform [2–5].Therefore, the spiral phase filtering using an SPP is known asradial Hilbert transform and the function, expðimyÞ is regarded asradial Hilbert phase mask where m is designated as the order ofthe radial Hilbert transform which is also the topological chargeof the vortex. The sign of the topological charge of a vortexfunction decides the helicity of the vortex. The LG spatial filter [6]gives sharper edge enhancement because it modifies both ampli-tude and phase of the field at the filter plane. The vortex basedspiral phase filters are also used in microscopy [7–9]. 1D Hilberttransform of a function is convolution of the function with thekernel �1/px which can be realized optically by using Hilbertmask at the Fourier plane in the filtering set-up. The output of thestandard 4f spatial filtering system contains the convolution ofthe object function and the Fourier transform of the kernel.Fourier transform of a function separable in polar coordinates isprovided by the sum of weighted Hankel transforms of allpossible orders. Generally the edge enhancing effect is isotropic,when a spiral phase plate is used as a filter. In certain

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arma).

applications, it may be preferred to have some edges to beemphasized more than others. Therefore, to enhance the edgesselectively, one has to break the symmetry of the radial Hilbertmask. Vortex based selective edge enhancement is possible byusing a fractional vortex mask [1,10–13] or by using an aniso-tropic vortex filter [14].

As aberrations [15–20] are indispensable part of opticalsystems, it is important to study the effect of aberrations on theradial Hilbert mask. In this paper we analyze the effect ofaberrations on the radial Hilbert mask, with regard to edgeenhancement. It is seen that small amount of astigmatism inthe vortex mask leads to selective edge enhancement, while otheraberrations disturb the image detrimentally. Different aberrationshave been introduced in the radial Hilbert mask which is used asa filter in the 4f system for edge enhancement. Further we presentthe balancing of aberrations and the effect on edge enhancement.Simulated and experimental results are presented.

2. Primary aberrations and aberrated vortex filter

The general aberration function for rotationally symmetricoptical system can be written as [15]

Wðr,yÞ ¼ Asr4þAcr3 cosðyÞþAar2 cos2ðyÞþAf r2þAdr cosðyÞ, ð1Þ

where As, Ac, Aa, Af and Ad are coefficients for spherical aberration,coma, astigmatism, field curvature (or defocus) and tilt (ordistortion) respectively.(r,y) are the polar coordinates on the exitpupil of the imaging system.

A vortex beam has a helical phase structure with a point ofundefined phase at its core [16–19] and this point is referred toas phase singularity. Accumulated phase variation around thesingularity is an integral multiple of 2p, and this integer is

M.K. Sharma et al. / Optics and Lasers in Engineering 50 (2012) 1501–15071502

referred to as topological charge m. Optical beam carrying such apoint of undefined phase with order of helicity m, can berepresented as

Uðr,yÞ ¼ expðimyÞ, ð2Þ

In general an aberrated vortex beam can be represented as

Uabðr,yÞ ¼ expðimyÞexpðiKWðr,yÞÞ, ð3Þ

where W(r,y) is aberration function representing the wave aber-ration. K¼2p/l, is the propagation constant.

The radial vortex phase mask of charge one, expðiyÞ enhancesthe edges of the image in all radial directions. The mask functioncan be seen as superposition of signum functions in all radialdirections, expressed in polar coordinates. In conventional Hilberttransform setup, a signum function is used as a filter that retardsthe negative frequency components of the object by p whileleaving the positive frequency components intact. The output ofthe 4f system is the convolution of the object and the Fouriertransform of the filter function. The anisotropy in the vortexphase disturbs the signum formation along some radial directionsand such a filter causes selective edge enhancement. The topolo-gical charge or the order of the radial Hilbert mask is consideredas 1 throughout the paper. The vortex phase mask, correspondingfiltered output and the 3D profile of the output for a circularaperture as an object are shown in Fig. 1(a)–(c) respectively. Thetransverse output intensity plot is shown in Fig. 1(d). The resultshown in Fig. 1(a)–(d) for comparison with the results obtainedwith aberrated vortex masks.

2.1. Spherical aberration

In a lens suffering from spherical aberration, marginal andparaxial rays focus at two different axial points. The distance

Fig. 1. (a) Phase profile of the pure vortex beam, (b) vortex filtered output, (c) 3D

profile of the output and (d) plot of the intensity profile in horizontal and vertical

directions.

between these two foci is a measure of spherical aberration in thelens. The spherical aberration can occur in the system even if theobject is placed on the optical axis.

The spherical aberration function can be written asW(r,y)¼Asr

4 where As are spherical aberration coefficients inunits of wavelength of light. Hence the vortex filter with sphericalaberration can be written as

Us ¼ expðiyÞexpðiKAsr4Þ ð4Þ

The spherical aberrated vortex phase mask, the filtered outputand its 3D profile for a circular object are shown in Fig. 2(a)–(c)respectively. It is quite clear from the figures that the sphericalaberration in the vortex phase mask can deteriorate the filteredoutput critically. Even a very small value of As leads to concentricrings around the output and the maximum intensity is lowered.The transverse profiles (horizontal and vertical) of the outputintensity patterns have been plotted and are shown in Fig. 2(d).The intensity in the enhanced edges becomes marginally lesswhen the spherical aberration is present because of the spread inthe edges. The intensity distribution for spherical aberration isrotationally symmetric as is clear from Fig. 2(d). If the value of As

is further increased the output intensity becomes even lesser.

2.2. Distortion/tilt

The distortion aberration may arise in the system because ofthe fact that the different areas of the imaging system (say lens)may have different focal lengths and different magnifications. Inabsence of any other aberrations, distortion causes a misshapingof the image as a whole. The aberration function for distortion is

Fig. 2. (a) Phase profile of the vortex beam with spherical aberration

(As¼1�10�8), (b) corresponding filtered output, (c) 3D profile of the output

and (d) plot of the intensity profile in horizontal and vertical directions for

spherical aberrated vortex filtered out put in horizontal and vertical directions.

M.K. Sharma et al. / Optics and Lasers in Engineering 50 (2012) 1501–1507 1503

given by Wðr,yÞ ¼ Adr cosðyÞ. Hence the vortex filter with distor-tion/tilt aberration can be written as

Ud ¼ expðiyÞexpðiKAdr cosðyÞÞ ð5Þ

It can be seen from Fig. 3(a) that the phase of the distortedvortex mask takes the form of fork type grating and the corre-sponding filtered outputs are the edge enhanced images whichappear as different diffraction orders of the fork grating. Theoutput transverse intensity is plotted in two directions normal toeach other and is shown in Fig. 3(d). The output intensity isslightly greater for the vortex filter with distortion/tilt in hor-izontal direction while in vertical direction it is comparable to thevortex filtered intensity. This is a kind of redistribution ofintensity in two normal directions at the imaging plane. Moreimportantly the increment in intensity occurs in the directionwhich is parallel to the fork fringes.

2.3. Astigmatism

Astigmatism is an off-axis point wavefront aberration, causedby the inclination of incident wavefronts relative to the opticalsurface. When an object point lies at an appreciable distance fromthe optical axis, the incident cone of rays strikes the lensasymmetrically, giving rise to astigmatism. Astigmatism is knownto modify the focal plane intensity distribution [20–22] of asingular beam.

The aberration function for astigmatism is given by Wðr,yÞ ¼Aar2 cos2ðyÞ where Aa are astigmatic aberration coefficients in

Fig. 3. (a) Phase profile of the distorted vortex beam (Ad¼0.05), (b) corresponding

filtered output, (c) 3D profile of the output and (d) plot of the intensity profile in

horizontal and vertical directions for distorted vortex filtered output in horizontal

and vertical directions.

units of wavelength of light. Hence the astigmatic vortex filter canbe written as

Ua ¼ expðiyÞexpðiKAar2 cos2ðyÞÞ ð6Þ

It is found that astigmatic aberration in the vortex phase canintroduce anisotropy and hence leads to selective edge enhance-ment in addition to edge blurring effect in either radial direction.We analyze the effect of astigmatism by introducing differentvalues of astigmatic coefficient in Eq. (6). The astigmatic vortexmasks corresponding to two different values of astigmatic coeffi-cients are shown in Fig. 4.1(a) and (d). The corresponding filteredoutput intensities are shown in Fig. 4.1(b) and (e). The figuresrepresent the selectively edge enhanced images for a circularaperture used as an object. Also it can be seen that as we increaseastigmatism in the filter function, the anisotropy also increasesand the lesser region of the image is edge enhanced. The edges inthe other directions are blurred due to astigmatism aberration.The direction of the enhancement and the blurring can bechanged by adding an additional offset angle to the phase. Thetransverse output intensity profiles along two orthogonal direc-tions (horizontal and vertical) are plotted, and are shown inFig. 4.2(a) and (b) for the two astigmatic vortex filters respec-tively. It can be seen that the edge intensity is increased in theoutput when astigmatic mask is used because of different focus-ing properties introduced by astigmatism in meridional andsagittal planes in the system. This increment in the intensity atcertain regions of the filtered output is a consequence of redis-tribution of intensity in different radial directions.

2.4. Coma

Coma is the first off-axis aberration. For an optical systemsuffering from coma, the magnifications from the different zonesof the optical system are different. The aberration function forcoma is given by Wðr,yÞ ¼ Acr3 cosðyÞ where Ac are coma aberra-tion coefficients in units of wavelength of light. Hence the vortexfilter with coma aberration can be written as

Uc ¼ expðiyÞexpðiKAcr3 cosðyÞÞ ð7Þ

The phase masks for the comatic mask, the intensity outputand the 3D pattern are shown in Fig. 5(a)–(c) respectively. Theshape of the phase mask is altered in presence of coma and thephase near to the center is not altered much. The output isdeteriorated in the presence of coma and for some higher valuesof comatic mask the output seriously loses its shape and sizewhich is expected as the irradiance distribution for coma issymmetric about the tangential plane, and looses symmetry inmeridional plane. Thus it has line symmetry in the tangentialplane. The diffraction focus lies on this line. The transverseintensity plot given in Fig. 5(d) also shows that the output ishorizontally symmetric while it is non-symmetric in perpendi-cular direction.

2.5. Defocus/field curvature

For an object point away from the axis, the distance betweensagittal and tangential foci increases as the object point movesaway from the axis. Then the tangential and sagittal foci of points,which are at different distances from the axis, lie on two surfacesof different curvature because of astigmatism. If the opticalsystem is free from astigmatism, these two surfaces wouldcoincide but the imaging surface could still be curved. This effectappears because of defocus or field curvature present in thesystem. The aberration function for defocus is given by W(r,y)¼Af r2. Hence the vortex filter with defocus aberration can be

Fig. 5. (a) Phase profile of the comatic vortex beam (Ac¼4�10�6), (b) corre-

sponding filtered output, (c) 3D profile of the output and (d) transverse output

intensity plot for comatic vortex filtered output in horizontal and vertical

directions with vortex filtered output intensity plot in horizontal direction.

Fig. 6. (a) Phase profile of the defocused vortex beam (Af¼1�10�4), (b) corres-

ponding filtered output, (c) 3D profile of the output and (d) plot of the intensity

profile in horizontal and vertical directions for defocused vortex filtered output

with vortex filtered output intensity plot in horizontal direction.

Fig. 4.1. (a) Phase profile of the astigmatic vortex beam Aa¼0.0003,

(b) Corresponding filtered output, (c) 3D profile of the output, (d) phase profile

of the astigmatic vortex beam Aa¼0.0005, (e) corresponding filtered output and

(f) 3D profile of the output. 4.2 (a) Transverse output intensity plot for vortex

filtered output in horizontal and vertical directions and (b) plot of the intensity

profile in horizontal and vertical directions for astigmatic vortex filtered output.

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written as

Uf ¼ expðiyÞexpðiKAf r2Þ ð8Þ

In Fig. 6(a)–(c) the phase profile and the filtered outputs areshown. The results are obvious consequences of defocus effects.The edges of the output image are broadened as expected by

M.K. Sharma et al. / Optics and Lasers in Engineering 50 (2012) 1501–1507 1505

defocus effect and also shown in Fig. 6(d). The output intensity issymmetrically distributed around the center.

3. The balanced aberrations

It is important to know the effect of balancing of the aberra-tions by one another on the edge enhancement [23–26]. Thefollowing balanced pairs are reported in perspective of leastconfusion and minimum variance.

(a)

Fig.(As¼

and

sphe

Balancing of spherical aberration with defocus.

(b) Balancing of astigmatism with defocus. (c) Balancing of coma with distortion/tilt.

3.1. Balancing of spherical aberration with defocus

The balancing of the spherical aberration in imaging systems isdone by appropriate amount of defocus (Ad¼�As) in the systembut as Fig. 7 which shows the filtered outputs and phase profilesof balanced spherical vortex mask for As¼0.00000001, suggestsnot much improvement in the filtered output with the balancedspherical vortex mask in comparison to a non-balanced sphericalvortex mask. The transverse output intensities for balanced andunbalanced spherical vortex mask have been plotted and as isshown in Fig. 7(d), the two profile exactly overlap hence showingalmost no improvement in the filtered output. Hence the sphe-rical aberrations effects cannot be effectively compensated bydefocus in spatial filtering masks.

7. (a) Phase profile of the balanced spherical aberrated vortex beam

5�10�8), (b) corresponding filtered output, (c) 3D profile of the output

(d) the horizontal output intensity plots for balanced spherical mask and for

rical vortex mask.

3.2. Balancing of astigmatism by defocus

The astigmatism present in the imaging system can bebalanced by defocus to get the minimum variance with respectto the point spread function of the imaging system. The defocus(or field curvature) of Af¼�Aa/2 is introduced in the astigmaticvortex filter. The phase profile of the filter for Aa¼0.0003 has beenshown in Fig. 8(a). The phase becomes smoother around thecenter in comparison to the phase of unbalanced astigmaticvortex filter. The balancing of the astigmatism results in thefourfold symmetry of the point spread function. The filteredoutput intensity, its 3D profile for a circular aperture with thebalanced astigmatic vortex filter have been shown in Fig. 8(b) and(c). It can be seen from the figures that the output comes out to befourfold symmetric as it is for point spread function of imagingsystem.

3.3. Balancing of coma with distortion/tilt

The coma is usually balanced with distortion (Ad¼�2Ac/3) i.e.tilt to get the minimum variance. We introduce same distortion inthe comatic mask (for Ac¼0.00005). The phase profile for such afilter is shown in Fig. 9(a). We see no change in the phase profileafter balancing. Similarly the filtered output intensity and its 3Dprofile are shown in Fig. 9(b), and (c) respectively. The figuressuggest almost no improvement on balancing. Hence so far thespatial filtering for edge enhancement is concerned not muchnoticeable aberration compensation is observed which is alsoevident from the transverse intensity profiles shown n Fig. 9(d)and (e).

Fig. 8. (a) Phase profile of the balanced astigmatic vortex beam (Aa¼4�10�4),

(b) corresponding filtered output, (c) 3D profile of the output and (d) the

horizontal output intensity plots for balanced astigmatic mask and for astigmatic

vortex mask.

Fig. 9. (a) Phase profile of the balanced comatic vortex beam (Ac¼5�10�5),

(b) corresponding filtered output, (c) 3D profile of the output, (d) plot of the

intensity profiles in horizontal and vertical directions for balanced comatic vortex

mask and for comatic vortex mask and (e) plot of the intensity profiles in vertical

directions for balanced comatic vortex mask and for comatic vortex mask.

Fig. 10. Experimental setup for the selective edge enhancement; the CGH

displayed on the SLM is shown separately; SF: Spatial filter; L1: Collimator;

S: Sample stage; M: Microscopic objective; L2: Lens to image Fourier transform on

the SLM with magnification of 4; SLM: Spatial light modulator to display the

computer generated hologram of the proposed functions in phase mode;

L3: Imaging lens; CMOS: camera to record the obtained images after filtering process.

Fig. 11.1. Experimental results when a circular aperture is used as an object, with

(a) Aa¼0.0003 and (b) Aa¼0.0005. 11.2 Experimental results when phase mask

has (a) spherical aberration, (b) tilt, (c) coma and (d) field curvature. 11.3Experimental results for balanced phase masks: (a) balanced spherical aberration,

(b) balanced astigmatism and (c) balanced coma.

M.K. Sharma et al. / Optics and Lasers in Engineering 50 (2012) 1501–15071506

4. Experimental results and discussion

We have implemented the phase masks corresponding to theaberrated vortex function with the help of reflective Spatial LightModulator (SLM), Holoeye LC-R 2500 with resolution 1024�768and pixel pitch19 mm. The object used is a circular aperture of size200 mm. The experimental setup is shown in Fig. 10. The object isilluminated by collimated beam from He–Ne laser (632.8 nm) andFourier transformed with the help of Newport 10� microscopicobjective and the Fourier transform is imaged on the SLM for 4�magnification by a lens of focal length 135 mm. The SLM is

operated in phase mode keeping the polarizer at angles 1701 toget the phase shift up to 2p. The computer generated holograms(CGH) corresponding to the aberrated functions are formed anddisplayed on the SLM. The CGHs corresponding to the proposedfunctions are formed in MATLAB keeping the resolution same asthat of the SLM. The incident light wave is then diffracted by thefork grating displayed on the SLM, and only the light diffracted atthe first diffraction order is used. The undesired diffraction ordersare blocked. Imaging is done with the help of a lens of focal length200 mm, kept in between the SLM and infinity-1 CMOS camera, ata distance of 200 mm from each. The images so recorded areshown in Fig. 11.1 for astigmatism, in Fig. 11.2 for other aberra-tions and in Fig. 11.3 for balanced masks, for a circular apertureused as an object. The experimental results are very much sameto those obtained by the simulation.

5. Conclusion

It has been shown that the astigmatic vortex filter, which isprepared by introducing astigmatic aberration in a vortex filter orin an SPP, provides selective edge enhancement. The selectiveness

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comes because of anisotropy as well as astigmatic effect presentin the filter. The effect of other aberrations is also analyzed inregard to edge enhancement. More over other aberrations excepttilt make images detrimental. The balancing of aberrations whichis known to suppress the effect of the aberrations in imagingseems ineffective so far the vortex spatial filtering is concerned.

Acknowledgment

Manoj Kumar Sharma thankfully acknowledges Council ofScientific and Industrial Research (CSIR) of India for SeniorResearch Fellowship.

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