real-time digital holographic beam-shaping system with a...

Real-time digital holographic beam-shapingsystem with a genetic feedback tuning loop

Joonku Hahn, Hwi Kim, Kyongsik Choi, and Byoungho Lee

A novel implementation of a real-time digital holographic system with a genetic feedback tuning loop isproposed. The proposed genetic feedback tuning loop is effective in encoding optimal phase holograms ona liquid-crystal spatial light modulator in the system. Optimal calibration of the liquid-crystal spatiallight modulator can be achieved via the genetic feedback tuning loop, and the optimal phase hologram canthen overcome the aberration of the internal optics of the system. © 2006 Optical Society of America

OCIS codes: 050.1970, 070.4560, 090.1000, 090.1970.

1. Introduction

In digital holography, designing phase hologramsthat produce the desired diffraction images is one ofthe most important engineering goals. The designand fabrication of phase holograms or holographic(diffractive) optical elements have been intensivelyinvestigated over the past two decades.1–3 Real-timedynamic digital holography using a liquid-crystalspatial light modulator (LCSLM) has recently be-come more attractive than surface-relief fixed-typeholographic (diffractive) optical elements. Real-timedynamic digital holographic beam-shaping technolo-gies are useful in a wide range of applications, includ-ing telecommunications, displays, optical storage,and optical information processing, and they are cur-rently being extensively studied.4–6

In general, the design and implementation of adigital holographic beam-shaping system is as fol-lows. Continuous phase holograms are designedwith the aid of design algorithms such as the iter-ative Fourier-transform algorithm (IFTA).7,8 Thequantized phase hologram is next encoded on theLCSLM. An incident beam on the LCSLM is dif-fracted (transformed) through the internal optics ofthe beam-shaping system and generates images inthe image plane. Modulating the phase of the inci-dent beam so as to have the designed value for eachpixel of the LCSLM is important for achieving effi-

cient and accurate beam shaping.9 In the phase-hologram design stage, the ideal mathematicalmodel for an optical system is assumed. In general,the internal optical system of the beam-shapingsystem is mathematically modeled based on the lin-ear canonical transform (LCT).8 However, in prac-tical systems, several physical factors, such asphase- and amplitude-modulation errors induced ineach cell of the LCSLM, misalignment of the opticalcomponents, and aberrations, may induce a differ-ence between ideal simulation results and experi-mental results obtained in a real system. In termsof implementing a practical system, analysis of theerrors caused by these physical factors and relevantfine-tuning are important for obtaining optimal dif-fraction images. In general, this fine-tuning is per-ceived as somewhat intricate since the origins of theerrors are various, the theoretical identification ofthe errors for optimal compensation is not easy, andthe optimization of many system parameters re-quires a novel system and may complicate the finalconfiguration of the system. In addition, the fine-tuning problem cannot be addressed at the designstage but is applied at the system-implementationstage. The fine-tuning technique should be robust,reliable, and reproducible.

In this paper the novel implementation of a real-time digital holographic beam-shaping system with aLCSLM, having a specially devised fine-tuning pro-cess referred to as the genetic feedback tuning loop, isproposed. The genetic feedback tuning loop is basedon a well-known genetic algorithm.7,10 The resultsshow that the proposed genetic feedback tuning loopis effective in encoding optimal phase holograms forovercoming several physical factors that degrade theperformance of a practical system.

The authors are with the School of Electrical Engineering, SeoulNational University, Kwanak-Gu Shinlim-Dong, Seoul 151-744,South Korea. B. Lee’s e-mail address is [email protected].

Received 11 May 2005; accepted 12 July 2005.0003-6935/06/050915-10$15.00/0© 2006 Optical Society of America

10 February 2006 � Vol. 45, No. 5 � APPLIED OPTICS 915

This paper is organized as follows. In Section 2 thecauses of the degradation of diffraction images inpractical digital holographic beam-shaping systemsare discussed. The objects to be treated for optimallytuning the system are identified and formulated. InSection 3 the proposed genetic fine-tuning loop is de-scribed. In Section 4 the real-time holographic beam-shaping system with the genetic feedback tuning loopis experimentally realized. In Section 5 concludingremarks and perspectives for the devised system areprovided.

2. Causes Degrading Diffraction Images in PracticalDigital Holographic Beam-Shaping Systems

In this section the causes behind the degradation ofdiffraction images in a practical digital holographicbeam-shaping system with a LCSLM are identifiedand analyzed. Figure 1 shows a schematic diagram ofa practical LCSLM holographic beam-shaping sys-tem, which consists of a laser, a beam expander, aLCSLM, two polarizers, and a Fourier lens with focallength f. The phase-modulation range (including themaximum phase modulation of the 2� phase shift)can be realized in the LCSLM by using appropriatepolarizers. The incident optical wave impinges on theback side of the LCSLM and then passes through theLCSLM with its phase modified. The modulated op-tical wave is transformed by the optical system toform a diffraction image in the image plane. In thisconfiguration the two main causes for the degrada-tion of the diffraction image are incorrect SLM cali-bration and aberrations present in the internal opticsof the beam-shaping system. Accordingly, in this pa-per we focus on the appropriate treatment for SLMcalibration and the aberration compensation of thebeam-shaping system.

First, we discuss SLM calibration. In the designstage a continuous phase hologram is obtained bythe IFTA. In practice, the possible phase-modulation levels at the pixels of the SLM arequantized, and the relationship between the phase-modulation level and the quantization number istypically nonlinear, which is referred to as thephase-modulation table of the SLM. The SLM with8 bit resolution controls the phase-modulation levelat each pixel by an integer called the encoding in-dex, which ranges from 0 to 255. The SLM should becalibrated by finding the relationship between thephase-modulation level and the encoding index, i.e.,a phase-modulation table. In addition, accuratecharacterization of the transmittance of the phase-modulation devices is important in the field of dif-fractive optics. The nonlinear transmittance modelshould be taken into account for fabricated diffrac-tive optical elements as well as for the LCSLM.11

Although the accurate modeling of the transmit-tance of a practical SLM is more complicated,12–15

the phase-modulation characteristics of the SLMcan be easily characterized by using experimentalapproaches. The first procedure of tuning is finding

the nonlinear phase-modulation table. This type ofcalibration is typically accomplished by interfero-metric methods. However, in our system the cali-bration is considered to be an optimization problem.It will be shown that this approach is also effectivefor SLM calibration. For the optimization scheme,the cost function to be maximized and unknownvariables to be optimized need to be identified. Inthis case of the SLM with 256 encoding indices, thenumber of unknown variables is 254, since the firstlevel and the last level are fixed as the phase mod-ulation of 0 and 2�, respectively. The phase-modulation level of a pixel on the SLM is controlledby a signal designed with an integer number. Whenthe phase modulation for an input integer n is de-noted by pn, the phase-modulation levels pns are theunknown variables to be found. In this paper we as-sume a monotonic relationship between the inputinteger and the phase-modulation level. Hence thecost function F is parameterized by 254 unknownvariables pn and the optimization problem is definedas

max F�p1, p2, p3, . . . , p254�

for 0 � p1 � p2 � · · · � p254 � 2�, (1)

the definition of which is completely described in Sec-tion 3.

We next discuss the aberration of the beam-shaping system and its compensation. Let the dis-tance from the hologram plane to the lens and thatfrom the lens to the image plane be d1 and d2, respec-tively, as shown in Fig. 1. Let the wavelength of theoptical wave be �. The transform of the optical systemFr� � is represented as the LCT or the generalizedFresnel transform8

F�x2, y2� � Fr�G�x1, y1��

��

� ��

�

h�x2, y2, x1, y1�G�x1, y1�dx1dy1,

(2)

where �x1, y1� and �x2, y2� denote the coordinates of apoint in the SLM plane and that in the image plane,respectively, and h�x2, y2, x1, y1� is the propagator ofthe Fresnel transform and takes the form

Fig. 1. Schematic of the digital holographic beam-shaping sys-tem.

916 APPLIED OPTICS � Vol. 45, No. 5 � 10 February 2006

h�x2, y2, x1, y1� ��j

��d1 � d2� ��d1d2

f �� exp� j�

��d1 � d2� ��d1d2

f

�1 �d1

f ��x22 � y2

2� � 2�x2x1

� y2y1� � 1 �d2

f ��x12 � y1

2�� .

(3)

The optical field that just passed through the SLMwith a phase hologram encoded is denoted byG�x1, y1�. In the phase-only modulation mode of theSLM, G�x1, y1� is represented as

G�x1, y1� � A exp�j�x1, y1��, (4)

where �x1, y1� is the phase hologram encoded on theSLM and A indicates a constant amplitude. In anaberration-free situation, the phase hologram de-signed by the IFTA may be directly used as the finalphase hologram to be encoded on the SLM. However,defocusing, misalignment of the optical elements,and aberrations cause the real transform to deviatefrom the ideal model of Eq. (3). These factors aresystem dependent.

The aberration can be theoretically analyzed basedon diffraction theory.16 The most general representa-tion of the aberration is based on Zernike polynomi-als. Let the aberration of the optical system berepresented by �x1, y1� and the mathematicalexpression of the transform of the real system read as

hreal�x2, y2, x1, y1� � h�x2, y2, x1, y1�exp�j�x1, y1��.(5)

The compensated optical field G��x1, y1� should takethe following form:

G��x1, y1� � A exp�j�x1, y1��exp��j�x1, y1��. (6)

The addition of a conjugate phase termexp��j�x1, y1�� for system aberration to the designedphase hologram will compensate for the distortion ofthe diffraction image induced by the aberration. Theaberration component of the optical system �x1, y1�is given by the weighted sum of Zernike polynomialsas

�x1, y1� � ��, ��

� A00 �1

�2�n�2

�

An0Rn0��

n�1

�

�m�1

n

AnmRnm��

� cos m� � �n�1

�

�m�1

n

Anm�Rnm��sin m�, (7)

where Anm and Anm� denote the coefficients of termshaving an x-axis symmetry and a y-axis symmetry,respectively, and Rn

m�� denotes the circular Zernikepolynomial of the order �m, n�. And � and � are, re-spectively, given by

� � ��x � xc�2 � �y � yc�2�1�2, (8)

� � tan�1y � yc

x � xc�, (9)

where �xc, yc� indicates the center of the Zernike poly-nomials. The second procedure for tuning is compen-sating for the internal aberration of the optics of thebeam-shaping system. In this case the unknown vari-ables to be optimized are coefficients and the centerposition of the Zernike polynomials. The cost functionF is then parameterized by the coefficients of theZernike polynomials and the center position. The op-timization problem is expressed as

max F�Anm, Anm�, xc, yc�.

The truncation order of the Zernike polynomial isdetermined by considering the sampling resolution ofthe SLM and the computation cost.

In this paper a powerful and general optimizationtechnique, a genetic algorithm, is applied to find thenonlinear phase-modulation table of the SLM and theaberration-compensated phase holograms to be en-coded on the SLM of the beam-shaping system. Sincethe inverse transform of the real optical system fromthe CCD plane to the SLM domain does not exist, anefficient and effective genetic optimization algorithmthat uses only the forward transform from the SLMdomain to the CCD plane is employed. The proposedfine-tuning technique of the digital holographicbeam-shaping system evaluates the qualities of thediffraction images on the CCD plane and feeds thisinformation back to the holograms on the SLM do-main with the genetic algorithm. The genetic algo-rithm enables all system parameters, i.e., nonlinearphase-modulation table and coefficients of Zernike

Fig. 2. Schematic of a real-time digital holographic beam-shapingsystem with a genetic feedback tuning loop.


polynomials for aberration compensation, to be auto-matically fine-tuned.

In the following sections, detailed methods related tothe two procedures of tuning are described. The exper-imental implementation of the real-time digital ho-lographic beam-shaping system with the genetic feed-back tuning loop is developed and the feasibility offinely tuning the beam-shaping system and finding theoptimal phase holograms is described. The nonlinearphase-modulation table of the SLM is extracted andapplied to the SLM to obtain an enhanced diffractionimage. The aberration of the internal optics of the dif-fractive beam-shaping system is compensated by im-proving the phase hologram through the geneticfeedback tuning loop. The resulting improved diffrac-tive images generated by the tuned beam-shaping sys-tem are presented.

3. Genetic Feedback Tuning Loop

In this section the proposed genetic feedback tuningloop for fine-tuning the real system is described. Figure2 shows a schematic diagram of the devised systemwith the genetic feedback tuning loop. As shown in Fig.2, the feedback loop is constructed between the CCDand the SLM through a computer. The CCD capturesthe energy distribution of the diffracted field at theimage plane. The captured image is transported to acomputer for evaluating the quality of the diffractionimage, which is part of the genetic optimization of thesystem parameters identified in Section 2. The systemparameters modified by the computer are sent to theSLM, which then modifies the diffraction image cap-tured by the CCD. The evaluation process in the com-puter is repeated. The genetic algorithm implementedin the computer gradually enhances the diffraction im-age by tuning the system parameters. This iterativefeedback loop is referred to as the genetic feedbacktuning loop.

In general, the genetic algorithm searches localoptima through two distinguished operations, cross-over and mutation. Conceptually, the crossover oper-ation is effective in convex optimization problems,since the crossover operation is mathematically anal-ogous to the linear combination of two chromosomesin a floating-point coding scheme. The mutation issimply a random search, making the small variationsin the present solution. If the present solution is nota local optimum, small variations will lead to an im-provement in the present solution. The mutation op-eration is more general in the genetic algorithm.

In our system two factors, i.e., the nonlinear phase-modulation relationship of the SLM and the dynamicaberration compensation, are focused on. It is ambig-uous that the crossover operation is effective, sincethe related optimization problem may not be convex.Thus, for convenience and low computation cost, onlythe mutation operation is employed in our applica-tion. Hence a somewhat simplified genetic algorithmscheme is devised for use in our system.

Figure 3 shows a flow chart of the simplified ge-netic algorithm implemented in the genetic feedbackloop shown in Fig. 2. It should be noted that the

expression in the flow chart corresponds to the case offinding the phase-modulation table of the SLM. First,the initial population P0 of a population size m isselected. The population is composed of subset xi,which has 254 elements in accordance with the en-coding indices. The unknown variable Pi, n, that is, theelement of xi, means the relative phase delay of the�n � 1�th encoding index. The cost values F�P0� ofchromosomes in the initial population are next cal-culated, and, the fittest chromosome with the largestcost value, i.e., the elite x�, is determined [see problem(1)]. The cost function F is the weighted linear com-bination of the standard evaluation factors of theobtained diffraction image, the definition of which iscompletely described in Section 4. The mutation op-eration is next applied to the initial population P0,and the modified population P0

�� is then obtained. Forthe mutation, an adaptive mutation operator forfloating-point coding is implemented, in which themutation probability and the system parameter fordetermining the degree of nonuniformity are tuned to�0.03 and 20 (see Chaps. 5 and 6 of Ref. 10). Thesystem parameter, which indicates the degree of non-uniformity, effectively controls the convergence speedof the genetic algorithm. It should be noted that theevaluation of each chromosome requires a completesignal flow along the path of encoding the phase ho-logram on the SLM, capturing the resulting diffrac-tion image by the CCD, and evaluating the cost valueof the chromosome. Thus, to achieve one generationof the genetic algorithm, several sequential repeti-tions of encoding and capturing should be made.

Fig. 3. Flow chart for the simplified genetic algorithm imple-mented in the genetic feedback tuning loop.


In the case of compensating for the internal aber-ration of the optics of the beam-shaping system, thecoefficients of the Zernike polynomials Anm, Anm� andthe center position �xc, yc� are substituted into Pi, n inthe flow chart shown in Fig. 3. Consequently, thenumber of the unknown variables is also changed.

4. Experimental Realization of a Digital HolographicBeam-Shaping System with a Genetic FeedbackTuning Loop

In this section an experimental realization of thedigital holographic beam-shaping system with thegenetic feedback tuning loop is presented and someexperimental results are shown. In Subsection 4.Athe structure of the cost function for evaluating theobtained diffraction images is devised. In Subsec-tion 4.B the phase modulation and amplitude trans-mission characteristics of the SLM used areinspected by means of an interferometric experi-ment, and these data are compared with data ob-tained using the genetic feedback loop. InSubsection 4.C the experimental evidence of theaberration compensation attained by the proposedmethod is given.

A. Cost Function for Evaluating Diffraction Images

In this paper the desired diffraction image to begenerated in the image plane, namely, the targetimage, is a simple, conventional binary image, thebright region of which is termed the signal regionand the region outside the signal region is called thenoise region. The binary target image is used toevaluate real diffraction images that have continu-

ous brightness. The brightness of diffraction imagescaptured by the CCD with an 8 bit resolution isquantized into 256, and this quantized brightnesslevel is referred to as a gray level.

The cost function is devised to minimize the devi-ation of a diffraction image from the target image andthe length of the boundary, which is the number ofedge pixels across the threshold in a diffraction im-age. We evaluate a diffraction image in a local processin which the partial cost of each pixel contributeslinearly to the cost of a diffraction image. This processrequires a low computation cost and is effective intuning our system with the genetic feedback tuningloop.

The deviation of a diffraction image is separatedinto the intensity of deviation from the target imagein the signal region and the variation in intensity inthe noise region. Minimizing the cost function mustincrease the transmission efficiency (brightness) inthe signal area and decrease the mean-square errorbetween the target image and the diffraction image.Let the gray level at position �x2, y2� in the real dif-fraction image captured by the CCD be denoted byGL�x2, y2�, then the normalized mean-square error inthe signal region is defined as

Mean-square error �

�SGL�x2, y2� � 255

255 �2

�S

1, (10)

where �s denotes the summation in the signal regionand the denominator �s 1 indicates the area of thesignal region. In experiments some bright spots oftenappear in a captured image and it is impossible toreduce the average of the gray levels in the noiseregion to zero. These bright spots increase the statis-tical variation value of the intensity distribution inthe noise region. The variation in the image reflectsthe quality of the image better than the deviationfrom zero in the noise region. The normalized varia-tion in the noise region is defined as

Fig. 4. Schematic of the interferometer for measuring the phasemodulation and amplitude transmission of the LCSLM

Fig. 5. Characteristics of the LCSLM measured by interferomet-ric methods: (a) phase modulation versus encoding index and (b)amplitude transmission versus encoding index.

Fig. 6. Comparison of the phase-modulation tables obtained atthree different stages of generation (1st, 43rd, and 299th) in SLMcalibration.


Normalized variation �

�NGL�x2, y2� � GLN

�

GLN� �2

�N

1,

(11)

GLN��

NGL�x2, y2��

N1, (12)

where �N denotes the summation in the noise regionand GLN� indicates the average of the gray levels in

the noise region. A decrease in the normalized vari-

ation will restrain the appearance of bright spots inthe noise region of a diffraction image. On the otherhand, there is an intricate problem associated withthe experiment. The measured system is sensitive toexternal noise, such as vibration, fluctuations in thelight source, and scattering. However, a stable sys-tem is essential to the operation of the genetic feedbackloop. Therefore, to achieve stability, the deviationvalue is compensated by a stabilization factor so thatthe deviation in the diffraction image with the samephase hologram and system parameters remains con-stant with time. For this, we use the initially selectedreference phase hologram and its deviation of a dif-fraction image as the reference value. In practice, thereference phase hologram does not produce the samedeviation value, and a stabilization factor is defined asthe ratio of a temporal deviation value to the referencevalue. We obtain a stabilization factor in every gener-ation and correct the deviation value by multiplying bya stabilization factor.

The boundary length in a diffraction image changesextensively when the hologram on the SLM is modifiedto compensate for the aberration of the internal optics

of the system. Since the internal aberration deforms adiffraction image, the boundary length increases gen-erally. Therefore, in practice, the boundary length de-creases as the aberration is compensated. On the otherhand, the boundary length is stationary as the phase-modulation table of the SLM changes. Therefore theboundary length must be considered to evaluate the qual-ity of the image only when the system is tuned for aber-ration compensation. The boundary length is defined as

Boundary length � �S�N

edge�x2, y2�, (13)

where �x2�, y2�� is the point adjacent to �x2, y2� in adiffraction image captured by the CCD and thethreshold is a fixed value (60 in our system).

The cost function for evaluating the diffraction im-age is given by the weighted sum of the three evalu-ation Eqs. (10), (11), and (13). The appropriateweights are necessary for the correct and stable con-vergence of the genetic algorithm and are selectedfrom experience. In our system we set the weight ofthe normalized variation as �1, the mean-square er-

edge�x2, y2� � �1 if GL�x2, y2� threshhold and GL�x2�, y2�� threshhold0 otherwise , (14)

Fig. 7. Diffraction images observed during the calibration of the SLM with the genetic feedback tuning loop at (a) the 1st generation stageand (b) the 299th generation stage (the stagnated state).

Table 1. Evaluation Parameters of Diffraction Images inSpatial Light Modulator Calibration

Generation 1st Generation 299th Generation

GLS�a 114.1 158.5GLN� 10.7 11.4Mean-square error 0.3806 0.2408Normalized variation 0.5742 1.198Stabilization factor 0.9896 1.006Cost value �38.23 �25.41

aGLS� denotes the average of the gray levels in the signal region.


ror as �100, and that of the boundary length as 0 forcalibrating the SLM or �0.0132 for the aberrationcompensation. Since the genetic algorithm is pro-grammed to find the maximum, the cost value isdefined as a negative value of the form.

B. Calibration of the Liquid-Crystal SpatialLight Modulator

Here we discuss in detail the calibration of the SLMused in our system. To calibrate the SLM, we obtainthe phase-modulation tables of the SLM by two meth-ods. The first is a classical interferometric experi-ment, and the second is the genetic feedback tuningloop. A comparison between the results for the twomethods is also presented in this section.

In this paper a twisted-nematic (TN) LC display(Sony LCX016AL-6 with 8 bit resolution) is used asthe SLM. With the SLM control program (HoloeyeLC2002), we set the contrast and brightness param-eters to 198 and 102, respectively, which are the pa-rameters for adjusting the range and mean value ofthe voltage impressed on the LC cell. A laser (Coher-ent DPSS Nd:YAG) with a wavelength of 532 nm isused as the light source, and the diffraction image iscaptured by a CCD (Kodak MegaPlus ES1.0�MV with8 bit resolution). The optical transmission pow-er is measured with an optical powermeter (Newport1835-C). Figure 4 shows a schematic diagram of theinterferometer used to measure the phase modula-tion and the amplitude transmission of the SLM,which is similar to the well-known Mach–Zehnderinterferometer. When the SLM is operated in thephase-modulation mode, two linear polarizers areplaced before and behind the SLM. The latter polar-izer is conventionally referred to as an analyzer. Therotation angle of the polarization is defined as thepositive angle between the axis of the polarizer andthe director axis of the TN LCSLM at the input face.In our system the rotation angles of the polarizer andanalyzer were selected as �p � 330° and �A � 10°,respectively. The fringes of the interference patternare captured by the CCD while varying the encoding

indices of the SLM. The phase shift can be calculatedfrom the movement of the fringe, the method forwhich follows that described in Ref. 15:

Phase shift�radiation� �2� � Shift of fringe

Period of fringe . (15)

On the other hand, the beam splitter placed behindthe analyzer divides the light into two separate arms,and the amplitude transmission (i.e., transmission ef-ficiency) can be measured by using an optical power-meter at the same time as the phase modulation ismeasured.

The measured phase-modulation and amplitude-transmission tables are plotted in Fig. 5. It can beseen that the phase modulation and amplitude trans-mission are nonlinear according to the encoding in-dex. The results obtained satisfy the generalproperties of a TN LCSLM. Phase modulation in-creases monotonically from 0 to 2�. It increases rel-atively rapidly around the 190th encoding index,where the amplitude transmission is at a minimum.Generally, in the system using a TN LCSLM with acombination of two linear polarizers, when the phaseof an incident optical wave is modulated, the trans-mission of the wave also changes, since amplitudetransmission is somewhat sensitive to polarizer set-tings. In this paper, because the phase modulation ofthe SLM is our main concern, the angles of the po-larizer and analyzer were set so that the phase mod-ulation would cover the full range from 0 (rad) to2� �rad�. However, it should be noted that our settingof the SLM and polarizers is not sufficient to attain aconstant amplitude transmission and that the mea-sured amplitude-transmission characteristic is anonflat curve as shown in Fig. 5(b).

Next, using the implemented beam-shaping sys-tem with the genetic feedback tuning loop, we inspectSLM calibration. The experimental setup is shown inFig. 2. The circular aperture with a 400 pixel diam-eter is defined, and the phase hologram calculatedwith the IFTA is encoded on the aperture area on the

Fig. 8. Phase holograms and aberration compensation: (a) phase holograms without compensation at the 1st generation stage, (b)aberration compensation obtained by the genetic feedback tuning loop, and (c) phase holograms with compensation at the 553rd generationstage (stagnated state).


SLM. The target image is a 400 � 400 pixel imagewith a T-shaped signal region.

In this experiment an initial population with fivechromosomes, P0 � �xi|i � 1, 2 . . . 5�, is created, andthe individual chromosome is given as

xi ��pi, n �n

255 � 2�|n � 1, 2 . . . , 254 . (16)

The initial population has the same five chromo-somes representing the linear phase-modulation ta-ble according to the encoding index. The mutationoperator is given by

xi� ��pi, n � ��t, 2� � pi, n� for rbinary � 0pi, n � ��t, pi, n � 0� for rbinary � 1, (17a)

where ��t, y� is defined as

��t, y� � y�1 � r�1�t�T�b�, (17b)

where rbinary is a randomly generated binary number,r is a randomly generated real number within (0, 1),t is the ordinal number of the present generation, andT is the ordinal number of the final generation in thegenetic algorithm. As seen in Eqs. (17a) and (17b), theupper and lower limits of the variables are 2� and 0,respectively. The mutation probability is set to 0.03,and the mutation parameter b, determining the de-gree of nonuniformity, is tuned to 20 (see Chaps. 5and 6 of Ref. 10).

The evolution stagnates at the 299th generation,although the maximum number of generations is1000. The phase-modulation tables obtained atthree different generation stages, the 1st, 43rd, and299th, are compared in Fig. 6. The phase-modulation curve proceeds to concave downwardforms, but the evolution stagnates at a state wherethe phase-modulation table obtained by the geneticfeedback tuning loop does not coincide with thatobtained in the interferometric experiment, sincethe SLM in our system has nonflat amplitude-transmission characteristics. The diffraction imagegenerated by the hologram encoded according to thephase-modulation table of the SLM obtained in theinterferometric experiment has a lower cost valuethan that obtained by the genetic feedback tuningloop. Figures 7(a) and 7(b) show the diffraction im-age observed at the 1st generation stage and that atthe 299th generation stage (the stagnated state).The mean-square errors of these two cases are de-termined to be 0.3806 and 0.2408, respectively. Thisindicates that the latter stage has a higher trans-mission efficiency than the former, and the averageintensities in the signal region also reflect this fact.On the other hand, the normalized variation in-creases from 0.5742 to 1.198. However, since theweight of the mean-square error is set 100 timeslarger than that of the normalized variation, thecost value is consequently improved from �38.23 to

Fig. 9. Diffraction images observed during compensating for aberration with the genetic feedback tuning loop: (a) at the 1st generationstage and (b) at the 553rd generation stage (the stagnated state).

Table 2. Results of Aberration Compensation: Coefficients of theZernike-Polynomials

Coefficients Values

A00 0.241605A02 �0.170412A04 0.249949A11 �0.215106A13 �0.205430A22 �0.059679A24 0.237322A33 0.230890A11= �0.245197A13= 0.245686A22= 0.248907A24= �0.079823A33= �0.044566

Table 3. Results of Aberration Compensation: Center Position ofZernike Polynomials

Center Position Values

xc 0.999728yc �0.048932


�25.41. Detailed data are summarized in Table 1. Asignificant improvement can be seen by the geneticfeedback tuning loop not only in transmission effi-ciency but also in uniformity.

C. Complementation of the Phase Hologram toCompensate for Aberration

Here we show that the genetic feedback tuning loopis effective in compensating for the aberration of theoptics in the digital holographic beam-shaping sys-tem. On the hologram plane, the wavefront of thebeam is spatially divided by the pixel size of theSLM, and the limit for the aberration compensationin the system is determined by the pixel size andthe number of pixels. Since the phase-modulation

table of the SLM has the intrinsic characteristic ona pixel-size scale, finding the optimum phase-modulation table is independent of compensatingfor the aberration. Therefore the phase-modulationtable obtained in Subsection 4.B can be used toencode the hologram on the SLM. As shown in Eq.(7), the coefficients Anm and Anm� and the center po-sition �xc, yc� of the Zernike polynomials are unknownvariables. Floating-point coding is adopted in the ge-netic algorithm. The order �m, n� of the circularZernike polynomial is limited in the interval 0 � n� 3 and 0 � m � 4. In practice, 15 variables are to beoptimized: 8 Anm ’s, indicating the coefficients ofterms with an x-axis symmetry; 5 Anm� ’s, indicatingthose with a y-axis symmetry; and 2 unknown vari-ables in the center position �xc, yc�.

In this experiment an initial population withfive chromosomes, P0 � �xi|i � 1, 2, . . . , 5�, is cre-ated, and the individual chromosome is given as

The initial population has the same five chromo-somes, representing a constant phase that leads to noaberration compensation. For encoding the Zernikepolynomials on an appropriate scale, the dimensionof the SLM plane is scaled so that the diameter of thephase hologram �400 pixels� is set to a unit lengthand the center of the circular aperture of the encodedphase hologram indicates the position of the coordi-

xi ��pl�1. . .8 �Anm

2�� 0, pl�9. . .13 �

Anm�

2�� 0, p14 � xc � 0, p15 � yc � 0�n � 0, 1, 2, 3, m � 0, 1, 2, 3, 4 . (18)

Fig. 10. Diffraction images in the tuned system from holograms (a) without and (b) with aberration compensation, which was previouslyobtained by the genetic feedback tuning loop.

Table 4. Evaluation Parameters of Diffraction Images inAberration Compensation

Generation 1st Generation 553rd Generation

GLN� 195.4 221.8GLS� 22.0 22.5Mean-square error 0.1494 0.0690Normalized variation 3.598 1.720Boundary length 13,130 10,645Stabilization factor 0.9932 0.9545Cost value �191.5 �148.6


nate �0, 0�. The coefficients of the Zernike polynomi-als are limited within the range from �0.25 to 0.25.Therefore the mutation is given by

xi� ��pi, l � ��t, 0.25 � pi, l� for rbinary � 0pi, l � ��t, pi, l � 0.25� for rbinary � 1, (19)

where ��t, y� holds the same form as shown in Eq.(17b). Here the mutation probability is set to 0.025,and the mutation parameter b is set to 20.

In the aberration-compensating experiment, toprove the effectiveness of the genetic feedback tuningloop on aberration compensation, we tilted the CCDat an angle of 10° and tilted the Fourier lens at anangle of �10°, which would effectively induce theinternal aberration in the beam-shaping system.

The evolution is stagnated at the 553rd generation,although the maximum number of generations is 1000.Figure 8 shows how the phase hologram is compen-sated by the genetic feedback tuning loop. Figure 8(a)shows the phase hologram encoded on the SLM, whichis designed without aberration compensation at the 1stgeneration stage, and Fig. 8(b) shows the aberrationcompensation obtained from the Zernike polynomialsby the genetic feedback tuning loop at the 553rd gen-eration stage. Tables 2 and 3 present the coefficientsand center position of the Zernike polynomials at the553rd generation stage (the stagnated state). Figure8(c) is the aberration-compensated phase hologram,which is the overlapped image of Figs. 8(a) and 8(b).Figures 9(a) and 9(b) show the diffraction image ob-served at the 1st generation stage and that at the553rd generation stage. The boundary length de-creases from 13,130 to 10,645. The mean-square errorand the normalized variation also decrease, and thecost value is consequently improved from �191.5 to�148.6. Detailed data are summarized in Table 4. Ascan be seen, the shape of the letter T becomes clearerafter aberration compensation.

To verify the effect of tuning the system with ab-erration compensation, another phase hologram de-signed with the IFTA algorithm, which regards asystem free from the aberration, is encoded on theSLM. Figure 10(a) shows the diffraction image, whichis distorted owing to aberration, which was not con-sidered in the IFTA algorithm. Figure 10(b) showsthe diffraction image for the phase data with aberra-tion compensation by adopting the compensationprofile of Fig. 8(b). Although the letters cannot beidentified in Fig. 10(a), the letters OEQE can be readclearly in Fig. 10(b). As seen in the results, the pro-posed genetic feedback tuning loop is effective for theaberration compensation of a digital holographicbeam-shaping system with a SLM.

5. ConclusionIn conclusion, the findings herein show that the pro-posed genetic feedback tuning loop can be used suc-

cessfully to properly calibrate the LCSLM of a real-time digital holographic beam-shaping system and toencode the aberration-compensated phase holo-grams, thus overcoming the aberration of the inter-nal optics of the system. The genetic feedback tuningloop is an adaptive, system-independent, and auto-matic mechanism for tuning the general digital holo-graphic beam-shaping system. It is expected that theproposed technique would be useful in refining morecomplex digital holographic beam-shaping systemsthan those dealt with in this paper.

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