1674 J. Opt. Soc. Am. A/Vol. 11, No. 5/May 1994

Optical-field correction with deformable mirrors

Kai-yun Wang, Yu Qi, and Jing-wen Sun

Southwest Computing Center, P.O. Box 106, Mianyang, Sichuan, China

Received October 20, 1992; revised manuscript received October 25, 1993; accepted October 25, 1993Optical-field correction with deformable mirrors can be accomplished by correction of both amplitude andphase. As a result of developments over the past 20 years, phase correction with deformable mirrors hasbecome a mature technology. We discuss simply the phase correction when it is concerned with field correc-tion. The basic principle of amplitude correction with deformable mirrors is that if a certain phase distri-bution is constructed at the deformable mirror, after a vacuum diffraction, a certain amplitude distribution canbe obtained. Some algorithms for implementing the principle have been put forward by several researchers[T. T. Karr, Proc. Soc. Photo-Opt. Instrum. Eng. 1221, 26 (1990); Wang Kai-yun et al., Proc. Soc. Photo-Opt. Instrum. Eng. 1628, 244 (1992)]. But there are two problems that need to be solved. The first is thatthe vacuum path is too long. The second is that the precisions of these algorithms are relatively low. Wedescribe a new algorithm, which not only yields a 1-2 order-of-magnitude reduction in the vacuum distancebut also improves the amplitude correction precision.

Key words: deformable mirror, amplitude correction, field correction, Fourier transform.

1. INTRODUCTION

Optical-field correction can be carried out with con-ventional deformable-mirror technology. Its principleis shown in Fig. 1. The first deformable mirror (DM,)imparts a phase to the optical field E (E0), which is syn-thesized to give E the beacon amplitude pattern at theend of the vacuum. The second deformable mirror (DM2 )adjusts the E (E2) phase to be the opposite of the beaconphase. Thus E3 is the conjugate field of the beacon. Thephase of the first deformable mirror can be calculated bythe small-signal algorithm' or by the paraxial theory ofamplitude correction.2 But there are two problems withthe algorithms that must be solved. The first is that thevacuum path is too long for application of the amplitudecorrection. The second is that the precisions of thesealgorithms are relatively low, since they are limited tolinearized theory. In this paper we describe a new al-gorithm that not only leads to a 1-2 order-of-magnitudereduction in the vacuum path but also improves theamplitude-correction precision.

The beacon field in Fig. 1 may also be a hypotheticalfield. Because the deformable mirrors are completelycontrolled by a computer, an arbitrary field distribution,which we require, could be obtained by means of suchan optical-field correction system. The main confine-ment for the system is that the field propagation mustbe paraxial.

Obviously, on account of reciprocity, it is entirely pos-sible that the field correction can be applied to imagecompensation.

2. THEORIES OF FIELD CORRECTION

Optical-field correction with deformable mirrors appliesa correction to both the amplitude and the phase. Itdiffers from the conventional phase-only correction ap-proach. Based on the paraxial equation of a monochro-matic wave, the optical-field correction is used to remove

the field distortion by means of deformable-mirror tech-nology. The paraxial wave equation in vacuum is

aE = V2E,-z 2Ko (2.1)

whereV2 = 2 /ax2 + 2 /ay2

and Ko = 2ir/Ao, with Ao as the wavelength of the opticalfield. The optical field E can be expressed as

E=eO, Q=A+iqb, (2.2)

where A is the log amplitude and k is the phase. We willfirst describe the small-signal algorithm and the paraxialtheory of amplitude correction.

Small-Signal Algorithm and Paraxial Theory ofAmplitude CorrectionSubstituting Eq. (2.2) into Eq. (2.1), we obtain the Rytovequation3 in vacuum:

a = 2K 2 + (Vat)2]- (2.3)

If the perturbation of v1 is small, we can linearize thisequation by neglecting the term q12. Then Eq. (2.3) be-comes

(2.4)a az= i V 2

Karr gave its solution in the Fourier frequency domain inEq. (20) of Ref. 1:

F(V2) = F(tfJi)exp(-iakz), (2.5)

where F denotes Fourier transform, z is the distance forvacuum diffraction, a = K/2Ko, K = k + k2

2, andkl and k2 are two arguments in the Fourier frequency

0740-3232/94/051674-06$06.00 © 1994 Optical Society of America

Wang et al.,

Vol. 11, No. 5/May 1994/J. Opt. Soc. Am. A 1675

E E E2 E3

7

_1BEAOM F DM2

Fig. 1. Field correction with two deformable mirror

domain. In fact, Eq. (2.5) is the complex expressthe two equations

F(A2) = F(Al)cos(akz) + F(kI)sin(akz),

F(02) = F(+f)cos(akz) - F(Ai)sin(akz).

Accordingly,

F(01)= [F(A2) - F(Al)cos(akz)]sin(akZ)

Suppose that when a beam of light passes throdeformable mirror, its phase is changed and its ampis not, i.e., Al = Ao. Equation (2.7) indicates thatdeformable mirror is controlled according to the reithis equation, an amplitude distribution exp(A2 ) given approximately at a distance z from the deformirror.

When the amplitude of E is uniform, Eq. (2.7) rEto

F(,0) = F(A2)/sin(akz).

Compressing-Beam Method of Amplitude CorrectionAn important question concerns the length of the vacuum

-ON path. Suppose that the optical wavelength AO is 1 j.m,ACON the actuator stroke of the deformable mirror is 8 Am (i.e.,

50.27 rad), and Eo is a uniform field. If the A2 distribu-tion is needed to be 0.8 X sin(2irx/0.3) (the perturbationwavelength is 0.3 m), then, from Eq. (2.11) as solved for

ilon of the vacuum path, z = 962 m. Such a long system is un-wieldy for applications.

For shortening the vacuum distance, which amplitude(2.6a) correction requires, we use the method shown in Fig. 2 to

compress the optical path. The two deformable mirrors(2.6b) play the same roles as in Fig. 1. The focus of the first

lens (Li) is coincident with the focus of the second lens(L2). L and L2 compress the beam of light to raise theperturbation frequency. L3 and L4 , which have the op-posite functions to L1 and L2, enlarge the beam to restore

(2.7) its size. If zO, z1 , z 3, and z 4 << Z2, the field diffractionsbetween DM, and L1 , L1 and L2, L3 and L4 and DM2 maybe considered to be geometric propagations. Thus

ugh alitudeif the

sult of-an bemable

educes

(2.8)

This is the small-signal algorithm put forward by Karr.'Since the derivation of the paraxial theory of ampli-

tude correction is complex, here we introduce directlyEqs. (2.12) and (2.13) from Ref. 2:

A2 = A ( ZV201+ aA a 2 a~j2Ko (Xl2ax ax ay

Z =2 aA 2 ( aA 0P2 = '1 Ki A + ax +K ay}

I'

_a~i 2

axa0y 2tay)

ay J(2.9a)

(2.9b)

For the convenience of using a Fourier transform, we ne-glect nonlinear terms and terms with a variable coeffi-cient. So the Fourier forms of Eqs. (2.9a) and (2.9b) are

F(A2) = F(Al) + F(0D)akz,

F(02) = FGl,) - F(Al)akz,

(2.10a)

(2.10b)

respectively. When akzI << 1, then, from Eqs. (2.6), wecan also obtain Eqs. (2.10a). Thus

F(/ ) - [F(A2) - F(AD] (2.11)akz

02(X,y = Ok(ax,ay),

A 2(x, y) = A1(ax, cry) + n a,

A3(x,y) = A 4(ax, cry) + ln a,

(2.12a)

(2.12b)

(2.12c)

where the compression ratio a = Do/D1 . Equation (2.11)gives

F(02) = [F(A3) - F(A2)]/akz 2. (2.13)

Then, from the frequency scaling property of Fouriertransforms, it follows that

F[-2(X,Y)] = F(k,k2) =01 (kl, k2/a)

AF'(k1,k *) - [A4 (ki/cr, k2/r) + F(ln a)]22 a2 2=[ k 2

A~(1,k) -[Af(ki/a,k 2/a) + F(In a)]a2

(2.14a)

(2.14b)

(2.14c)

where the superscript F denotes Fourier frequency do-main. Inserting Eqs. (2.14) into Eq. (2.13) and replacingK by K/a, ki by k/a, and k2 by k2 /a, we obtain

1(ki, k2) = [AF4(k1, k 2 ) - AlF(kl, k 2 )]/ l 2akz 2. (2.15)

Under otherwise identical conditions, if a = 5 in Fig. 2,the vacuum distance Z2 is only 1/25 of z in Fig. 1. There-fore the method of compressing the beam of light greatlyreduces the size of the correction system.

The derivation from Eqs. (2.12) to Eq. (2.15) is basedon the case in which the optical-field distribution is un-

DM1 L, L4 DM2

Do

I" .- 1 '. J. 1.i- - :1ZF ZC Z Z Z

Fig. 2. Compressing-beam principle of field correction.

Wang et al.

1676 J. Opt. Soc. Am. A/Vol. 11, No. 5/May 1994

limited. A rigorous analysis must take account of theaperture restriction function. But the numerical resultsindicate that such an approximate treatment is valid.

Iterative AlgorithmEquation (2.1) is the same type as that of Eq. (2.4).Thus, according to Eq. (2.5), which is the solution ofEq. (2.4), we obtain the exact solution for Eq. (2.1):

F(E2) = F(El) exp(-iakZ); (2.16)

that is,

F[exp(A2 + i 2)] = F[exp(A1 + i'b)]exp(-iakz) -

(2.17)

Equations (2.6) and (2.10), which are approximate solu-tions of Eq. (2.1), are hard to avoid, bringing about someerrors. Equation (2.17) is the exact solution of Eq. (2.1),but it cannot be used directly for the amplitude correc-tion. The main reason is that, although A1 and A2 areknown and Eq. (2.17) can be separated into two real equa-tions, we cannot obtain the phase 01 directly with thesetwo real equations, since they are trigonometric equationswith variable coefficients. Thus we use the iterative so-lution of Eq. (2.17) to approach the exact solution of 01by degrees. We will discuss the relationship of the er-ror terms of El and E2 in advance. Let the approximatesolution Vi be the sum of the exact term tP, and the errorterm 8t:

= p + 0; (2.18)we can also write

For the first iteration we obtain the initial value of 02from Eqs. (2.10).

-1[ F(A, 2 ) - F(A~l)sc1'021 = 'Pp2 + 6P021 = F i -,akz

(2.22)

where Ap1 = AO, Ap2 is the log amplitude of E2, andsc = 1 + (akz) 2 . Substituting Eq. (2.22) into Eq. (2.17)gives the first iterative solution of Pil:

011 = 'pl + 35'Pi = Im(clg{F-'[F(E 2 l)exp(akz)]}),(2.23)

where E21 = exp(Ap2 + i 21) and cg stands for complexlogarithmic computation. From Eq. (2.21) we obtain

IjE 2 (0, 02l) 2IEp22

= ISEl(8All, 3,0k1)I21Epi12 . (2.24)

For the second iteration, '022 is

022 = Pp2 + 5022 = Im(clg{F1 l[F(Ell)exp(-iakz)]}),

where El, = exp(Ap1 + i,,). In the same way in whichwe were led to Eq. (2.24), we obtain

15E2 (3A22 , 0 22)12 Ep 22 = j3El(0, 34ll)j 2jEPl . (2.25)

Thus the second iterative solution of 01 is

012 = Ppl + 8012 = Im(clg{F-'[F(E 2 2)exp(iakz)]}),(2.26)

where E22 = exp(Ap2 + i 22).error terms of El and E2 is

The relation between the

E = E(1 + E), (2.19)

where E, = exp(qip). Thus

18E12= lexp(3q/) -112

= 1 - 2 exp(GA)cos(8'P) + exp(28A).

Let -7r c '0 < ir; then only one pole is found for I5E12:

8A= 0, '0 = 0.

It is a minimal point, and at this point the minimum valueof 16E12 is 0. For this reason, if 80 does not changeand 18AI degrades, 1'E12 will reduce. From Eq. (2.16)we have

F(E, 2 ) = F(Epl)exp(- iakz). (2.20)

Combining Eqs. (2.16) and (2.19) and subtracting Eq.(2.20) give

F(8E2 E, 2 ) = F(5ElEpl)exp(-iakz).

Squaring the equation, and from Parseval's theorem,4 wehave

I8E 2 12IEp2 12 = 8E12

1EP112 , (2.21)

where the overbar denotes spatial average.

18E2 (0, 6'P22 )I2IEP 2 12 = 18E 1 (A 1 2 , 80 1 2 )I2IEp,1

2 . (2.27)

Since 8EJ2 will degrade with the reduction of 18A, fromEqs. (2.24), (2.25), and (2.27), we obtain

18E 1 (A 11 , 50ll)I 2 > 16E1 (0, 5'102) 2 ,

I5E 2 A2 2 , '0 22)12 > I8E2 (0, 50 22)I2 .

Thus, provided that the number of iterations n is largeenough, 18E112 and 18E 2 12 will approach 0, and 8501n willapproach 0, too. That is,

0pl =lim ¢kin -

Therefore the iterative process is convergent. The re-quirement for the convergence is that the paraxial waveequation in vacuum should be satisfied. The amplitudecorrection precision of this method after the first itera-tion is much higher than in the cases of Eqs. (2.7) and(2.11). With increments in the number of iterations theprecision will be improved progressively. When we usethe compressing-beam method of amplitude correction asshown in Fig. 2, all the terms akz in this section must bereplaced by

a 2 (Z2 + Z/Ca + Z3/)ak- (2.28)

The term (Z + z3)/a, which is absent from Eq. (2.15),could improve the correction precision somewhat. Wedo not give a proof of this statement, for the function(z + Z3)/c is much smaller than the term Z2.

Wang et al.

Vol. 11, No. 5/May 1994/J. Opt. Soc. Am. A 1677

tion. It can also eliminate errors of the phase correctionin terms of theory.

The theory and the technology of the phase correc-tion are mature. Therefore we will deal mainly with theamplitude correction. For the quantitative analysis arelative error function is introduced to express the pre-cision of the amplitude correction:

Pa= Lf (E21 -' E2 1)2ds/f(IE21 - IE42ds] , (3.1)

where s is the integral variable, 1E21 is the field ampli-tude (which we need before the second deformable mirror),and 1E2'1 is the diffracted field amplitude. With 1E21 and1E1I, 'Pi could be calculated by the use of correction algo-rithms. 01 and 1ERG form E1 , and 1E2'

1 can be obtainedfrom Eq. (2.16). The precision is improved with the re-duction of Pa. We think that the correction is invalid if

8. 00

0.25 6.00(b)

Fig. 3. One-dimensional simulations of intensity (or amplitude)correction using the system shown in Fig. 1: (a) small-signalalgorithm; (b) paraxial theory. The correction precisions of (a)and (b) are 1.2215 and 1.1255, respectively, and z = 1 km,D = 0.4 m, N = 12,800, and DF = 1.2 m. 1, El12 ; 2, 1E2 12;3, 1E2 '12 ; 4, 0.01801; 5, 0.0180b2.

4.00

2.00

0.00

-2.

So far, we have discussed the amplitude correction ofthe optical field. The phase correction of the optical fieldis simple. Its method consists in inserting the known logamplitude AO (note that Al = AO) of the incident field Eoand 'Pn, which is obtained after the nth iteration, intoEq. (2.17). Then we obtain the phase of the field E2:

(a)

8 .00

6.00

4.00

2.0002 = Im(clg{F-`[F(Ao + 0P.)exp(-iakz)]}). (2.29)

0.00

-2.00 0 We can implement the phase correction by controlling thesecond deformable mirror with the difference between thephase that we want for E3 and 02. Thus the field cor-rection is now completed. When the compressing-beammethod is used for the amplitude correction, the term akz

in Eq. (2.29) must be replaced by expression (2.28).

2.00

1.00

3. DISCUSSION

In the preceding sections three algorithms for the fieldcorrection were introduced:

(1) Small-signal algorithm [Eqs. (2.6b) and (2.7)];(2) Paraxial theory [Eqs. (2.10) and (2.11)];(3) Iterative algorithm [Eqs. (2.22), (2.23), and (2.29)].

The first two algorithms are approximate solutions forthe paraxial wave equation in vacuum [Eq. (2.1)]. Foreither the amplitude correction or the phase correctionthey would produce some errors. The third algorithmcan greatly improve the precision of the amplitude correc-

0.00

25 0.00(b)

0.25

'_Z0.25 0.00 0.25(c)

Fig. 4. One-dimensional simulations of intensity (or ampli-tude) correction using the system shown in Fig. 2 with aniterative algorithm. The number of iterations is 1. Thecorrection precision is 0.7028, 2 = 50 m, z = 6 m, DO =0.4 m, D1 = 0.08 m, N = 80,000, and DF = 0.6 m. (a) Eland E2 : 1, Eu12 ; 2, 1E2 12 ; 3, 0.05'pu; 4, 0.0502. (b) E 3 :1, E312 ; 2, 0.0303 . (c) E1 and E 4 : 1, El12 ; 2, 1E412 ; 3, 1E4

12 ;4, 0.0201; 5, 0.0204.

2.003-

1.00

0.00

-1. - .25

2.00 .3-

1 .00 _

0.00

-1. -0.25

(a)

o 00

-ei

Wang et al.

- 1 .0 n

1678 J. Opt. Soc. Am. A/Vol. 11, No. 5/May 1994

0.25

Yf 0.00

-0.25-o0.

0.25

Yt0.00

-0.250.25

2.00

1.00

-.0%.

0.70

0.60

Pa 0.50

0.40

n1 n I-

0.25

x

(a)

0.00

x(b)

0.25

(c)

- 0.00 10.00 20.00 30.00 40.00NUMBER OF ITERATIONS

(d)Fig. 5. Two-dimensional simulations of intensity (or amplitude)correction using the system shown in Fig. 1 with an iterativealgorithm. The number of iterations n is 40. The correctionprecision is 0.3240, and z = 1000 m, D = 0.4 m, N = 256 256,and the Fourier transform area = 0.5 X 0.5 m. (a) Intensitypattern (E 212) that is needed. (b) Intensity pattern (E 2'1

2)after correction. (c) 1E11

2 and 1E212 on the X axis: 1, IE1 12;2, 1E212; 3, IE2 '1 2. (d) Variation of Pa with the number ofiterations.

Pa > 1, and the correction is almost perfect if P < 0.4.Figures 3(a) and 3(b) are one-dimensional simulations of

amplitude corrections using two approximate algorithms(the small-signal algorithm and the paraxial theory, re-spectively). There are no compressing lenses in the cor-rection system. The field propagation from El to E2 iscommensurate with the single-slit diffraction. Conser-vation of energy was considered. The basic conditionsare the length of the vacuum path (z = 1 km), the width ofthe slit (D = 0.4 m), the optical wavelength (Ao = 1 utm),the number of calculated lattice points (N = 12,800),and the width of the region for calculating the Fouriertransform (DF = 1.2 m). The amplitude correction pre-cisions of the small-signal and the paraxial theory algo-rithm are 1.2215 and 1.1255, respectively. The small-signal algorithm, which has nonzero singular points, mayexcessively magnify some high-frequency components.

The amplitude-correction process with the compressing-beam method iterative algorithm is shown in Fig. 4. Thenumber of iterations is 1. Here z1 = Z3 = 6 m, Z2 = 50 m,D = 0.4 m, D1 = 0.08 In, a = 5, A = 1 tm, DF = 0.6 m,and N = 80,000. The amplitude-correction precision is0.7028. Under the condition of not demanding a higherperformance of the deformable mirror than in the case ofFig. 1, this method greatly shortens the vacuum path andsignificantly improves the correction precision. In thesimulation the field propagations from El to E4, except forphase transforms of lenses, are calculated with Eq. (2.16).The amplitude-correction process indicates that the hy-pothesis of Eq. (2.12) is approximately valid.

An example of a two-dimensional amplitude correc-tion with an iterative algorithm is given in Fig. 5. Thenumber of iterations is 40. The simulation conditionsare D = 0.4 In, N = 256 X 256, Z = 1 km, A = 1 um,and Fourier transform area = 0.5 X 0.5 m. Figures 5(a)and 5(b) are intensity contour plans of the required field(E2) and the diffraction field (E2'), respectively. Theamplitude-correction precision is 0.3240. Figure 5(c)shows distributions of El and E2 on the X axis. FromFigs. 5(a) and 5(b) there seems to be some differences be-tween 1E21

2 and E2'12 . However, E2 '1 2 is close to E212 in

Fig. 5(c). High-frequency fluctuation of E2'12 is causedby the diffraction effect of finite aperture. Such a fluc-tuation could not be removed completely. The relationof the amplitude-correction precision to the number ofiterations is given in Fig. 5(d). With an increase in thenumber of iterations the correction effectiveness becomesbetter step by step.

4. CONCLUSIONThe correction method in this paper is, in fact, a controlmethod. The method can be applied to controlling boththe amplitude and the phase of a near optical field as theparaxial condition is satisfied in the field propagation.Since Eq. (2.16) is also applicable to the analysis of theFraunhofer diffraction, the amplitude correction might beused for controlling the intensity of a far optical field invacuum. The main constraint is diffraction limitation.

For a small-aperture system the vacuum path requiredby the compressing-beam field correction will be relativelyshort. For instance, if Do = 0.1 m and a = 5, a lengthof approximately 5 m is sufficient for the path. Thus itis not difficult to conduct a non-real-time experiment onthe field correction. Whether a real-time correction ex-

i . . . . . . . .

Wang et al.

Vol. 11, No. 5/May 1994/J. Opt. Soc. Am. A 1679

periment can be carried out depends on the calculationspeed because the iterative algorithm includes a lot ofFourier transforms and related calculations. There aretwo ways of increasing calculating speed. The first is tofind a method for speeding up the convergence of the itera-tive algorithm in order to reduce the number of iterations.The second is to use computer hardware for implement-ing the necessary calculation.

In the real system the performance of the deformablemirror (such as the stroke and the space of actuators) willproduce an important impact on the precision of the fieldcorrection. We have not dealt with this problem.

Phase-only correction to strong thermal blooming willinduce the so-called phase compensation instability. Ifwe use the method of optical field correction to compen-sate for thermal blooming, such an instability might beeliminated. This problem needs further research.

REFERENCES

1. T. J. Karr, "Instabilities of atmospheric laser propagation," inPropagation of High-Energy Laser Beams Through the Earth'sAtmosphere, P. B. Ulrich and L. E. Wilson, eds., Proc. Soc.Photo-Opt. Instrum. Eng. 1221, 26-55 (1990)

2. Wang Kai-yun, Sun Jin-wen, and Zhang Wei, "Paraxial theoryof amplitude correction," in Intense Laser Beams, P. B. Ulrichand R. C. Wade, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1628, 244-252 (1992).

3. J. W. Strohbehn, "Modern theories in the propagation of op-tical waves in a turbulent medium," in Laser Beam Propaga-tion in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag,Berlin, 1978).

4. D. F. Elliot and K. Ramamohan Rao, Fast Transforms: Al-gorithms, Analysis, Applications (Academic, New York, 1982),p. 24.

ACKNOWLEDGMENT

We thank You-pin Li for helpful discussions.

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