2550
J. Opt. Soc. Am. A / Vol. 16, No. 10 / October 1999
V. F. Canales and M. P. Cagigal
Photon statistics in partially compensated wave fronts Vidal F. Canales and Manuel P. Cagigal Departamento de Fisica Aplicada, Universidad de Cantabria, Los Castros S/N, 39005 Santander, Spain Received April 8, 1999; accepted May 21, 1999 To retrieve the high-spatial-frequency information of atmospherically distorted wave fronts, it is desirable to use an adaptive optics system with a high degree of compensation. However, in the visible only partial compensation is attainable. We analyze the photoevent statistics corresponding to wave fronts with partial compensation. It is shown that the photoevent statistics evolve from a Bose–Einstein distribution to a Laguerre distribution as the number of corrected Zernike polynomials increases. Furthermore, the photoevent statistics are also analyzed as a function of the position at the image plane. © 1999 Optical Society of America [S0740-3232(99)00210-0] OCIS codes: 010.1080, 010.1330.
1. INTRODUCTION Random aberrations at the aperture of the telescope that are due to atmospheric turbulence determine the angular resolution of ground-based telescopes. Compensation of the wave-front degradation before detection (adaptive optics) and extraction of diffraction-limited information from the image series (speckle interferometry) are the two techniques that can overcome this limitation. Although adaptive optics systems with a large number of subapertures in the wave-front sensor and a large number of actuators in the deformable mirror provide the best results, they are complicated and expensive. In contrast, simpler adaptive optics systems have a great potential for application. In this paper imaging with a partially compensated adaptive optics system (fewer than one actuator per atmospheric coherence diameter1–4) is analyzed. The introduction of a proper description of the phase distribution in the partially compensated wave front allows us to predict the statistics of the light intensity in the image plane. The statistics of the point-spread function (PSF) core have already been described5 as a function of the telescope diameter, the Fried parameter (which takes into account the atmospheric conditions), and the residual phase variance in the wave front after compensation, which can be obtained theoretically.6 However, to evaluate the light intensity probability density function (PDF) necessitates a numerical integration that is time consuming and does not allow easy interpretation of the PSF core behavior. In a previous paper7 it was proposed that the intensity PDF be described as a Rician distribution function so that the effect of compensation performed over the wave front could be easily included. This distribution is characterized by two parameters; if they are properly estimated, the Rician distribution provides the same results as those obtained by numerical integration. To estimate these parameters, we developed a model for the image0740-3232/99/102550-05$15.00
formation process that is based on that proposed by Goodman8 for the case of noncompensated wave fronts. The difference we introduced was that our model takes into account the effect of partial compensation on the light intensity statistics. This model has been extended to yield the intensity statistics over the whole image plane; therefore an explicit dependence on the observation point coordinates has been introduced. From the light intensity statistics described by the Rician distribution, expressions for the photoelectron probability distribution function as a function of the degree of compensation and as a function of the position at the image plane have been developed. The photoelectron statistics at the PSF core evolve from a Bose–Einstein distribution to a Laguerre distribution when the degree of compensation increases. A similar evolution occurs when, for a determined degree of compensation, the observation point changes from the border to the center of the PSF. A simplified model has been developed for the case of low compensation or for points far from the PSF core with high compensation. A simulation procedure was established to test the theoretical proposals. Theoretical and simulated values of the photoelectron probability density were compared as a function of the degree of compensation and of the position. In all cases the fit was fairly good, which confirms that the theoretical expressions proposed really describe the photoelectron statistics at the image plane of an adaptive optics system operating under conditions of partial compensation.
2. THEORY A. Wave-Front Description Let the wave front be described by ⬁
⌽ 共 r, 兲 ⫽
兺 a Z 共 r, 兲 , i
i⫽1
© 1999 Optical Society of America
i
(1)
V. F. Canales and M. P. Cagigal
Vol. 16, No. 10 / October 1999 / J. Opt. Soc. Am. A
where a i are coefficients of the corresponding Zernike polynomials (Z i ). The effect of partial compensation on the wave front is that some of the coefficient vanish. The residual distortion in the compensated wave front may be estimated from the Noll6 expression for the average phase variance over the wave-front surface once the first j Zernike terms have been corrected: ⬁
兺
⌬j ⫽
i⫽j⫹1
具 兩 a i 兩 2 典 ⫽ coef共 j 兲
冉冊 D
冉 冊 1
r0
,
exp关 ⫺ 2 / 共 2⌬ j 兲兴 .
(3)
The characteristic function corresponding to the PDF of the phase, M ( ), can be evaluated as its Fourier transform: M 共 兲 ⫽
冕
冉
冊
⌬ j 2 exp共 j 兲 P 共 兲 d ⫽ exp ⫺ . (4) 2 ⫺⬁ ⬁
B. Light Intensity Statistics To describe the light intensity at the PSF core, we generalize the model for speckle that was developed by Goodman8 so that it can be applied to the case of partial compensation. The light intensity at the PSF core is obtained from the complex amplitude of the field, which results from the sum of contributions from many elementary areas (cells) in the wave front. As an approximation, let the phase screen consist of independent correlation cells so that each cell contributes with a phasor whose amplitude is proportional to the cell area. We assume that the amplitude and the phase of each elementary phasor are independent of each other and are independent of the amplitude and the phase of any other cell. Let A r and A i be the real and the imaginary parts of the resultant field at the PSF core. Since they are obtained as the sum of a large number of elementary phasor contributions, it is possible to consider A r and A i asymptotically Gaussian.8 The joint PDF of the real and the imaginary parts of the field is found to approach asymptotically5 P共 Ar , Ai兲 ⫽
1 共 2 r 2 兲 1/2
⫻
1 2 i 2
⫺共 冑I cos ⫺ 具 A r 典 兲 2
exp 1/2
1 共 2 i 2 兲
冋
exp 1/2
2 r2 ⫺共 冑I sin 兲 2 2 i2
册
,
(6)
P共 I 兲 ⫽
冕
⫺
P 共 I, 兲 d .
(7)
To estimate probability densities from Eqs. (5)–(7) it is necessary to evaluate 具 A r 典 and variances ( r 2 , i 2 ). It is possible to obtain them from the phase characteristic function given in Eq. (4). We stipulate that the amplitude of a cell is given by ␣ k /N. This normalization provides an exact fulfillment of Parseval’s law when the image obtained with a lens is analyzed. The expressions obtained are8
具 A r 典 ⫽ ¯␣ M 共 1 兲 , r 2 ⫽ 共¯ ␣ 2 /2兲关 1 ⫹ M 共 2 兲兴 ⫺ 共 ¯␣ 兲 2 M 2 共 1 兲 /N, i 2 ⫽ 共¯ ␣ 2 /2兲关 1 ⫺ M 共 2 兲兴 /N,
(8)
where N is the number of coherent cells in the entrance pupil of the system. An approximate estimate of the number of cells is given by N ⬇ (D/r 0 ) 2 , as Goodman8 proposed for the case of noncompensated wave fronts. C. Rician Probability Density Function Now, our objective is to find a simple expression to replace Eqs. (6) and (7). When the variance of the real part tends to that of the imaginary part, Eq. (7) tends to the Riciansquare noncentral distribution (Bessel distribution). For partial correction these two variances differ slightly, and a proper estimation of the distribution parameters is necessary to allow us to describe the light intensity statistics by using the Rician distribution,7 given by P共 I 兲 ⫽
冉
冊冉 冊
a 冑I I ⫹ a2 exp ⫺ I0 . 2 2 2 2 2 1
(9)
We perform the parameter estimation by equating expressions corresponding to the mean value and variance of the light intensity obtained when Eq. (7) is used with those obtained when the Rician distribution is used. It is possible5 to obtain expressions for the two parameters involved in the Rician distribution function as a function of 具 A r 典 and variances ( r 2 , i 2 ): a 2 ⫽ 关 具 A r 典 4 ⫹ 2 具 A r 典 2 共 i 2 ⫺ r 2 兲 ⫺ 共 i 2 ⫺ r 2 兲 2 兴 1/2, 2 2 ⫽ r2 ⫹ i2 ⫹ 具 A r典 2 ⫺ 关 具 A r典 4
exp关 ⫺共 A r ⫺ 具 A r 典 兲 2 /2 r 2 兴
⫹ 2 具 A r 典 2 共 i 2 ⫺ r 2 兲 ⫺ 共 i 2 ⫺ r 2 兲 2 ] 1/2. exp共 ⫺A i 2 /2 i 2 兲 ,
册
where the marginal PDF of the intensity is given by
(2)
1/2
2⌬j
2 共 2 r 2 兲 ⫻
冋
1
1
5/3
where the angle brackets denote the ensemble average and coef( j) is the corresponding coefficient given by Noll for the compensation degree j. The phase distribution function is assumed to be a zero-mean Gaussian function, of variance ⌬ j , that decreases when the number of corrected polynomials increases: P共 兲 ⫽
P 共 I, 兲 ⫽
2551
(5)
where r 2 , i 2 are the variances of the real and the imaginary parts, respectively; i.e., we have a Gaussian bivariate distribution function with a correlation coefficient equal to zero. The joint PDF of I and , bearing in mind that the Jacobian for the transformation from the variables (A r , A i ) to (I, ) is 1/2, is given by
(10)
From Eqs. (8) it can be found that a 2 ⬇ M 共 I 兲 2 ⫽ exp共 ⫺⌬ j 兲 , 2 2 ⫽ 具 I 典 ⫺ a 2 ⬇ 具 I 典 ⫺ M 共 1 兲 2,
(11)
where it has been assumed that the system is operating with low or medium correction and, consequently, that M (2) ⫽ M (1) 4 and 1/N are negligible. The limit of the number of corrected polynomials depends on the value of
2552
J. Opt. Soc. Am. A / Vol. 16, No. 10 / October 1999
V. F. Canales and M. P. Cagigal
M (2). As an example, for a telescope with diameter 3.84 m and a typical value of r 0 ⫽ 10 cm, (D/r 0 ) ⫽ 38.4, M (2) is negligible when the number of Zernike corrected polynomials is smaller than 90. Under this approximation relations (11) allow us to express Eq. (9) as P共 I 兲 ⫽
冉
冊冉
冊
a 冑I I⫹a exp ⫺ I0 2 . ¯I ⫺ a 2 ¯I ⫺ a 2 ¯I ⫺ a 2 1
2
P共 n 兲 ⫽ (12)
The dependence of the distribution on the correction degree is now evident, since it is now an explicit function of M (1). For systems with a modest number of actuators the attained compensation is not high, and the intensity PDF does not differ much from the exponential, which is characteristic of speckled images. Thus Eq. (12) can be approximated by the analytic function
冉
I ⫹ a2 P共 I 兲 ⬇ exp ⫺ ¯I ⫺ a 2 ¯I ⫺ a 2 1
冊冋 冉 冊 册 1⫹
a 冑I
2
¯I ⫺ a 2
.
(13)
For the value (D/r 0 ) ⫽ 38.4, this approximation is valid when 20 Zernike polynomials have been corrected. It is easy to see that this distribution tends to the exponential that corresponds to speckle when the correction is so weak that a 2 ⬇ M (1) 2 tends to zero. D. Extension to the Whole Image Plane The PDF of the light intensity at the PSF core has been obtained as a function of the degree of compensation. The result is the PDF given by Eq. (12), which is a function of the parameter a 2 ⬇ M (1) 2 . This PDF can be easily extended to other points at the image plane. Using the displacement theorem,9 it is possible to change the reference sphere of the wave front and, consequently, to change the observation point at the image plane. Hence the wave-front variance is that given by Eq. (2) plus two terms that depend on the distance from the image center to the observation point of coordinates (X, Y): ⌬ j 共 X, Y 兲 ⫽ ⌬ j 共 0, 0 兲 ⫹
冉 冊 冉 冊 XD 2f
2
⫹
YD
2
2f
,
(14)
where f is the focal length of the system. Hence it is possible to obtain the PDF of any point on the image plane by using Eq. (12) but taking into account that I is the mean intensity at point (X, Y) and that the residual phase variance must be evaluated from Eq. (14). E. Photon Statistics To obtain the photon statistics at any point of the image plane, it is enough to evaluate the Poisson transform of P(I) given by Eq. (12). The result is10 共¯I ⫺ a 2 兲 n P共 n 兲 ⫽ 共 1 ⫹ ¯I ⫺ a 2 兲 n⫹1
冉
a2
冊冋
⫻ exp ⫺ Ln 1 ⫹ ¯I ⫺ a 2
⫺a 2 / 共¯I ⫺ a 2 兲 1 ⫹ ¯I ⫺ a 2
册
.
whole image plane, since the mean intensity ¯I and a 2 are functions of the point coordinates (X, Y). It is interesting to show that for low correction relation (13) can be used, and its Poisson transform is
(15)
Hence the photon statistics can be expressed as a Laguerre distribution. This distribution function shows a clear dependence on the degree of correction performed on the original wave front through the value of a 2 . The photoevent distribution given by Eq. (15) is useful for the
冕
0
⫻
冉
I ⫹ a2 exp ⫺ ¯I ⫺ a 2 ¯I ⫺ a 2
⬁
1
I n exp共 ⫺I 兲 n!
dI.
冊冋 冉 冊 册 1⫹
a 冑I
2
¯I ⫺ a 2
(16)
This expression can be divided into two parts. The first is the Poisson transform of an exponential distribution, that is, a Bose–Einstein distribution. The second part is the Poisson transform of a 4 2 distribution, which provides a negative-binomial distribution: 共¯I ⫺ a 2 兲 n 共 n ⫹ 1 兲a2 P共 n 兲 ⫽ 1⫹ 共 1 ⫹ ¯I ⫺ a 2 兲 n⫹1 共¯I ⫺ a 2 兲共 1 ⫹ ¯I ⫺ a 2 兲 ⫻ exp
冉 冊 ⫺a 2
¯I ⫺ a 2
冋
册
.
(17)
In this equation it is easy to see that P(n) tends to the Bose–Einstein distribution for low correction, that is, when a 2 tends to zero.
3. SIMULATION The procedure we followed to simulate wave fronts with different degrees of compensation is that proposed by Roddier.11 The wave front is decomposed into Zernike polynomials, which allows us to control the degree of correction merely by assigning zero value to the coefficient of the corresponding corrected polynomial. Wave fronts are simulated with 560 Zernike polynomials. The original (uncompensated wave front) value of D/r 0 was 38.4. The number of samples at the aperture is 128 ⫻ 128, and the same sampling is used in the image plane. In the simulation procedure we assumed that the atmosphere produced changes in the phase of the electromagnetic field. Once the wave front has been obtained, the simulation procedure is used to estimate the field at a series of points at the image plane and, from them, the light intensity. The probability distribution function of the light intensity, P(I), was obtained from a series of 10,000 experiments. The photoelectron probability distribution function was obtained by introduction of the light intensity value obtained in each experiment into a Poisson generation subroutine.12 The analysis was performed as a function of the compensation degree and as a function of the position at the image plane.
4. RESULTS The first series of simulated experiments was used to test whether the residual phase variance at the compensated wave front followed the dependence on position predicted by Eq. (14). The light intensity PDF P(I, X, Y) was evaluated for a series of points at the image plane and then, by use of Eq. (12) and the experimental intensity mean value, the value of ⌬ j was fitted. Figure 1 shows the evolution of the residual phase variance ⌬ j as a function of the position at the image plane for the case of
V. F. Canales and M. P. Cagigal
D/r 0 ⫽ 38.4 and a compensation of 80 Zernike polynomials. The values of ⌬ j , obtained by our fitting the simulated P(I, X, Y) to the theoretical PDF given by Eq. (12), follow a parabolic behavior, as is predicted by Eq. (14). The good fit between both series of data suggest that Eq. (14) can be used to predict the residual phase variance, which is to be used to evaluate a 2 to predict theoretically, in turn, the evolution of the intensity probability density as a function of the position at the image plane. A second series of experiments was performed to test the photoelectron statistics at the PSF core as a function of the degree of compensation of the wave front and as a function of the position at the image plane. Figure 2 shows the evolution of the photoelectron statistics for the PSF core as a function of the degree of compensation performed on the original wave front. A quite good fit can be seen between theoretical values given by Eq. (15) and those obtained by simulation. For low compensation, that is, a low value of a 2 , Eq. (17) also provides very good
Fig. 1. Values of the residual phase variance of the compensated wave front as a function of the position of the observation point at the image plane. Solid curve, theoretical values; circles, simulated values obtained by our fitting the light intensity PDF obtained from simulation to the theoretical one described by Eq. (12).
Vol. 16, No. 10 / October 1999 / J. Opt. Soc. Am. A
2553
results, since it tends to the Bose–Einstein distribution, which describes the statistics for noncompensated speckle. In fact, Eq. (17) shows how the photoelectron statistics evolve from the Bose–Einstein to the Laguerre distribution as a function of the compensation level. Figure 3 shows the behavior of the photoelectron statistics as a function of the position at the image plane. Theoretical values are evaluated by use of Eq. (15). The dependence on the position can be introduced through the dependence of a 2 on ⌬ j , which is evaluated from Eq. (14). The photoelectron statistics are shown for the case of 80 corrected Zernike polynomials, since it allows us to see the evolution from small residual phase variance (with the reference sphere centered at the PSF core) to high residual phase variance (with reference sphere centered at points far from the PSF core). In this case the fit between theoretical and simulated values is quite good.
5. CONCLUSIONS In conclusion, theoretical expressions have been developed for the photoelectron statistics at the image plane when partially compensated wave fronts are detected. Starting from the theoretical procedure proposed by Goodman for estimating the PDF of the light intensity in noncompensated systems, we have proposed to replace the original distribution by a Rician distribution so that, with proper definition of its parameters, it will provide correct estimates of P(I) as a function of the compensation degree. We have extended the resulting PDF to the whole image plane by a convenient definition of the residual phase variance, ⌬ j . For low correction an analytic expression of the intensity PDF has been obtained, which permits easy interpretation and fast determination of the statistics. Starting from the light intensity PDF, we have developed theoretical expressions for the photoevent PDF. The degree of compensation has been introduced in a way that lets us understand the process easily.
Fig. 2. Photoelectron probability distribution at the PSF core for (a) 11, (b) 21, (c) 41, and (d) 81 corrected polynomials. theoretical exact values; dots, simulated values; triangles, theoretical approximated values.
Solid curves,
2554
J. Opt. Soc. Am. A / Vol. 16, No. 10 / October 1999
V. F. Canales and M. P. Cagigal
Fig. 3. Photoelectron probability distribution as a function of the position at the image plane for 80 corrected polynomials. Solid curves, theoretical exact values; dots, simulated values; (a) corresponds to the PSF core; (b)–(f ) correspond to points with a separation of f/4D.
The proposed PDF series has been tested by computer simulation. The final conclusion is that quite good descriptions of the photoelectron distribution function can be obtained with the simple expressions we propose. The dependence on the correction degree is quite clear and makes the distribution evolve from a Bose–Einstein distribution function (corresponding to speckle) to a Laguerre distribution function (corresponding to higher degrees of compensation).
2.
3. 4. 5. 6.
ACKNOWLEDGMENT
7.
This work has been supported by Direccion General de ˜ anza Superior e Investigacion Cientifica under Ensen project PB 97-0355.
8. 9.
M. P. Cagigal may be reached at
[email protected]. 10.
REFERENCES 1.
J. Y. Wang and J. K. Markey, ‘‘Modal compensation of atmospheric turbulence phase distortion,’’ J. Opt. Soc. Am. 68, 78–87 (1978).
11. 12.
F. Rigaut, G. Rousset, J. C. Fontanella, J. P. Gaffard, F. Merkle, and P. Lena, ‘‘Adaptive optics on a 3.6-m telescope: results and performance,’’ Astron. Astrophys. 250, 280–290 (1991). M. C. Roggemann, ‘‘Limited degree-of-freedom adaptive optics and image reconstruction,’’ Appl. Opt. 30, 4227–4233 (1991). M. C. Roggemann and B. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996). M. P. Cagigal and V. F. Canales, ‘‘Speckle statistics in partially corrected wave fronts,’’ Opt. Lett. 23, 1072–1074 (1998). R. J. Noll, ‘‘Zernike polynomials and atmospheric turbulence,’’ J. Opt. Soc. Am. 66, 207–211 (1976). V. F. Canales and M. P. Cagigal, ‘‘Rician distribution to describe speckle statistics in adaptive optics partial correction,’’ Appl. Opt. 38, 766–771 (1999). J. W. Goodman, Statistical Optics (Wiley-Interscience, New York, 1985). M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford UK, 1993). B. E. A. Saleh, Photoelectron Statistics (Springer-Verlag, Berlin, 1978). N. Roddier, ‘‘Atmospheric wavefront simulation using Zernike polynomials,’’ Opt. Eng. 29, 1174–1180 (1990). W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1989).
