PROPAGATION OF ADAPTIVELY CORRECTED LASER BEAMS THROUGH A TURBULENT ATMOSPHERE

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PROPAGATION OF ADAPTIVELY CORRECTED

LASER BEAMS THROUGH A TURBULENT

ATMOSPHERE

L. Bissonnette

To cite this version:

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PROPAGATION OF A D A P T I V E L Y CORRECTED LASER BEAMS THROUGH A TURBULENT ATMOSPHERE L.R. Bissonnette.

Centre de Recherches pour Za De'fense, VaZcartier C. P. 880, CourceZette, @&bee, Canada GOA 1RO.

R6sumd.- Get article ddcrit un modble mathdmatique pour la solution du problbme de propagation de faisceaux laser se d6plasant dans l'atmosphbre turbulente et corrig6s par une optique adaptable. Celle-ci est simulde 2 l'aide d'une relation math6matique simple mais suffisamment gdndrale pour re- presenter la plupart des systbmes existants. La mdthode permet de prddire les profils de l'intensitd moyenne et de la variance de l'intensitd pour des niveaux arbitraires de scintillation. On prd- sente quelques solutions typiques pour des faisceaux de longueur d'onde de 3,8 et 10,6 um. Ces r6- sultats illustrent les performances, aprss propagation dans la turbulence, d'un systbme d'optique adaptable en fonction de la longueur d'onde et du nombre d'dlGments actifs.

Abstract.- This paper describes a mathematical model for solving the propagation problem of laser beams travellingin atmospheric turbulence and corrected by adaptive optics. The modeling of the adaptive optics is mathematically simple but sufficiently general to encompass the majority of the existing systems. The method allows the prediction of the average irradiance and the irradiance variance beam profiles for arbitrary scintillation levels. Typical solutions are presented for 3;8

and 10.6 ym laser beams. These results illustrate the performance, after propagation in turbulence, of an adaptive system as a function of the wavelength and the number of active elements.

1.0 INTRODUCION Section 2.0 briefly reviews the propagation

model. Section 3.0 describes the modeling of adap- Adaptive optics has now been proven quite

tive optics and section 4 . 0 presents and compares efficient in compensating for laser beam spreading

typical calculation results at X = 3.8 and 10.6pm. induced by turbulence. The many aspects of this

new technology are well reviewed in Ref. 1 and in

a Topical Issue of the J. Opt. Soc. Am. (Ref. 2 ) . 2 . 0

PROPAGATION MODEL

However, the existing computational codes concern Our mathematical model for beam propagation mainly the simulation of the optical systems and is fully described in Refs. 3-5. Only the most propagation is most often reduced to computer gen- important features are recalled here.

erated input signals. This paper addresses itself to the intrinsic propagation problem. It is con- cerned with the prediction of the average irradiance and the irradiance variance profiles of laser beams propagating in turbulence and corrected by adaptive optics. It is an extension of the mathematical model developed in Refs. 3-5. The principal advantages of this model are that it is applicable at arbitrary scintillation levels and that the solutions can be calculated by straightforward finite difference techniques.

First, the instantaneous random scalar elec- tric field E of the assumed monochromatic wave propagating in the z-direction under negligible polar- ization effects is written as follows:

E = A cxp [jk(z

+

Q) - jot], (1)

where A is a complex amplitude, k$ is a phase func- tion, k=w/c is the optical wave number, w is the optical frequency of the source, c is the speed of light in free space, and j = a . Second, Eq. 1 is substituded for E in the scalar wave equation which, under the paraxial approximation and after separa-

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C9-4 16 JOURNAL DE PHYSIQUE

tion for A and $, becomes

where

N

is the instantaneous random index of refrac- tion, V is the gradient operator in the plane normal to the o-z axis, and V

_{- -}

= V $. In deriving Eq. 2, it was also assumed that

I

N-1

I

<< 1, which is well verified in atmospheric turbulence. Third, the random functions are written as sums of an average and a fluctuating part, i.e.

N = < N > + n ; < n > = O , (4)

V = < V > + v ; < v > =o, (5) A = < A > + a ; < a > = O , (6) where the pointed brackets denote ensemble averaging. Finally, the equations for the first- and second-order statistical amplitude moments, i.e. <A>, <aa> and

<sax>,

are derived by standard techniques from Eqs. 2-6.

The resulting moment equations contain more unknowns than there are equations. This is the classical closure problem which affects the treat- ment of turbulent phenomena governed by nonlinear or quasi-linear stochastic equations such as Eqs. 2 and 3. For the present application, this problem was solved in Refs. 3-5 where the unknown moments are mathematically related to the first- and second- order amplitude moments by the following equations:

1

< +a*v > = - ? ~ e a I { ~ ( z ) ): V < aa* 1, (10)

lim ~ 2 < a ( r + + p ) a ( r - ~ p ) > = - T ( z ) < a a > . (11)

P - 0 2

The functions K(z)

,

R(z) and T(z) form the

-

basis of our propagation model. The coefficients ~ ( z ) and R(z) were derived in a fully consistent

manner from the governing stochastic Eqs. 2 and 3. The principal hypothesis is that v and a are only

_-

weakly correlated. For a wave focused at z = F, K(z) and R(z) are given by

Since <v(z,r)v(u,s)>

_-

_{- -}

_-

is independent of the beam irradiance profile, K(z) and R(z) depend on wavelength, focal distance and medium properties alone.

Equation 11 is not essential since the equation for the one-point variance <aa> can be easily generalized to an equation for the two-point covariance <a(rl)a(r2)>. However, this greatly increase the level of complexity by adding at least two independent variables in the governing partial differential equation. To avoid this, we worked out, from semi-empirical considerations, the simple relation given by Eq. 11. The coefficient T(z) is

F - 212

T(Z) =

-

- C z / z A , (13) where C = 0.15 - 0.01j is an empirical constant and zA is the propagation scale characteristic of turbulence fading. For Kolmogorov's turbulence,

2 - 1 / ~ 1 2 / I l k 1 / 1 1

A - n (14)

where Cn is the index structure constant. Compari- son with data in Refs. 3-5 showed that Eqs.11 and 13 give excellent agreement on the widely different laboratory and atmospheric scales. Thus, in view of the simplicity and accuracy of this model, the need for the universal empirical constant C is considered acceptable.

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imation of Gaussian statistics for the complex amplitude a (Ref. 6), is given by

c12 = < aa* >'

+

< aa > < aa >* + 2 < A > < A >* < aa* >

+ < A > < A > < a a > * + < A > * < A > * < a a > . (16)

2

Thus, the profiles of <I> and oI are calculated from the solutions of the moment equations for <A>, a a > and <aa*> which are mathematically closed by the constitutive relations, Eqs. 7-11. It is worth noting that the solution for the average irradiance does not depend on the semi-empirical Eq. 11.

3.0 ADAPTIVE OPTICS MODEL

In this section, we derive the coefficients K(z) and R(z) for an adaptively corrected beam propagating in homogeneous Kolmogorov's turbulence. Equation 12 shows that K(z) and R(z) are functions of the covariance of v. The stochastic equation for v is easily derived from Eqs. 2 and 5. For an originally spherical phase front focused at z = F, it is given by

a v

r

-

a z +

(z,

.Vv +(&) +v.Vv = Vn ₍₁₇₎

which is the paraxial form of the eikonal equation of geometrical optics. Hence, v(z, r) is the com- ponent in the transverse plane of the unit vector parallel to the geometrical ray passing through the point (z, r) or, in other words, v(z, r) is the vector angle subtended by the geometrical ray at (z,r). The angle Ivl is generally small, typically smaller than 1 mrad, and the nonlinear term in Eq. 17 can be neglected with good accuracy. The solution of the linearized Eq. 17 for the outgoing phasefront whith the boundary condition v(0, r) = vo(r) is

-

In response to some corrective criterion, the

in the transmitter plane so that vo(r)

_{- -}

#

0 contrary to the cases studied in Refs. 3-5. Therefore, our task here consists in determining the statistics of vo(r). To begin, it is assumed that the system derives its information from the radiation reflected by an unresolved glint on the target illuminated at focal distance by the outgoing beam. The quantity of interest is the state of distortion of the phase of the returned signal, measured by the vector v, and the geometry is that of a spherical wave cen- tered on the target at z = F. Solving the linear- rized Eq. 17 for this spherical wave, we find at the transmitter plane

The ideal adaptive system then operates as follows.

A sensor measures in some form or other the random angle vt(r).

_..

_-

The latter is fed by a processor to a phasor array that continously changes the phasefront of the outgoing laser beam to make vo(r)

_{- -}

= -v (r).

-t

-

Hence, the reversibility of the ray paths predicts that the phasefront of the outgoing beam will grad- ually become undistorted and near-spherical as z+F.

In practive, however, the phase sensor and corrector have a discrete number of active elements. Therefore, the system cannot reproduce the function -v (r) on every point of the aperture of the

..t

-

outgoing beam and some form of spatial averaging must be taken into account. The true relation between the returned function vO(z) and the theo-

*

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C9-4 18 JOURNAL DE PHYSIQUE

where yoi(r) is the angle returned by the ith ele- particular system under study. A convenient ap- ment, v (r) is a random function respresenting the proach is to consider a homogeneous and isotropic

-b

noise of the system, T ( p ) is a correlation coeffi- noise with Gaussian correlation. This gives con- cient between the theoretical and the returned sistent results that are indicative of the system wavefront angle, W(p)

_-

is a window or weighting func- errors. However, because of the lack of space, we tion, and oi is the surface of the ith active ele- will not discuss these results here and we will

ment

.

simply take v = 0. Also, it is not possible to

-b

reproduce all the intermediate derivation steps, The expression for the covariance of v, i.e.

only the resulting model equations are given. They <v(z, r)

_{- -}

v (m, s)>, is easily derived from Eqs. 18-

_-

are

a

b2f b

( % + e d )

< a a * > -

-

l ~ < a a * > - ~ R e P I [ ~ ( q ) ] D b < a a * >

q - f a p ( f - 7 ) ) 2

The coefficients T(rl), K ( q ) and R ( Q ) which model 20. The calculations are performed for homogeneous the medium as well as the optical system are given Kolmogorovls turbulence. The simplifying hypotheses by

are that the correlation length of the turbulent T ( q ) = (0.15 - 0.01 j) f - 7 1 2

f - 7 , q . (25)

refractive index is much smaller than the propaga-

K ( q ) = K I ( q ) - K z ( 7 ) - K ~ ( T ) + K 4 ( q ) , (26) tion distances of interest and that the noise v is

-b R ( q ) = R l ( q )

-

R Z ( T I ) - R3(7)) + R A q ) , ₍₂₇₎

uncorrelated with the index n. Both these approxi-

where mations are well justified in practice. Finally,

there remains to compute the spatial averages (28)

over the surface of the active elements. For these

yy'

( f - x)* H( ,)

R , ( q ; f , 7 0 , q l ) = 2.43 d y d x n i 1 k ( f - ? ) ( f +; - 2 ~ ) ' (29) integrations to be manageable analytically, we as- o o

sume that the averaging effect of the active ele- _with _PI₌_q,

₊

_0.199

_-

_{( f}_-_{x ) ,}

f* 7 1 ;

ments is invariant with position on the aperture

( f - x)*

P 2 =

P3 = q 0 + 0 . 3 3 5 7 qx; L q = q O ,

and that it applies to a circular region of effec- f

tive raduis r given by Y I = f; Y , = q ; ~3 = y4 = Y ,

e

where S is the total area of the adaptive aperture

eY

and m a the number of active elements. G ( Y ) = y2

1;

+

37

+

[ 5/12 - Y - 3 y 2 ] --+U516, Y )

t

,

The contribution due to noise depends on the _ey

H(Y) = Y (1

+ [

516 - Y

]

p

r(516. Y ) ,

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fined by Eq. 6.5.20 of Ref. 7. The non-dimensional independent variables are

7 = ZIZ,; p = r / ~ . , _{(30 a,b)}

where zA is defined by Eq. 14 and ro is the repre- sentive radius of the source irradiance profile. The similarity paramaters are

f = F/z,; b = z, /krH , ₍₃₁₎

q0 = 0.123 kP;/zA, (32)

where

Lo

is the inner scale of turbulence, and

7 % = hz/zA (33)

which characterize respectively the laser beam, the turbulent medium and the adaptive optical system.

Equations 22-29 are sufficient to solve for the average irradiance, Eq. 15, and the irradiance variance, Eq. 16, of laser beams propagating in turbulence and corrected by adaptive optics. Solu- tions can be computed after the initial profile and the various similarity parameters have been speci- f ied

.

4.0 RESULTS AND DISCUSSION

For the present calculations, we consider two beams, one at 3 . 8 ~ and the other at 10.6um, propagating under the same experimental conditions. The atmospheric turbulence is assumed homogeneous with a very strong index structure constant Cn =

10'~ and an inner scale

Lo

= 0.Olm. The influence of the inner scale on the solutions is small and the exact value of L is not important

0

so long as it remains smaller than or of the order of lcm. The sources are Gaussian with an e-folding irradiance diameter 2ro = 0.5m and the outgoing beams are focused at F = 3km. The active optical

2

system has a total surface S = 7t/4 m and contains a variable number of active elements.

Sample results are shown in

FIG. 1. Predicted Strehl number plotted versus propagation distance normalized by the focal length; m is the number of active elements.

-6 -1/3. Parameters: X = 10.6pm; C, = 10 m

,

2

Lo

= 0.01,; rO = 0.25m; F = 3km; and S = n/4m

.

Fig. 1 where the on-axis Strehl numbers calculated for 10.6pm are plotted versus propagation distance for various numbers of active elements. The Strehl number is defined as the ratio of the average irradiance obtained under the specified conditions to the diffraction-limited irradiance. A value of unity means that the diffraction limit is attained. When no correction is applied, Fig. 1 reveals that the Strehl number continuously de- creases with propagation distance up to the focal plane where it reaches a value as low as 0.0082. With correction, the ratio decreasgs a little more rapidly at the beginning but much more slowly as z = F is approached. If the number of active elements.is sufficiently large, the ratio passes through a minimum at an intermediate propagation distance to rise again toward the focal plane. Ultimately, the diffraction limit is attained at

z = F and little improvement is gaiked by further increasing the number of active elements.

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c9-420 JOURNAL DE PHYSIQUE VI

-

[ % P ' =z Q 1 1 1 0 LOO 2 0 0 N U n B E R OF A C T I V E E L E n E N T S

FIG. 2. Predicted on-axis focal plane irradiance plotted versus the number of active elements. U : A = 10.6pm; o:

A

= 3.8pm. The horizontal lines represent the respective diffraction limits. Other parameters as in Fig. 1.

h = 3 . 8 and 10.6pm. In this figure, the on-axis focal plane average irradiance normalized by the on-axis transmitter irradiance is plotted versus the number of active elements. The conditions of operation are the same at both wavelengths so that the comparison is performed in terms of absolute quantities. Figure 2 shows a substantial reduction of the turbulence-induced beam spreading at both wavelengths. The degree of improvement grows with the number of active elements until the correspon- ding diffraction profile is attained. Before this limit is reached, the longer 10.6um wavelength is more advantageous in terms of target power density. But, if the number of active elements can be further increased, the advantage shifts to the 3.8pm beam since its diffraction profile is obviously sharper. For the conditions of Fig. 2, the cross-over point occurs at approximately 60 elements. This number is greater if system noise is taken into account, but smaller for weaker turbulence.

The on-axis focal plane scintillation

strength at 3 . 8 and 10.6um is plotted in Fig. 3 versus the number of active elements. The scintilla-

1 10 1 0 0 2 0 0

N U n B E R OF A C T I V E E L E n E N T S

FIG. 3 . Predicted focal plane scintillation strength, oI/<I>, plotted versus the number of active elements. 0: A = 10.6pm; o: A = 3.8pm. The values calculated for no correction are 1.26 and 1.09 for 10.6 and 3.8vm respectively. Other parameters as in Fig. 1.

tion strength is defined as the ratio oI/<I>. These results show that the scintillation remains strong even when the average irradiance beam profile is near diffraction limited; a reduction of at best 25% relative to the no-correction level is found. Such residual irradiance fluctuations are experimentally observed in Fig. 6 of Ref. 8. Theoretically, a non-zero scintillation level is not incompatible with a near diffraction-limited beam profile; irradiance fluctuations can indeed propagate on a smooth phasefront. Moreover, it can be shown from the model equations that correction for both beam spreading and scintillation would require the adaptive system not only to reproduce the conjugate of the phase of the spherical wave retro-reflected by the target glint, but to return its full spatial complex conjugate including the amplitude modula- tions. This is possible with the nonlinear optical technique of phase conjugation which has been dem- onstrated in recent years (Ref. 9). However, much developments are still required for the method to be technically applicable to high-power laser beams.

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elements just needed to achieve diffraction-limited propagation model in itself, only the boundary con- beam spreading. Therefore, the latter condition is ditions need be modified, albeit at the expense of the optimum operating condition for the type of greater algebraic complexity. Nevertheless, Eq. 20 adaptive systems studied here. Increasing the num- is a reasonable approximation and certainly gives a ber 6f active elements not only produces no further useful estimate of the corrective effect of adaptive gain in power density, but it worsens the level of optics.

residual scintillation.

6.0 REFERENCES 5.0 CONCLUSION

The results of Figs. 1-3 show that the propagation model described in this paper can efficient- ly predict the average irradiance and the irradiance variance profiles of adaptively corrected laser beams. The cases studied demonstrate the potential of adaptive optics to correct for turbulence beam spreading. Gain in focal plane power density of the order of 20 dB relative to uncorrected propagation appears possible with a practical number of active elements even in strong turbulence. Gains of this magnitude are certainly advantageous whatever the application. The present model should thus be very useful for the design of practical systems and for scaling test results to expected operational conditions.

One aspect of adaptive optics not fully ac- counted for in the proposed model is phase retriev- al. Here, we have assumed, as shown by Eq. 20, that the phasefront angle returned by the adaptive system

is the spatial average, over the effective surface area of the active elements, of a quantity which is linearly related to the true angle of the conjugate phasefront of the spherical wave reflected by the

Hardy, John W., Proc. IEEE, Vol. 66, NO. 6, pp 651-697 (June 1978).

Journal of the Optical Society of America, Vol. 67, No. 3, pp 269-409 (March 1977).

Bissonnette, L. R., "Average Irradiance and Irradiance Variance of Laser Beams in Turbu- lent Media", DREV R-4104/78 (May 1978). Bissonnette, L. R., Wodelling of Laser Beam Propagation in Atmospheric Turbulenceff, pp 73-94, Proceedings of the Second Internation- al Symposium on Gas-Flow and Chemical Lasers, John F. Wendt editor, Hemisphere Publishing Corporatfon, Washfrikton, D.C. (1979).

Bissonnette, L. R. "Focused Laser Beams in Turbulent Media", DREV R-4178/80 (in press). Bissonnette, L. R. and Wizinowich, P. L., Appl. Opt., Vol. 18, NO. 10, pp 1590-1599

(May 1979).

Abramowitz, M. and Stegun, I. A., *%andbook of Mathematical Functions", Dover Publica-- tiono, New York (1965).

Pearson, James E., Appl. Opt., Vol 15, No. 3, pp 622-631 (March 1976).

Yariv, Amnon, IEEE J. Quantum Electron., QE- 14, No. 9, pp 650-660 (Sept. 1978); see also comments in QE-15, No. 6, pp 523-525 (June 1979).