effects of polarization on wavefront shape in a lc-slm diffractive feedback system

$: Effects of polarization on wavefront shape in a LC-SLM diffractive feedback system$
Optics Communications 256 (2005) 261–271

www.elsevier.com/locate/optcom

Effects of polarization on wavefront shape in aLC-SLM diffractive feedback system

C.Y. Wu a, A.R.D. Somervell a,*, T.H. Barnes a, T.G. Haskell b

a Department of Physics, The University of Auckland, Private Bag 90219, Auckland, New Zealandb Industrial Research Ltd., Private Box 31310, Lower Hut, New Zealand

Received 21 March 2005; accepted 1 July 2005

Abstract

The effect of polarization on the phase distribution across a wavefront is discussed for a liquid crystal spatial lightmodulator based nonlinear diffractive single-feedback system. Theoretical analysis and numerical simulations show thatwhen the polarization of the input light beam and interference effects are considered the form of the spectral phase ratiois altered considerably. It is found that this ratio can become zero or even negative for phase distributions with a widerange of spatial frequencies. This may have implications for high-resolution adaptive optics applications where, by care-ful choice of the polarization angle, very good suppression of phase distortions can be obtained. This can be achieved atrelatively low feedback intensities and before the pattern formation threshold is reached.� 2005 Elsevier B.V. All rights reserved.

Keywords: Adaptive optics; Phase-distortion suppression; Nonlinear feedback control system; Spontaneous pattern formation

1. Introduction

In the last decade optical nonlinear feedbackdiffractive systems have generated a great deal ofinterest, mostly stemming from there their abilityto spontaneously form a variety of stable and

0030-4018/$ - see front matter � 2005 Elsevier B.V. All rights reserv

doi:10.1016/j.optcom.2005.07.006

* Corresponding author. Tel.: +6493737599x8890; fax:+6493737445.

E-mail address: [email protected] (A.R.D. Som-ervell).

unstable patterns. In the most simple single-feed-back system [1,2] light is incident on a thin Kerr-type nonlinear medium which alters the phasedistribution across the wavefront and the beamthen travels through a region of free space. Thewavefront distortions, either induced by the non-linear medium or already present on the incidentbeam, are then converted to intensity variationsby the resulting diffraction. The intensity of the dif-fracted beam is used to control the phase inducedby the nonlinear medium thus forming a nonlin-ear feedback loop. Both liquid crystal spatial light

ed.

mailto:[email protected]

LC-SLMread-side write-side

Afocal lens system with spatial filter

image plane

diffractionlength, LE (r,t)out

E (r,t)in

E (r,t)fb

BS

Pout P

Fig. 1. Sketch of the feedback phase distortion suppressionsystem.

262 C.Y. Wu et al. / Optics Communications 256 (2005) 261–271

modulators (LC-SLM) and thin Kerr mediums arecommonly used as the nonlinear element.

Much of the recent research in this area hasbeen aimed towards investigating techniques forcontrolling or manipulating patterns formedabove the pattern formation threshold [3–5]. How-ever, some work has also been done examining theinfluence of these systems on wavefront shape be-low the pattern formation threshold [6,7]. It hasbeen shown that these systems have the ability toreduce small wavefront distortions over a limitedspatial bandwidth.

Usually, it is assumed that the phase modula-tion introduced by the nonlinear medium is polar-ization independent or equivalently, in the case ofa LC-SLM, that the direction of polarization is thesame as the liquid crystal molecule directors. Oneexception to this is the work of Tschudi et al. [8]where pattern formation in a LC-SLM based dif-fractive feedback system was studied includingpolarization effects and as well as the more compli-cated phase modulation characteristics of the LCmaterial. It was found that by including the polar-ization modulation effects of a LC-SLM an evenricher range of behaviour was seen and that thethreshold for pattern formation was changedsignificantly.

Here, we study the effects of polarization onwavefront shape in LC based systems. An equa-tion for the ratio of the input and output phasespectra valid for small wavefront deviations isdeveloped for any polarization allows for spatialfiltering in the feedback loop. It is found that thespectral ratio is changed significantly when com-pared to a similar system with no polarizationmodulation. If the system parameters are chosenappropriately wavefront distortions can be re-duced to near zero over a wider range of spatialfrequencies. Furthermore, the possibility of phaseconjugation over a limited range of spatial fre-quencies is predicted.

2. The optical system

The optical system is sketched in Fig. 1. A beamof linearly polarized light ~Einð~rÞ with uniformamplitude, E0 and phase distribution, hinð~rÞ is inci-

dent on the read side of a liquid crystal spatial lightmodulator (LC-SLM). The beam reflected fromthe LC-SLM consists of two orthogonal polariza-tion components. One these undergoes phase mod-ulation whilst the other is unaffected by the LClayer. After reflection from the LC-SLM the beamis incident on a beam splitter, BS, and the reflectedcomponent passes through the linear analyzer,Pout, forming the output beam. Pout is orientatedat the same angle as the polarization componentmodulated by the LC layer.

The transmitted component passes through alinear analyzer, P, in the feedback loop. It is thenincident on a Fourier lens block that creates a(spatially filtered) image of write side of theLC-SLM the at a plane located a distance, L, be-fore the write side of the LC-SLM. The beam thenpropagates through free space before reaching atthe write side. The Fourier block is composed of4 lenses and a filter is located in the Fourier trans-form plane of the first lens. We denote the ampli-tude and phase response of this filter as Að~qÞ andP ð~qÞ, respectively.

As mentioned above, this system was firstdiscussed in [8] with regard to pattern formation.Unlike other diffractive feedback systems discussedfor phase distortion suppression the intensity of thebeam controlling the LC-SLM is a result of bothdiffraction and interference. The interferenceoccurs between the two orthogonal polarizationcomponents incident on the read side of theLC-SLM and the relative amplitude of the two

C.Y. Wu et al. / Optics Communications 256 (2005) 261–271 263

interfering beams is selected by the polarization ofthe input beam and by the orientation of the linearanalyzer, P.

In the following, we develop an expression forthe response of the system to small phase distor-tions in the linear approximation. We note thatthis has been done previously for similar systemsbut without the inclusion polarization effects [9].

3. Linear analysis

Assume that the electric field of the input beamis polarized at an angle h1 then

~Einð~rÞ ¼ E0

cos h1sin h1

� �exp½ihinð~rÞ�; ð1Þ

where hinð~rÞ is the phase distortion on the inputbeam. After reflection from the phase modulatorthe electric field can be written

~Eð~r;~tÞ ¼ V~Einð~rÞ

¼ E0

exp½iUð~r; tÞ� 0

0 1

� �cosh1sinh1

� �exp½ihinð~rÞ�;

ð2Þ

where Uð~r; tÞ is the phase distribution introducedby the LC-SLM to one polarization component.Here, we use a Kerr-type model and assume thatthe phase Uð~r; tÞ is proportional to the intensitydistribution incident on the write side of the LC-SLM.

The output field is

~Eoutð~r;~tÞ ¼ r1E0 cos h1 exp½iðU þ hinÞ� expðiuoÞ¼ r1E0 cos h1 expfi½hð~r; tÞ þ �h�g expðiuoÞ;

ð3Þ

where r1 depends on the reflectivity at BS andother system losses.

We have defined

Uð~r; tÞ � uð~r; tÞ þ �h and

hð~r; tÞ � uð~r; tÞ þ hinð~rÞ; ð4Þ

where uð~r; tÞ is the spatially varying part of themodulation phase induced by LC-SLM while �hdenotes a spatially uniform phase offset inducedby the spatial average of the feedback intensity.

We are interested in developing an expressionfor the spectral ratio: T ð~qÞ � Hð~s;1Þ=Hinð~sÞwhere Hinð~sÞ is the spectrum of the input phasedistribution and Hð~s;1Þ is the spectrum of thesteady-state phase distribution after reflectionfrom the LC-SLM, i.e., the Fourier transformof hð~r; tÞ.

Assume that the spatially varying phase distor-tions are sufficiently small so that we can use thefirst order expansion so that exp(ih) @ 1 + ih andexp(ihin) @ 1 + ihin then in the linear approxima-tion the intensity at the write side of the LC-SLM can be written

j~Efbð~r; tÞj2 ¼ aA2ð0Þða2 þ b2 þ 2a � b cos �hÞ

� 2iaAð0ÞbZ

½b sinWþ a sinðWþ �hÞ

� Að~sÞHð~s; tÞ expði2p~r~sÞ d~s

� 2iaAð0ÞaZ

½b sinðW� �hÞ þ a sinW�

� Að~sÞHinð~sÞ expði2p~r~sÞ d~s; ð5Þ

where W � �pkL~s2 þ Pð~sÞ � P ð0Þ, ~s is spatial fre-quency, a ” sinh1sinh2, b ” cosh1cosh2 and a ”(t1E0)

2 is the intensity of the feedback beam beforethe polarizer in the feedback loop. This dependson the intensity of the input beam, E2

0, and lossesfrom beam splitters etc which are grouped togetherin a single term. For simplicity, we have assumedradial symmetric spatial filters, i.e., Að~sÞ ¼ Að�~sÞand Pð~sÞ ¼ P ð�~sÞ. A(0) and P(0) are the on-axis(dc) amplitude and phase of the spatial filters.

If the LC-SLM is modelled as a thin Kerr slicethe phase modulation introduced onto the polar-ization component corresponding to the directionof the LC molecule directors satisfies the equa-tion

soUð~r; tÞ

ot� D2r2

?Uð~r; tÞ þ Uð~r; tÞ ¼ KIwð~r; tÞ þ h0;

ð6Þ

where r2? � o2

ox2 þ o2

oy2, K is the nonlinear coeffi-cient, D the diffusion coefficient of the LC layer,and Iwð~r; tÞ the optical intensity incident on thewrite side plane of the LC-SLM and h0 is thephase offset introduced by the LC-SLM whenthe feedback intensity is zero. Combining (4)and (6) we obtain


so

otþ 1� D2r2

?

� �hð~r; tÞ � ð1� D2r2

?Þhinð~rÞ

¼ S1ð~r; tÞ ð7Þ

with

S1ð~r; tÞ � Kj~Efbð~r; tÞj2 � ð�h� h0Þ¼ KaA2ð0Þ½b2 þ a2 þ 2a � b cos �h� � ð�h� h0Þ

� 2iKaAð0ÞbZ

½b sinWþ a sinðWþ �hÞ�

� Að~sÞHð~s; tÞ expði2p~r~sÞ d~s� 2iKa2

� Að0ÞaZ

½b sinðW� �hÞ þ a sinW�

� Að~sÞHinð~s; tÞ expði2p~r~sÞ d~s ð8Þ

in the linear approximation.Converting hð~r; tÞ and hinð~rÞ into spatial spec-

tra Hð~s; tÞ and Hinð~sÞ in (6), when t! 1, oh/ot! 0, we can find the stationary solutionHð~s;1Þ and calculate the spectral ratio of theresidual and input phase distribution which isgiven by

0 2000 40000.5

1

1.5

2

s

T(s

)

0 2000 40001

2

3

4

5

s (a=0.5)

Qd(

s)

0 2000

0.5

1

1.5

2

s

0 2000

1

2

3

4

5

s (a=

Fig. 2. The phase distortion suppression coefficient of the

T ð~sÞ �Hð~s;1ÞHinð~sÞ

¼ Gð~sÞ� 2KaAð0ÞAð~sÞa½b sinðW� �hÞþ a sinW�Gð~sÞþ 2KaAð0ÞAð~sÞb½asinðW� �hÞþb sinW�

;

ð9Þwhere �h is the solution of

�h ¼ KaA2ð0Þða2 þ b2 þ 2a � b cos �hÞ þ h0 ð10Þand

Gð~sÞ � 1þ 4p2D2~s2. ð11ÞThe denominator of Eq. (9) is also important be-cause a zero or negative value for some~s indicatesof onset of pattern formation and is similar to theequation used to find pattern formation thresholdsin [8], but also includes the effects of spatial filter-ing in the feedback loop.

Eq. (10) is a transcendental equation relatingthe constant phase �h to the feedback amplitude,a, and the polarization angles h1 and h2. This phaseshift is a result of interference between the two

0 4000

0 4000

0.72)

0 2000 4000-2

-1

0

1

2

s

0 2000 4000-2

0

2

4

6

s (a=1.2)

system shown in Fig. 1 for the case of h1 = h2 = 0.


polarization components and has been discussed innumerous publications concerning feedback inter-ferometry [10] where diffraction is not present. Iteffects on pattern formation in diffractive feedbacksystems has been discussed in detail in [8]. In gen-eral, �h becomes multi-stable at high intensities (un-less either h1 or h2 are orientated zero or 90� to theLC molecule director), i.e., a given intensity canproduce more than one value of �h.

4. Behaviour for special cases

We first discuss the behaviour of the above sys-tem when the polarization of the feedback beam isaligned with the LC molecule directors (h2 = 0)and then when the feedback is from the polariza-tion component perpendicular to the directors(h2 = p/2). The first case corresponds to the dif-fractive feedback system discussed in previous pa-pers where polarization effects are unimportantand is, therefore, mentioned only very briefly here.

0 2000 40000

0.5

1

1.5

s

T(s

)

0 2000 40001

2

3

4

5

s (a=0.5)

Qd(

s)

0 200-0.5

0

0.5

1

1.5

2

s

0 2001

2

3

4

5

s (a=

Fig. 3. The phase distortion suppression coefficient of the

The second case corresponds to a system withoutfeedback, i.e., the beam controlling the phase mod-ulator is unaffected by the phase modulator. Inboth of these situations, there are no interferenceeffects thus the spatially uniform phase, �h, playsno part in the phase response.

4.1. Feedback with no polarization effects

For the first case, we set h1 = h2 = 0, so thata = 0, b = 1. In this case, the suppression coeffi-cient degenerates to

T ð~sÞ ¼ Gð~sÞGð~sÞ þ 2KaAð0ÞAð~sÞ sinW ; ð12Þ

which is the same as the expression of the spectralcoefficient for phase transformation defined in [9]where polarization was not taken into account.In Fig. 2, T(s) and the denominator of Eq. (12)are plotted (upper and lower plots, respectively)for feedback intensities a = 0.5, 0.72, 1.20 (corre-sponding to E0 = 1, 1.2, 1.55, respectively, when

0 4000

0 4000

0.72)

0 2000 4000-2

-1

0

1

2

s

0 2000 40001

2

3

4

5

s (a=1.62)

system shown in Fig. 1 for the case of h1 = h2 = p/2.


T = 0.707). We have used K = �1 and assumedthat there is no filtering in the feedback loop,i.e., Að~sÞ ¼ 1; P ð~sÞ ¼ 0. In these and all the otherresults presented in this paper typical values havebeen used for other parameters: D = 80 lm,k = 550 nm, L = 0.5 m and we have assumed thatthe focal length of the lenses in the Fourier lensblock were 0.2 m.

Suppression of phase distortions is obtained forspatial frequencies for which K sinW > 0 whileamplification occurs if K sinW < 0. For the param-eters used to find Fig. 2 the first suppression bandoccurs for spatial frequencies up to about1900 m�1 and it can be seen that the maximumsuppression obtained improves as the intensity ofthe feedback beam is increased. In the rightmostplots shown in Fig. 2 Qdð~sÞ becomes negative at~s � 2250 m�1 indicating the feedback intensityhas reached the pattern formation threshold. Thebest suppression obtained before the threshold is

Fig. 4. Numerical simulation showing the suppression of phase distorand h1 = h2 = p/2. (a) and (c) are the absolute value of input and outpoutput phase across the center row only; (c) is a the center row of thesolid is the phase spectrum predicted from the linear theory; (f) is the sthat from Eq. (13) (solid).

reached is about 0.4 (at ~s � 1500 m�1). To im-prove the suppression further spatial filtering isrequired (plus an increase in the intensity of thefeedback beam).

Examining Eq. (12) shows that in order to ob-tain a suppression close to zero at some~s, the feed-back intensity, a, must become large so thatGð~sÞ � j2KaAð0ÞAð~sÞ sinWj. Effectively, perfectsuppression (over the first suppression band) isonly obtained when the intensity of the feedbackbeam approaches infinity.

4.2. Direct control of the LC-SLM with thedistorted beam

Next, we consider the behaviour when the beamcontrolling the LC-SLM is not influenced by thephase modulator itself, i.e., only the polarizationcomponent that is unaffected by the LC-SLM isincident on the write side of the LC-SLM. When

tions for a = 0.78 (corresponding to E0 = 1.25 when T = 0.707)ut phase, respectively; (b) and (d) are, respectively, the input andinput and output phase spectrum (dashed and dotted) while theuppression obtained from the numerical simulation (dotted) and


h1 = h2 = 90�, so that a = 1, b = 0, Eq. (9)becomes

T ð~sÞ ¼ GðsÞ � 2KaAð0ÞAð~sÞ sinWGðsÞ . ð13Þ

Note that because this is not a feedback system it isalways stable thus spontaneous pattern formationetc is not expected.

Examining Eq. (13) shows that, unlike the pre-vious case, T ð~sÞ may be zero for some ð~sÞ at finitefeedback beam amplitudes, i.e., ideal correctioncan be obtained for some spatial frequencies.Moreover, T ð~sÞ may become negative indicatingphase conjugation at some spatial frequencies.This is reflected in Fig. 3 which shows T ð~sÞ forfeedback intensities of a = 0.5, 0.72 and 1.62.Again we have assumed that no filtering is donein the feedback loop and that K = �1. The resultsobtained for a = 0.5 are similar to the results ob-tained at the pattern formation threshold in thefeedback system in Section 4.1. At a = 0.72 a sup-pression coefficient of zero is obtained at somespatial frequencies. However, the bandwidth over

Fig. 5. Numerical simulation showing the system response at a = 1.6

which good suppression is obtained is not particu-larly large.

When the feedback intensity, a, is further in-creased to 1.62 it is seen that the sign of the phaseis changed for some spatial frequencies indicatingthat phase conjugation may be possible.

Figs. 4 and 5 show the results of simulationsperformed by numerically solving the nonlineardynamic Eq. (6) [11]. The input phase distributionhinð~rÞ was a realization of 2D zero-mean randomfunction with uniform distribution. This was pre-filtered by a low-pass spatial filter so that the spec-trum of the input phase distribution is reduced tozero outside the first suppression band. The feed-back field Efbð~r; tÞ was calculated numericallyusing Fresnel diffraction theory and a 2D FFTprocedure. In the simulations presented here resultswere obtained over a 128 · 128 grid.

In Fig. 4, the input amplitude is a = 0.78, andall the other parameters are the same as those usedabove. Plots (a) and (d) show images representingthe absolute value of the phase distribution of theinput beam and output beam, respectively. Plots

2 and h1 = h2 = p/2. Plots (a)–(f) are similar to those in Fig. 4.


(b) and (e) show a slice of the phases along the cen-ter horizontal line. Clearly significant correction isobtained. Plot (c) show a line of the amplitude theinput phase spectrum Hinð~sÞ (dashed-dotted), andthe corresponding line of the corrected outputphase Hð~s; tÞ (solid). To compare with the outputphase predicted by the linear theory T ð~sÞ �Hinð~sÞis also shown (dotted line). Finally, in plot (f) thesuppression coefficient T ð~sÞ calculated using Eq.(13) is given (solid) together with the ratio foundfrom one line of the numerically simulated spectra,Hð~s; tÞ=Hinð~s; tÞ. Plots (c) and (f) show that the lin-ear theory accurately models the system responsefor small scale phase distortions. These simula-tions were repeated with different realizations ofthe input phase and similar results were alwaysobtained.

Other simulations were done using a = 1.62 andthe results are shown in Fig. 5. Here, the inputphase distribution is the same as that used forFig. 5 and again the phase spectrum is accurately

0 1000 20000

1

2

30

0 10000

1

2

3π/4

0 1000 20000

1

2

33π/4

0 10000

1

2

3π

0 1000 20000

1

2

33π/2

0 10000

1

2

37π/4

Fig. 6. Phase transfer function, T(s) (solid) and stability curve (dottshown in the top left of each plot).

modelled by the linear theory. Although phaseconjugation occurs for some spatial frequenciesthe band over which this occurs is rather limited.We note that it may be possible to improve thebandwidth over which good phase conjugation isachieved by placing an appropriate spatial filterin the feedback loop.

5. Other polarization angles

When the angle of the polarization analyzer isnot zero or p/2 with respect to the LC moleculedirectors the interference between the modulatedand non-modulated polarization components be-comes important. Here, we discuss the caseh1 = h2 = p/2 where the contrast of the interfer-ence fringes is largest and �h affects the behaviourmost strongly. �h itself is a complicated functionof both the feedback intensity and of the offsetphase h0 and above a critical intensity can exhibit

2000 0 1000 20000

1

2

3π/2

2000 0 1000 20000

1

2

35π/4

2000 0 1000 20000

1

2

32π

ed) for a = 1.3, h1 = h2 = p/4, for various phase shifts �h (value


multi-stability and hysteresis. In the following,though, we only consider cases for which the feed-back intensity is below the threshold for multi-stability and we note that �h may take on any valueby adjusting the offset phase. In practice, this couldbe done fairly easily by introducing a variableretarder into the feedback loop such as a Soleil–Babinet compensator.

To illustrate the effects of �h on the phase trans-fer function, T(s) (Eq. (10)) is plotted using feed-back amplitudes of a = 1.3 and a = 2 in Figs. 6and 7, respectively. The stability curves are alsoplotted using dashed lines. The curves are onlyplotted over a spatial frequency range that corre-sponds (approximately) to the first band for whichT(s) < 1. Similarly, to the cases described above,there are other higher spatial frequency bandswhere the phase is also amplified or suppressed.Also shown using dotted points in Figs. 7 and 8are the phase suppression spectra obtained fromnumerically solving the nonlinear dynamic Eq.(6) using a small amplitude, random input phase

0 1000 2000

0

1

2

30

0 1000

0

1

2

3π/4

0 1000 2000

0

1

2

33π/4

0 1000

0

1

2

3π

0 1000 2000

0

1

2

33π/2

0 1000

0

1

2

37π/4

Fig. 7. Phase transfer function, T(s) (solid) and stability curve (dotted)in the top left of each plot).

distribution. These simulations give excellentagreement with the linear theory.

When a = 1.3 the system is well below the pat-tern formation threshold for all �h and whena = 2 the system is on the verge of becomingunstable (this first occurs at s = 0 for �h ¼ p=2).For both cases, it can be seen that the shape ofthe phase transfer function varies significantly with�h, i.e., the minima and the spatial frequency atwhich the minima occur are strongly dependanton �h. When �h is near p the system has almost noeffect on the phase distribution. Physically, thiscorresponds to destructive interference betweenthe two orthogonal polarization components thusthe amplitude of the feedback beam is nearly zero.At other values of �h, it is clear that good suppres-sion of phase distortions is obtained well beforethe threshold for pattern formation pattern isreached. Unlike the case described in Section 4.1where the polarizers were aligned parallel to thecomponent modulated by the LC layer, near zerosuppression is obtained at a relatively small

2000 0 1000 2000

0

1

2

3π/2

2000 0 1000 2000

0

1

2

35π/4

2000 0 1000 2000

0

1

2

32π

for a = 2, h1 = h2 = p/4, for various phase shifts �h (value shown

Fig. 8. Numerical simulations showing suppression of phase distortions with a = 2, h1 = h2=45� and �h ¼ 1.6p. Absolute value of theinput (a) and output (b) phase distribution. (c) comparison of T(s): solid curve obtained from Eq. (9), dots from the numericalsimulation. Plots (d)–(f) are similar to (a)–(c), respectively, but obtained using a phase distribution five times as large.


feedback amplitude and before the pattern forma-tion threshold is reached. Moreover, the band-width over which good suppression is obtained issignificantly larger than that obtained for the situ-ation discussed in Section 4.2.

To illustrate the phase suppression capabilitymore clearly simulations were carried out using asmall random phase distribution for the input fieldand the output phase distribution was calculatednumerically using the full nonlinear Eq. (6) as be-fore. The results are shown in the top plots ofFig. 8 where a = 2, h1 = h2 = p/4 rad and �h ¼ 0.Clearly, the suppression obtained is very goodand the corrected wavefront is very close to a planewave. Also, the linear theory closely matches thatobtained from the nonlinear equation.

Also shown in bottom plots of Fig. 8 are the re-sults for the same random phase distribution butwith the amplitude increased so that the maximumamplitude of the distortions are between ±1 rad.In this case, it can be seen by examining plots 8cand 8f that the agreement between the results pre-

dicted by the theory for small phase distortionsand that using the full nonlinear theory is not asgood. Nevertheless, the corrected wavefront ismuch closer to a plain wave than the original.

It is important to note that the systems dis-cussed here are not capable of correcting verylow spatial frequency aberrations. However, itmay be possible to use this system in conjunctionwith a conventional low resolution adaptive opticssystem thus allowing both low and high-frequencyaberrations to be removed from a wavefront.

6. Conclusion

The effects of polarization on a LC-SLM opti-cal feedback system on wavefront shape haveinvestigated and a general expression for the phasesuppression coefficient was developed. It wasshown that by choosing the polarization of theinput beam correctly the suppression obtained isimproved when compared to a system which no


polarization modulation. Good suppression wasobtain at relatively low feedback intensities, unlikethe situation when the polarization of the inputbeam is polarized in the same direction as thatmodulated by the LC-SLM. Moreover, good cor-rection was obtained well below the threshold forspontaneous pattern formation.

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effects of polarization on wavefront shape in a lc-slm diffractive feedback system

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