Probability of the residual wavefront variance of an adaptive optics system and its application Jian Huang,1,* Chao Liu,2,3 Ke Deng,1 Zhousi Yao,1 Hao Xian,2,3 and Xinyang Li2,3 1
School of Astronautics and Aeronautics, University of Electronic Science and Technology of China, Chengdu 610054, China 2 The Key Laboratory on Adaptive optics, Chinese Academy of Sciences, Chengdu 610209, China 3 Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu 610209, China * [email protected]
Abstract: For performance evaluation of an adaptive optics (AO) system, the probability of the system residual wavefront variance can provide more information than the wavefront variance average. By studying the Zernike coefficients of an AO system residual wavefront, we derived the exact expressions for the probability density functions of the wavefront variance and the Strehl ratio, for instantaneous and long-term exposures owing to the insufficient control loop bandwidth of the AO system. Our calculations agree with the residual wavefront data of a closed loop AO system. Using these functions, we investigated the relationship between the AO system bandwidth and the distribution of the residual wavefront variance. Additionally, we analyzed the availability of an AO system for evaluating the AO performance. These results will assist in designing and probabilistic analysis of AO systems. ©2016 Optical Society of America OCIS codes: (010.1080) Active or adaptive optics; (010.1330) Atmospheric turbulence; (000.5490) Probability theory.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
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13–23 (2005). 17. H. Jian, D. Ke, L. Chao, Z. Peng, J. Dagang, and Y. Zhoushi, “Effectiveness of adaptive optics system in satellite-to-ground coherent optical communication,” Opt. Express 22(13), 16000–16007 (2014). 18. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66(3), 207–211 (1976). 19. G. A. Tyler, “Bandwidth considerations for tracking through turbulence,” J. Opt. Soc. Am. A 11(1), 358–367 (1994). 20. R. F. Stapelberg, Handbook of Reliability, Availability, Maintainability and Safety in Engineering Design (Springer, 2009).
1. Introduction When wavefront disturbances become unavoidable, adaptive optics (AO) helps improve the performance of its “parent” optical system owing to its ability to correct the wavefront distortion. As a result of this, AO systems have been widely used in many fields, such as astronomical imaging [1,2], human retina imaging [3,4], free-space laser transmission and atmospheric communication [5–7], and laser fusion [8]. In conventional AO, the limitations on phase conjugation stem from many sources. The wavefront measurement quality, the insufficient control loop bandwidth, the fitting errors associated with deformable mirrors, and the wavefront surface contour reproduction quality all affect the phase-conjugation process [9], with the limitation of the insufficient control loop bandwidth of an AO system closed loop being the most significant. Many researchers have developed expressions relating the parameters of AO systems to their residual wavefront and Strehl ratio [10–15], and the wavefront variance average owing to the insufficient control loop bandwidth, obtained by using the power spectrum method, is widely used for design and analysis of AO systems. However, the power spectrum method does not yield the temporal probability of the residual wavefront variance, which is extremely important for calculating the availability of an AO system in its “parent” optical system, especially for the “parent” systems for which the information acquisition bandwidth is larger than the bandwidth of an AO system closed loop. The work represented here is mainly motivated by the studies of AO systems for improving the performance of optical atmospheric communication, in which one frame with insufficient wavefront compensation would yield an error in the order of millions of bits. Different from the AO systems that are used for imaging, the evaluation and design of an AO system to be used in high-speed optical communication applications must be based on the probability distribution of the wavefront variance of the AO system [16,17]. Fundamentally, the results of the probability analysis of AO systems should hold for all AO systems and applications. In the following part of this paper, we describe our studies on the probability density function (PDF) of the residual wavefront variance, PDF of the Strehl ratio of an AO system, and AO system availability. 2. Probability of the AO residual wavefront variance The probability of the residual wavefront variance of an AO system in the time domain provides more information on the AO system performance than the average index. The probability of the residual wavefront variance could be obtained by calculating the statistical characteristics of the Zernike coefficients of the residual wavefront error, for two scenarios: 1) calculations for instantaneous exposure of all wavefront error frames, and 2) calculations for long-term exposure, obtained by summing over many wavefront error frames. If we assume that the atmosphere is locally isotropic and uniform, the Zernike coefficients of the residual wavefront error of an AO system must obey the Gaussian distribution owing to the insufficient bandwidth (3 dB cut-off frequency) of AO control loops. In addition, the bandwidth of a tilt mirror loop is lower than the bandwidth of a deformable mirror loop, owing to the lower response frequency of the tilt mirror, which causes the root mean square (RMS) of the Zernike coefficients of the residual tilt to be larger than that of other higherorder terms. Therefore, the probability of the residual wavefront variance can be more efficiently analyzed by independently considering the tilt error and the high-order error.
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Received 5 Jan 2016; revised 1 Feb 2016; accepted 1 Feb 2016; published 3 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.002818 | OPTICS EXPRESS 2819
As the X- and Y-axes can be chosen arbitrarily, the Zernike coefficients of residual tilts in the X and Y directions have the same variance σ zt2 for isotropic and uniform atmosphere in the long term. Then, the Zernike coefficients of the residual tilts in the X and Y directions, a2 and a3, obey the same Gaussian distribution N ( 0, σ zt2 ) . As a result, the variance of the instantaneous residual tilt can be expressed as:
σ t2 = a22 + a32 From the statistical theory, the quantity σ with two degrees of freedom:
2 t
σ
2 zt
(1) is distributed as a χ random variable 2
σ t2 σ zt2 = ( a22 + a32 ) σ zt2 ~ χ 2 ( 2 )
(2)
The PDF of σ t2 is: σ2 exp − t 2 (3) 2σ 2σ zt The probability of the residual higher-order wavefront error can be obtained in the same manner. The power spectra of all high-order Zernike modes for the Kolmogorov turbulence are constant for low frequencies, and follow the same −17/3 power law for high frequencies, and the variance of each higher mode is nearly the same [18]. Therefore, if these higher-order modes are corrected by the same deformable mirror loop, the residual high-order Zernike modes must have the same variance σ zh2 owing to the insufficient control loop bandwidth when the fitting error of the deformable mirror is ignored. Then, the Zernike coefficients of the higher-order residuals ak obey the same Gaussian distribution N ( 0, σ zh2 ) . Therefore, the 1
f (σ t2 ) =
2 zt
variance of instantaneous residual higher-order wavefront can be expressed as: n+3
σ h2 = ak2
(4)
k =4
Then, the quantity σ h2 σ zh2 is distributed as a χ 2 random variable: n
σ h2 σ zh2 = ak2 σ zt2 ~ χ 2 ( n )
(5)
k =4
and the PDF of σ h2 is:
(σ ) σ f (σ ) = exp − ( 2σ ) Γ ( n / 2 ) 2σ 2 h
2 n / 2 −1 h
2 zh
n/ 2
2 h 2 zh
(6)
where n is the number of the degrees of freedom of the χ 2 distribution, defined by the number of higher-order Zernike modes that reconstruct the residual wavefront, and Γ ( ⋅) is the gamma function. The probability of the variance of the residual wavefront error could be examined by using the results of an AO system. Here, we analyzed the probability pattern of the residual wavefront variance of an AO system composed of 127 elements, for a 1.8-m-diameter telescope operated by the Adaptive Optics Laboratory of Chinese Academy of Sciences. This AO system has two closed loops for independently controlling its tilt and deformable mirrors, and the frame frequency of the Hartmann sensor is 2000 Hz at the wavelength of 0.55 µm. Each measurement consisted of the 10000 frames wavefront data acquired during 5 seconds for different star beacons. The measurement conditions are listed in Table 1.
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Received 5 Jan 2016; revised 1 Feb 2016; accepted 1 Feb 2016; published 3 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.002818 | OPTICS EXPRESS 2820
Table 1. Measurement Conditions Star number
Star Magnitude M
elevation angle (degree)
Spectrum type
Fried parameter r0 (cm)
HIP28360 HIP25423 HIP23015 HIP18532 HIP14576
1.9 1.6 2.69 2.9 2.09
65 63 55 42 33
ARV B73 K32 B0.5 B8V
11.8 9.3 8.5 7.9 7.2
From the residual wavefront error, the distributions of the Zernike coefficients have been calculated for different conditions; in these calculations, the averages of the Zernike coefficients have been firstly removed as static errors. The calculated histograms of the Zernike coefficients for the different conditions reveal Gaussian distributions, which implies that the residual wavefront error is mainly caused by the insufficient bandwidth. The RMS values for the 35 Zernike coefficients of residual wavefront errors are shown in Fig. 1.
RMS of coefficients (λ)
1
M = 2.09 r = 7.2 0
0.9
M = 2.9
0.8
M = 2.69 r = 8.5
r = 7.9 0
0
0.7
M = 1.6
r = 9.3
M = 1.9
r = 11.8
0 0
0.6 0.5 0.4 0.3 0.2 0.1 0 0
5
10 15 20 25 Zernike terms of residual wavefront error
30
35
Fig. 1. RMS values for the Zernike coefficients of the AO system.
Our findings are as follows. First, the RMS values of tilt X (second term) and tilt Y (third term) are larger than the other higher-order terms, mainly stemming from the lower bandwidth owing to the lower response frequency of the tilt mirror. Second, the RMS of tilt X (second term) is smaller than that of tilt Y (third term), which may be due to the lateral wind in the Y direction, yielding a stronger high-frequency spectral component for tilt Y; in our measurements, the turbulence was not strictly isotropic and uniform. Third, the RMS values of higher-order terms are nearly the same and independent of the turbulence strength; apparently, these high-order residuals arise mainly owing to the 3 dB cut-off frequency of the deformable mirror loop. In these five data sets of the Zernike coefficients, the data for M = 2.9 and M = 1.9 cases are accordant with the assumptions of isotropy and uniformity that we want to discuss in this paper. Thus, it is reasonable to validate Eq. (3) and Eq. (6) by using only the data for M = 2.9 and M = 1.9. For the M = 2.9 data, we calculated σ zt = 0.179λ and σ zh = 0.0415λ ; for the M = 1.9 data, we obtained σ zt = 0.205λ and σ zh = 0.0431λ . The PDFs of the residual tilt variance were calculated from the wavefront data and by the theoretical analysis of Eq. (3). The results are compared in Fig. 2.
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Received 5 Jan 2016; revised 1 Feb 2016; accepted 1 Feb 2016; published 3 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.002818 | OPTICS EXPRESS 2821
15 M = 1.9, σ2 χ2(2) zt M = 1.9, σ2 of AO closed loop t M = 2.9, σ2 χ2(2) zt M = 2.9, σ2 of AO closed loop t
PDF
10
5
0 0
0.1
0.2
0.3 0.4 0.5 residual tilt variance σ2 ( λ 2) t
0.6
0.7
0.8
Fig. 2. Comparison of the PDFs of the residual tilt variance obtained from the measured wavefront data and theoretical analysis.
As shown in Fig. 2, the theoretical analysis results strongly agree with the wavefront data for the closed loop AO system. In the same manner, the PDFs of the residual high-order wavefront error variance were calculated from the wavefront data and by the theoretical analysis of Eq. (6) with n = 32 degrees of freedom. Considering the finite-size samples, the PDF curves for the wavefront data were smoothed by using a moving average filter. The results are compared in Fig. 3. 30 M = 1.9, σ2 χ2(33) zh M = 1.9, σ2 of AO closed loop h
25
M = 2.9, σ2 χ2(33) zh M = 2.9, σ2 of AO closed loop h
PDF
20
15
10
5
0 0
0.02
0.04 0.06 0.08 0.1 residual high-order wavefront variance σ2 (λ 2)
0.12
0.14
h
Fig. 3. Comparison of the PDFs of the residual high-order wavefront variance obtained from the measured wavefront data and theoretical analysis.
The difference between the RMS values of high-order Zernike modes causes a mismatch between the PDF curves of the measured high-order wavefront variance and the theoretical results; the error may be due to the effect of a finite number of deformable mirror actuators. The fitting error becomes notable when the deformable mirror responds quickly. Ignoring the fitting error of the deformable mirror, the variance of the total residual wavefront error can be calculated by summing the variances of the residual tilt and high-order Zernike modes:
#256902 © 2016 OSA
Received 5 Jan 2016; revised 1 Feb 2016; accepted 1 Feb 2016; published 3 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.002818 | OPTICS EXPRESS 2822
σ 2 = σ t2 + σ h2 (7) According to Eq. (7), the probability of the residual wavefront variance can be described as a sum of two mutually independent χ 2 -distributed random variables, and its PDF is calculated by the convolution of the two PDFs of σ t2 and σ h2 :
(σ ) σ exp − f (σ ) = ( 2σ ) Γ ( n / 2 ) 2σ σ
n / 2 −1
2
2
2
h
2
n/2
2
0
=
h
zh
2
(
2
)
zh
2
n/2
2
2
zt
zt
σ exp − (σ Γ (n / 2) 2σ σ
2
1
2σ zt 2σ zh
2
1 σ −σ 2σ exp − 2σ 2
2
zt
2 h
0
)
n / 2 −1
2 h
d σ
2 h
(8)
1 1 exp − − σ d σ 2σ 2σ 2
2
zh
zt
2
2
h
h
if we define t = (1 2σ zh2 − 1 2σ zt2 ) σ h2 , Eq. (8) becomes:
( )=σ
f σ
1 1 2 − 2 n /2 Γ ( n / 2 ) 2σ zh 2σ zt
1
2
2 zt
( 2σ ) 2 zh
1 1 − 2 2 2 σ 2 σ zh zt σ zt2 ( 2σ zh2 ) n /2
ϒ n / 2,
=
−n/2
1 1 2 2 σ 2 − 2 σ 2 σ zh zt
σ 2 exp − 2 2σ zt
tn
2 −1
exp ( − t ) dt
0
(9)
2 −n/2 σ 2 1 − 1 exp − σ 2σ 2 2σ 2 2σ 2 zh zt zt
where ϒ ( ⋅) is the lower incomplete gamma function, defined as: x
1 t a −1 exp ( −t ) dt (10) Γ ( a ) 0 The PDFs of the residual wavefront variance are calculated from Eq. (9), and the AO system closed loop data. The calculation results are shown in Fig. 4. ϒ ( a, x ) =
10
M M M M
9 8
= 1.9, = 1.9, = 2.9, = 2.9,
theoratical result AO closed loop theoratical result AO closed loop
7
PDF
6 5 4 3 2 1 0 0
0.1
0.2 0.3 0.4 residual wavefront variance σ2 (λ 2)
0.5
0.6
Fig. 4. Comparison of the PDFs of the residual wavefront variance obtained from the measured wavefront data and theoretical analysis.
According to the results in Fig. 4, the theoretical analysis of the PDF of the residual wavefront variance agrees well with the closed loop AO system data. In addition, in the worst case of the 127 elements AO compensation analyzed in this work, the larger residual
#256902 © 2016 OSA
Received 5 Jan 2016; revised 1 Feb 2016; accepted 1 Feb 2016; published 3 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.002818 | OPTICS EXPRESS 2823
wavefront variance is mainly produced owing to the insufficient bandwidth of the tilt control loop. The value of the lower incomplete gamma function ϒ ( ⋅) in Eq. (9) quickly approaches 1 with increasing the wavefront variance. Then, the instantaneous wavefront variance of the AO system is simply described as an exponential function for large wavefront variance values that are of concern for many researchers: −n/ 2
1 1 2 − 2 2 2 σ σ zt f (σ 2 ) = 2zh σ zt ( 2σ zh2 ) n / 2
σ2 exp − 2 2σ zt
(11)
3. Probability distribution of the Strehl ratio
Researchers also adopt the Strehl ratio for evaluating the AO system performance because it is related to the wavefront variance after compensation. Instantaneous Strehl ratios and longterm exposure Strehl ratios can all be used for evaluating the AO system performance, depending on the AO application scenarios. The instantaneous Strehl ratio is determined by the residual high-order wavefront error as: S = exp ( −σ h2 )
(12)
Considering that Eq. (12) as a monotonic function, the PDF of the instantaneous Strehl ratio can be easily calculated from Eq. (6): f ( S ) = fσ 2 ( − ln S ) h
d ( − ln S ) dS
ln S ( − ln S ) = 2 n / 2 −1 exp 2 σ zh 2 Γ ( n / 2) S 2σ zh n / 2 −1
1
(13)
The long-term exposure Strehl ratio, Sl, is affected by the total residual wavefront error, and can be calculated as the expected value of the residual high-order wavefront error and the residual angular tilt [9]: Sl =
exp − E (σ h2 ) 2.22σ α d 1+ λ
2
(14)
where E ( ) denotes the expected value of a random variable, σ α is the RMS of the angular tilt in the X or Y directions. On the other hand, we know that the angular tilt α obeys the Rayleigh distribution, and its PDF can be written as: f (α ) =
α2 α exp − 2 σα 2σ α
(15)
The angular tilt variance is: Var (α ) = 0.429σ α2 (16) Considering that the relationship between the variance of the residual tilt and the tilt angle is:
π dα σ = 2λ 2 t
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2
(17)
Received 5 Jan 2016; revised 1 Feb 2016; accepted 1 Feb 2016; published 3 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.002818 | OPTICS EXPRESS 2824
and the variance of a random variable is defined as: Var (X) = E X − E ( X )
2
(18)
the expected value of the residual tilt variance is computed from Eqs. (15)–(18) as: 2
2
πd πd 2 2 E (σ t2 ) = E (α ) = Var (α ) + E (α ) 2λ 2λ 2
π 2 2.22σ α d πd 2 = 0.429σ α + σ α = 2 λ 2λ Thus, Eq. (14) can be rewritten as: Sl =
2
exp − E (σ h2 )
(19)
(20)
1 + E (σ t2 )
The denominator in Eq. (20) equals to the first order approximation of Maclaurin’s series of exp E (σ t2 ) . Thus, the long-term exposure Strehl ratio can be written as: Sl = exp − E (σ t2 ) exp − E (σ h2 ) = exp − E (σ 2 )
(21)
Therefore, the long-term exposure Strehl ratio is determined by the expected value of the total residual wavefront error after AO compensation. Obviously, each sample of σ 2 has the same expected value and variance. According to the central limit theorem, for a finite number N of frames with σ 2 , the arithmetic mean 1 N σ 2 = σ k2 will be approximately Gaussian-distributed. The expected value and variance N k =1 of σ 2 are:
(
E σ2
(
Var σ 2
) = E (σ ) 2
(22)
) = N1 Var (σ ) 2
(23)
From Eq. (2) and Eq. (5), the temporal variance of the residual wavefront variance is: f Var (σ ) = 4σ = 23 TG f t −3dB 2 t
4
4 zt
(24)
and: Var (σ h2 ) = 2nσ zh4 =
10 3
2 fG n f d − 3dB
(25)
and: Var (σ 2 ) = Var (σ t2 ) + Var (σ h2 ) Then, the PDF of the wavefront variance average σ
2
(26)
for a finite number N of frames is
approximately a Gaussian function:
#256902 © 2016 OSA
Received 5 Jan 2016; revised 1 Feb 2016; accepted 1 Feb 2016; published 3 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.002818 | OPTICS EXPRESS 2825
f
(σ ) 2
= exp − δ 2π 1
(σ
−μ
2
2δ 2
)
2
(27)
where μ = E (σ 2 ) and δ 2 = Var (σ 2 ) N . Here, we emphasize that the expected value of a random variable is calculated for an infinite-size samples, while the average of the variable is calculated for a finite-size samples. Hence, from Eq. (21) and Eq. (27) the N frames long-term exposure Strehl ratio is lognormally distributed: ( ln S nl + μ )2 (28) exp − 2δ 2 2πδ For the wavefront data for M = 2.9 and M = 1.9, the PDFs of the long-term exposure Strehl ratios are calculated from the measured wavefront data and by the theoretical analysis of Eq. (28) for N = 10. The wavelength was changed from 0.55μm to 1.55μm for calculating the PDF curves, and the curves were smoothed by using a moving average filter. The results are compared in Fig. 5. f ( Snl ) =
Snl
8 M M M M
7 6
= = = =
1.9, 1.9, 2.9, 2.9,
lognormal distribution AO closed loop lognormal distribution AO closed loop
PDF
5 4 3 2 1 0 0.2
0.3
0.4 0.5 0.6 0.7 N frames long-exposure Strehl ratio S nl
0.8
0.9
Fig. 5. Comparison of the long-term exposure Strehl ratios obtained from the measured wavefront data and theoretical analysis for N = 10.
The long-term exposure Strehl ratios calculated from the wavefront data were in a good agreement with the theoretical analysis results. The dispersion of the long-term exposure Strehl ratios decreased quickly with increasing N, and theoretically reached an asymptote for N>100. 4. Application of the calculated probability results
The probability density function of the residual wavefront variance provides novel methods for determining the AO system bandwidth and evaluating the AO system performance. By combining the results with the power spectrum method, the relationship between the PDFs of the residual wavefront variance and the control loop bandwidths of the tilt and deformable mirrors can be obtained. Furthermore, a novel availability index can be introduced for evaluating the AO system performance and for determining to what extent a certain AO system satisfies the requirement of its “parent” optical system.
#256902 © 2016 OSA
Received 5 Jan 2016; revised 1 Feb 2016; accepted 1 Feb 2016; published 3 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.002818 | OPTICS EXPRESS 2826
4.1 Determining the control loop bandwidth based on the probability analysis For a tilt-control system with the bandwidth (3 dB cut-off frequency) of ft-3dB, the angular tilt variance after compensation is [9]: 2
f λ Var (α ) = TG f t − 3dB d
2
(29)
where fTG is the tilt Greenwood frequency [9,19] and d and λ denote the beam diameter and the wavelength, respectively. From Eq. (16) and Eq. (29), the angular tilt variance related to the AO system parameters is: 1 λ fTG σα = 0.429 d ft − 3dB In addition, Eq. (19) can be rewritten as:
2
2
2
(30)
2
πd πd 2 2 E (σ ) = E (α ) = Var (α ) + E (α ) 2λ 2λ 2 t
2
fTG π 2 πd 2 = 0.429σ α + σ α = 11.5 2 2λ ft − 3dB On the other hand, from Eq. (2) we have: E (σ t2 ) = 2σ zt2
2
(31)
(32)
Finally, the parameter σ can be determined directly from the tilt Greenwood frequency and the bandwidth of the tilt mirror control loop, and is: 2 zt
2
f σ = 5.75 TG (33) f t −3dB In the same manner, the expected value of the higher-order wavefront variance limited by the insufficient bandwidth of the deformable mirror control loop, E (σ h2 ) , is [9]: 2 zt
f E (σ ) = G f d −3dB
53
2 h
(34)
where fd-3dB is the bandwidth of the deformable mirror control loop, and fG is the Greenwood frequency [15]. According to Eq. (5), we have: E (σ h2 ) = nσ zh2
(35)
The parameter σ zh2 can be determined directly from the Greenwood frequency and the bandwidth of the deformable mirror control loop, and is: 53
1 f σ = G (36) n f d −3dB The relationship between the PDF of the residual wavefront variance and the AO system bandwidth could be obtained by combining Eq. (9) with Eq. (33) and Eq. (36). This allows obtaining the PDF pattern of the AO wavefront variance, rather than the average variance. 2 zh
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Received 5 Jan 2016; revised 1 Feb 2016; accepted 1 Feb 2016; published 3 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.002818 | OPTICS EXPRESS 2827
Furthermore, we can design the PDF pattern by controlling the control loop bandwidth of the tilt mirror and deformable mirror, which yields a better analysis of the AO system performance. 4.2 Estimating the availability of an AO system Usually, the extent to which an AO system is in an operable state of satisfying the requirement of its “parent” optical system is implicit in its design stage. Using the probability distribution of the residual wavefront variance, we can introduce the concept of availability for more precisely characterizing this issue; this would help to evaluate the price-performance ratio of an AO system. Simply put, the availability is the fraction of time a system spends in a functioning condition, and its value is usually computed as the ratio of the up time over the total time of a system [20]. The result is often referred to as a “mission capable” rate. Thus, the availability of an AO system could be calculated by integrating the corresponding PDF in the interval between zero and a given wavefront variance threshold, C, determined by the “parent” optical system. It could be defined as: C
A = f ( x)dx
(37)
0
Here, we only discuss the estimation of the system availability by using the residual wavefront variance variable. Estimations using the Strehl ratio can be transformed into the form of Eq. (37). Based on the aforementioned definition of availability, the common imaging AO system maybe a low-availability system, with the availability of 1%, implying that 1% of images are effective; this availability may be sufficient for satisfying the requirement for getting clear details of stars. For a laser power transmitting system, 90% availability may be desired. For a high-speed laser communication system, the availability should reach an extremely high level, i.e., “five nines” high availability standard. If the variable x in Eq. (37) denotes the instantaneous wavefront variance, the instantaneous availability of an AO system can be calculated by substituting Eq. (9) into Eq. (37). From the M = 2.9 and M = 1.9 data, the availability values owing to the insufficient bandwidth for different wavefront variance thresholds are calculated from Eq. (9) and Eq. (37). The 1-A curves are shown in Fig. 6. 0
10
M = 1.9 M = 2.9
-2
1-A
10
-4
10
-6
10
-8
10
0
0.2
0.4
0.6
0.8
1
1.2
residual wavefront variance threshold C (λ 2)
Fig. 6. The estimated availability of the 127 elements AO system, for different wavefront variance thresholds. at the 0.55μm wavelength. The availability A was calculated from Eq. (9) and Eq. (37).
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Received 5 Jan 2016; revised 1 Feb 2016; accepted 1 Feb 2016; published 3 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.002818 | OPTICS EXPRESS 2828
Only by considering the effect of the insufficient bandwidth on the residual wavefront error, the value of 1-A is log-linear with respect to the variance threshold, with the slope −1 ( 2σ zt2 ) for large wavefront variances, in accord with Eq. (11). From the results in Fig. 6, the availability of the AO system dramatically decreases with decreasing the wavefront variance threshold. By averaging the M = 2.9 and M = 1.9 availability, the instantaneous availability values for the 127 elements AO system are 36% and 1.35% for C = 0.1λ 2 and 0.05λ 2 , respectively. On the other hand, the requirement of high availability may impose significant challenges on the AO system. For example, the mean wavefront variance of 1 rad2 suits state-of-the-art imaging AO systems, but building an AO system for optical communication, which requires the wavefront variance probability to be under 10−6 for variances above 1 rad2 may be technically and financially challenging. If the variable x in Eq. (37) denotes the wavefront variance average, by normalizing the PDF of σ 2 , the N frames long-term exposure availability of an AO system can be calculated as: 1 A = P N
N
σ i =1
2 i
C−μ < C ≈ Φ N δ
(38)
where Φ ( ⋅) is the cumulative distribution function of a standard normal distribution. Finally, by combining Eqs. (24)–(26), Eq. (31), Eq. (34), and Eq. (38), the long-term exposure availability of an AO system can be estimated directly by using the AO system parameters. 5. Conclusions and discussion
The probability analysis of an AO system can contribute to a better analysis of the AO system performance, compared with the power spectrum method. We have derived analytical formulas for the PDFs of the residual wavefront variance and the Strehl ratio, measured instantaneously and in the long term, subject to the insufficient control loop bandwidth. The formulas are quite useful for designing and probabilistically evaluating AO systems. For many applications, the availability of an AO system can be calculated by using these PDFs. The results derived in this paper should be considered as a conservative estimate, based on the assumptions of isotropy and uniformity of the Kolmogorov atmosphere that limit applicability to practical scenarios. More comprehensive probability-based studies of AO systems should be performed in future. The error sources, such as the fitting error of a deformable mirror, the Hartmann sensor noise, the angular anisoplanatic aberration, and the non-Kolmogorov turbulence should be investigated together. The availability provides a better link between the AO system and its “parent” optical system. To improve the availability of an AO system, decreasing the residual wavefront error and relaxing the budget of the “parent” optical system can help. The tradeoff among all factors, such as technical parameters, system requirements, time and money constraints, should be carefully considered for different AO applications because of the complexity of system engineering. Acknowledgments
We gratefully acknowledge the suggestions of Professor Wenhan Jian, Institute of Optics and Electronics, Chinese Academy of Sciences. The work is supported by National Natural Science Foundation of China (NSFC) (No.61405023).
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Received 5 Jan 2016; revised 1 Feb 2016; accepted 1 Feb 2016; published 3 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.002818 | OPTICS EXPRESS 2829
School of Astronautics and Aeronautics, University of Electronic Science and Technology of China, Chengdu 610054, China 2 The Key Laboratory on Adaptive optics, Chinese Academy of Sciences, Chengdu 610209, China 3 Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu 610209, China * [email protected]
Abstract: For performance evaluation of an adaptive optics (AO) system, the probability of the system residual wavefront variance can provide more information than the wavefront variance average. By studying the Zernike coefficients of an AO system residual wavefront, we derived the exact expressions for the probability density functions of the wavefront variance and the Strehl ratio, for instantaneous and long-term exposures owing to the insufficient control loop bandwidth of the AO system. Our calculations agree with the residual wavefront data of a closed loop AO system. Using these functions, we investigated the relationship between the AO system bandwidth and the distribution of the residual wavefront variance. Additionally, we analyzed the availability of an AO system for evaluating the AO performance. These results will assist in designing and probabilistic analysis of AO systems. ©2016 Optical Society of America OCIS codes: (010.1080) Active or adaptive optics; (010.1330) Atmospheric turbulence; (000.5490) Probability theory.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
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13–23 (2005). 17. H. Jian, D. Ke, L. Chao, Z. Peng, J. Dagang, and Y. Zhoushi, “Effectiveness of adaptive optics system in satellite-to-ground coherent optical communication,” Opt. Express 22(13), 16000–16007 (2014). 18. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66(3), 207–211 (1976). 19. G. A. Tyler, “Bandwidth considerations for tracking through turbulence,” J. Opt. Soc. Am. A 11(1), 358–367 (1994). 20. R. F. Stapelberg, Handbook of Reliability, Availability, Maintainability and Safety in Engineering Design (Springer, 2009).
1. Introduction When wavefront disturbances become unavoidable, adaptive optics (AO) helps improve the performance of its “parent” optical system owing to its ability to correct the wavefront distortion. As a result of this, AO systems have been widely used in many fields, such as astronomical imaging [1,2], human retina imaging [3,4], free-space laser transmission and atmospheric communication [5–7], and laser fusion [8]. In conventional AO, the limitations on phase conjugation stem from many sources. The wavefront measurement quality, the insufficient control loop bandwidth, the fitting errors associated with deformable mirrors, and the wavefront surface contour reproduction quality all affect the phase-conjugation process [9], with the limitation of the insufficient control loop bandwidth of an AO system closed loop being the most significant. Many researchers have developed expressions relating the parameters of AO systems to their residual wavefront and Strehl ratio [10–15], and the wavefront variance average owing to the insufficient control loop bandwidth, obtained by using the power spectrum method, is widely used for design and analysis of AO systems. However, the power spectrum method does not yield the temporal probability of the residual wavefront variance, which is extremely important for calculating the availability of an AO system in its “parent” optical system, especially for the “parent” systems for which the information acquisition bandwidth is larger than the bandwidth of an AO system closed loop. The work represented here is mainly motivated by the studies of AO systems for improving the performance of optical atmospheric communication, in which one frame with insufficient wavefront compensation would yield an error in the order of millions of bits. Different from the AO systems that are used for imaging, the evaluation and design of an AO system to be used in high-speed optical communication applications must be based on the probability distribution of the wavefront variance of the AO system [16,17]. Fundamentally, the results of the probability analysis of AO systems should hold for all AO systems and applications. In the following part of this paper, we describe our studies on the probability density function (PDF) of the residual wavefront variance, PDF of the Strehl ratio of an AO system, and AO system availability. 2. Probability of the AO residual wavefront variance The probability of the residual wavefront variance of an AO system in the time domain provides more information on the AO system performance than the average index. The probability of the residual wavefront variance could be obtained by calculating the statistical characteristics of the Zernike coefficients of the residual wavefront error, for two scenarios: 1) calculations for instantaneous exposure of all wavefront error frames, and 2) calculations for long-term exposure, obtained by summing over many wavefront error frames. If we assume that the atmosphere is locally isotropic and uniform, the Zernike coefficients of the residual wavefront error of an AO system must obey the Gaussian distribution owing to the insufficient bandwidth (3 dB cut-off frequency) of AO control loops. In addition, the bandwidth of a tilt mirror loop is lower than the bandwidth of a deformable mirror loop, owing to the lower response frequency of the tilt mirror, which causes the root mean square (RMS) of the Zernike coefficients of the residual tilt to be larger than that of other higherorder terms. Therefore, the probability of the residual wavefront variance can be more efficiently analyzed by independently considering the tilt error and the high-order error.
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Received 5 Jan 2016; revised 1 Feb 2016; accepted 1 Feb 2016; published 3 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.002818 | OPTICS EXPRESS 2819
As the X- and Y-axes can be chosen arbitrarily, the Zernike coefficients of residual tilts in the X and Y directions have the same variance σ zt2 for isotropic and uniform atmosphere in the long term. Then, the Zernike coefficients of the residual tilts in the X and Y directions, a2 and a3, obey the same Gaussian distribution N ( 0, σ zt2 ) . As a result, the variance of the instantaneous residual tilt can be expressed as:
σ t2 = a22 + a32 From the statistical theory, the quantity σ with two degrees of freedom:
2 t
σ
2 zt
(1) is distributed as a χ random variable 2
σ t2 σ zt2 = ( a22 + a32 ) σ zt2 ~ χ 2 ( 2 )
(2)
The PDF of σ t2 is: σ2 exp − t 2 (3) 2σ 2σ zt The probability of the residual higher-order wavefront error can be obtained in the same manner. The power spectra of all high-order Zernike modes for the Kolmogorov turbulence are constant for low frequencies, and follow the same −17/3 power law for high frequencies, and the variance of each higher mode is nearly the same [18]. Therefore, if these higher-order modes are corrected by the same deformable mirror loop, the residual high-order Zernike modes must have the same variance σ zh2 owing to the insufficient control loop bandwidth when the fitting error of the deformable mirror is ignored. Then, the Zernike coefficients of the higher-order residuals ak obey the same Gaussian distribution N ( 0, σ zh2 ) . Therefore, the 1
f (σ t2 ) =
2 zt
variance of instantaneous residual higher-order wavefront can be expressed as: n+3
σ h2 = ak2
(4)
k =4
Then, the quantity σ h2 σ zh2 is distributed as a χ 2 random variable: n
σ h2 σ zh2 = ak2 σ zt2 ~ χ 2 ( n )
(5)
k =4
and the PDF of σ h2 is:
(σ ) σ f (σ ) = exp − ( 2σ ) Γ ( n / 2 ) 2σ 2 h
2 n / 2 −1 h
2 zh
n/ 2
2 h 2 zh
(6)
where n is the number of the degrees of freedom of the χ 2 distribution, defined by the number of higher-order Zernike modes that reconstruct the residual wavefront, and Γ ( ⋅) is the gamma function. The probability of the variance of the residual wavefront error could be examined by using the results of an AO system. Here, we analyzed the probability pattern of the residual wavefront variance of an AO system composed of 127 elements, for a 1.8-m-diameter telescope operated by the Adaptive Optics Laboratory of Chinese Academy of Sciences. This AO system has two closed loops for independently controlling its tilt and deformable mirrors, and the frame frequency of the Hartmann sensor is 2000 Hz at the wavelength of 0.55 µm. Each measurement consisted of the 10000 frames wavefront data acquired during 5 seconds for different star beacons. The measurement conditions are listed in Table 1.
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Received 5 Jan 2016; revised 1 Feb 2016; accepted 1 Feb 2016; published 3 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.002818 | OPTICS EXPRESS 2820
Table 1. Measurement Conditions Star number
Star Magnitude M
elevation angle (degree)
Spectrum type
Fried parameter r0 (cm)
HIP28360 HIP25423 HIP23015 HIP18532 HIP14576
1.9 1.6 2.69 2.9 2.09
65 63 55 42 33
ARV B73 K32 B0.5 B8V
11.8 9.3 8.5 7.9 7.2
From the residual wavefront error, the distributions of the Zernike coefficients have been calculated for different conditions; in these calculations, the averages of the Zernike coefficients have been firstly removed as static errors. The calculated histograms of the Zernike coefficients for the different conditions reveal Gaussian distributions, which implies that the residual wavefront error is mainly caused by the insufficient bandwidth. The RMS values for the 35 Zernike coefficients of residual wavefront errors are shown in Fig. 1.
RMS of coefficients (λ)
1
M = 2.09 r = 7.2 0
0.9
M = 2.9
0.8
M = 2.69 r = 8.5
r = 7.9 0
0
0.7
M = 1.6
r = 9.3
M = 1.9
r = 11.8
0 0
0.6 0.5 0.4 0.3 0.2 0.1 0 0
5
10 15 20 25 Zernike terms of residual wavefront error
30
35
Fig. 1. RMS values for the Zernike coefficients of the AO system.
Our findings are as follows. First, the RMS values of tilt X (second term) and tilt Y (third term) are larger than the other higher-order terms, mainly stemming from the lower bandwidth owing to the lower response frequency of the tilt mirror. Second, the RMS of tilt X (second term) is smaller than that of tilt Y (third term), which may be due to the lateral wind in the Y direction, yielding a stronger high-frequency spectral component for tilt Y; in our measurements, the turbulence was not strictly isotropic and uniform. Third, the RMS values of higher-order terms are nearly the same and independent of the turbulence strength; apparently, these high-order residuals arise mainly owing to the 3 dB cut-off frequency of the deformable mirror loop. In these five data sets of the Zernike coefficients, the data for M = 2.9 and M = 1.9 cases are accordant with the assumptions of isotropy and uniformity that we want to discuss in this paper. Thus, it is reasonable to validate Eq. (3) and Eq. (6) by using only the data for M = 2.9 and M = 1.9. For the M = 2.9 data, we calculated σ zt = 0.179λ and σ zh = 0.0415λ ; for the M = 1.9 data, we obtained σ zt = 0.205λ and σ zh = 0.0431λ . The PDFs of the residual tilt variance were calculated from the wavefront data and by the theoretical analysis of Eq. (3). The results are compared in Fig. 2.
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Received 5 Jan 2016; revised 1 Feb 2016; accepted 1 Feb 2016; published 3 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.002818 | OPTICS EXPRESS 2821
15 M = 1.9, σ2 χ2(2) zt M = 1.9, σ2 of AO closed loop t M = 2.9, σ2 χ2(2) zt M = 2.9, σ2 of AO closed loop t
10
5
0 0
0.1
0.2
0.3 0.4 0.5 residual tilt variance σ2 ( λ 2) t
0.6
0.7
0.8
Fig. 2. Comparison of the PDFs of the residual tilt variance obtained from the measured wavefront data and theoretical analysis.
As shown in Fig. 2, the theoretical analysis results strongly agree with the wavefront data for the closed loop AO system. In the same manner, the PDFs of the residual high-order wavefront error variance were calculated from the wavefront data and by the theoretical analysis of Eq. (6) with n = 32 degrees of freedom. Considering the finite-size samples, the PDF curves for the wavefront data were smoothed by using a moving average filter. The results are compared in Fig. 3. 30 M = 1.9, σ2 χ2(33) zh M = 1.9, σ2 of AO closed loop h
25
M = 2.9, σ2 χ2(33) zh M = 2.9, σ2 of AO closed loop h
20
15
10
5
0 0
0.02
0.04 0.06 0.08 0.1 residual high-order wavefront variance σ2 (λ 2)
0.12
0.14
h
Fig. 3. Comparison of the PDFs of the residual high-order wavefront variance obtained from the measured wavefront data and theoretical analysis.
The difference between the RMS values of high-order Zernike modes causes a mismatch between the PDF curves of the measured high-order wavefront variance and the theoretical results; the error may be due to the effect of a finite number of deformable mirror actuators. The fitting error becomes notable when the deformable mirror responds quickly. Ignoring the fitting error of the deformable mirror, the variance of the total residual wavefront error can be calculated by summing the variances of the residual tilt and high-order Zernike modes:
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Received 5 Jan 2016; revised 1 Feb 2016; accepted 1 Feb 2016; published 3 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.002818 | OPTICS EXPRESS 2822
σ 2 = σ t2 + σ h2 (7) According to Eq. (7), the probability of the residual wavefront variance can be described as a sum of two mutually independent χ 2 -distributed random variables, and its PDF is calculated by the convolution of the two PDFs of σ t2 and σ h2 :
(σ ) σ exp − f (σ ) = ( 2σ ) Γ ( n / 2 ) 2σ σ
n / 2 −1
2
2
2
h
2
n/2
2
0
=
h
zh
2
(
2
)
zh
2
n/2
2
2
zt
zt
σ exp − (σ Γ (n / 2) 2σ σ
2
1
2σ zt 2σ zh
2
1 σ −σ 2σ exp − 2σ 2
2
zt
2 h
0
)
n / 2 −1
2 h
d σ
2 h
(8)
1 1 exp − − σ d σ 2σ 2σ 2
2
zh
zt
2
2
h
h
if we define t = (1 2σ zh2 − 1 2σ zt2 ) σ h2 , Eq. (8) becomes:
( )=σ
f σ
1 1 2 − 2 n /2 Γ ( n / 2 ) 2σ zh 2σ zt
1
2
2 zt
( 2σ ) 2 zh
1 1 − 2 2 2 σ 2 σ zh zt σ zt2 ( 2σ zh2 ) n /2
ϒ n / 2,
=
−n/2
1 1 2 2 σ 2 − 2 σ 2 σ zh zt
σ 2 exp − 2 2σ zt
tn
2 −1
exp ( − t ) dt
0
(9)
2 −n/2 σ 2 1 − 1 exp − σ 2σ 2 2σ 2 2σ 2 zh zt zt
where ϒ ( ⋅) is the lower incomplete gamma function, defined as: x
1 t a −1 exp ( −t ) dt (10) Γ ( a ) 0 The PDFs of the residual wavefront variance are calculated from Eq. (9), and the AO system closed loop data. The calculation results are shown in Fig. 4. ϒ ( a, x ) =
10
M M M M
9 8
= 1.9, = 1.9, = 2.9, = 2.9,
theoratical result AO closed loop theoratical result AO closed loop
7
6 5 4 3 2 1 0 0
0.1
0.2 0.3 0.4 residual wavefront variance σ2 (λ 2)
0.5
0.6
Fig. 4. Comparison of the PDFs of the residual wavefront variance obtained from the measured wavefront data and theoretical analysis.
According to the results in Fig. 4, the theoretical analysis of the PDF of the residual wavefront variance agrees well with the closed loop AO system data. In addition, in the worst case of the 127 elements AO compensation analyzed in this work, the larger residual
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Received 5 Jan 2016; revised 1 Feb 2016; accepted 1 Feb 2016; published 3 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.002818 | OPTICS EXPRESS 2823
wavefront variance is mainly produced owing to the insufficient bandwidth of the tilt control loop. The value of the lower incomplete gamma function ϒ ( ⋅) in Eq. (9) quickly approaches 1 with increasing the wavefront variance. Then, the instantaneous wavefront variance of the AO system is simply described as an exponential function for large wavefront variance values that are of concern for many researchers: −n/ 2
1 1 2 − 2 2 2 σ σ zt f (σ 2 ) = 2zh σ zt ( 2σ zh2 ) n / 2
σ2 exp − 2 2σ zt
(11)
3. Probability distribution of the Strehl ratio
Researchers also adopt the Strehl ratio for evaluating the AO system performance because it is related to the wavefront variance after compensation. Instantaneous Strehl ratios and longterm exposure Strehl ratios can all be used for evaluating the AO system performance, depending on the AO application scenarios. The instantaneous Strehl ratio is determined by the residual high-order wavefront error as: S = exp ( −σ h2 )
(12)
Considering that Eq. (12) as a monotonic function, the PDF of the instantaneous Strehl ratio can be easily calculated from Eq. (6): f ( S ) = fσ 2 ( − ln S ) h
d ( − ln S ) dS
ln S ( − ln S ) = 2 n / 2 −1 exp 2 σ zh 2 Γ ( n / 2) S 2σ zh n / 2 −1
1
(13)
The long-term exposure Strehl ratio, Sl, is affected by the total residual wavefront error, and can be calculated as the expected value of the residual high-order wavefront error and the residual angular tilt [9]: Sl =
exp − E (σ h2 ) 2.22σ α d 1+ λ
2
(14)
where E ( ) denotes the expected value of a random variable, σ α is the RMS of the angular tilt in the X or Y directions. On the other hand, we know that the angular tilt α obeys the Rayleigh distribution, and its PDF can be written as: f (α ) =
α2 α exp − 2 σα 2σ α
(15)
The angular tilt variance is: Var (α ) = 0.429σ α2 (16) Considering that the relationship between the variance of the residual tilt and the tilt angle is:
π dα σ = 2λ 2 t
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2
(17)
Received 5 Jan 2016; revised 1 Feb 2016; accepted 1 Feb 2016; published 3 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.002818 | OPTICS EXPRESS 2824
and the variance of a random variable is defined as: Var (X) = E X − E ( X )
2
(18)
the expected value of the residual tilt variance is computed from Eqs. (15)–(18) as: 2
2
πd πd 2 2 E (σ t2 ) = E (α ) = Var (α ) + E (α ) 2λ 2λ 2
π 2 2.22σ α d πd 2 = 0.429σ α + σ α = 2 λ 2λ Thus, Eq. (14) can be rewritten as: Sl =
2
exp − E (σ h2 )
(19)
(20)
1 + E (σ t2 )
The denominator in Eq. (20) equals to the first order approximation of Maclaurin’s series of exp E (σ t2 ) . Thus, the long-term exposure Strehl ratio can be written as: Sl = exp − E (σ t2 ) exp − E (σ h2 ) = exp − E (σ 2 )
(21)
Therefore, the long-term exposure Strehl ratio is determined by the expected value of the total residual wavefront error after AO compensation. Obviously, each sample of σ 2 has the same expected value and variance. According to the central limit theorem, for a finite number N of frames with σ 2 , the arithmetic mean 1 N σ 2 = σ k2 will be approximately Gaussian-distributed. The expected value and variance N k =1 of σ 2 are:
(
E σ2
(
Var σ 2
) = E (σ ) 2
(22)
) = N1 Var (σ ) 2
(23)
From Eq. (2) and Eq. (5), the temporal variance of the residual wavefront variance is: f Var (σ ) = 4σ = 23 TG f t −3dB 2 t
4
4 zt
(24)
and: Var (σ h2 ) = 2nσ zh4 =
10 3
2 fG n f d − 3dB
(25)
and: Var (σ 2 ) = Var (σ t2 ) + Var (σ h2 ) Then, the PDF of the wavefront variance average σ
2
(26)
for a finite number N of frames is
approximately a Gaussian function:
#256902 © 2016 OSA
Received 5 Jan 2016; revised 1 Feb 2016; accepted 1 Feb 2016; published 3 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.002818 | OPTICS EXPRESS 2825
f
(σ ) 2
= exp − δ 2π 1
(σ
−μ
2
2δ 2
)
2
(27)
where μ = E (σ 2 ) and δ 2 = Var (σ 2 ) N . Here, we emphasize that the expected value of a random variable is calculated for an infinite-size samples, while the average of the variable is calculated for a finite-size samples. Hence, from Eq. (21) and Eq. (27) the N frames long-term exposure Strehl ratio is lognormally distributed: ( ln S nl + μ )2 (28) exp − 2δ 2 2πδ For the wavefront data for M = 2.9 and M = 1.9, the PDFs of the long-term exposure Strehl ratios are calculated from the measured wavefront data and by the theoretical analysis of Eq. (28) for N = 10. The wavelength was changed from 0.55μm to 1.55μm for calculating the PDF curves, and the curves were smoothed by using a moving average filter. The results are compared in Fig. 5. f ( Snl ) =
Snl
8 M M M M
7 6
= = = =
1.9, 1.9, 2.9, 2.9,
lognormal distribution AO closed loop lognormal distribution AO closed loop
5 4 3 2 1 0 0.2
0.3
0.4 0.5 0.6 0.7 N frames long-exposure Strehl ratio S nl
0.8
0.9
Fig. 5. Comparison of the long-term exposure Strehl ratios obtained from the measured wavefront data and theoretical analysis for N = 10.
The long-term exposure Strehl ratios calculated from the wavefront data were in a good agreement with the theoretical analysis results. The dispersion of the long-term exposure Strehl ratios decreased quickly with increasing N, and theoretically reached an asymptote for N>100. 4. Application of the calculated probability results
The probability density function of the residual wavefront variance provides novel methods for determining the AO system bandwidth and evaluating the AO system performance. By combining the results with the power spectrum method, the relationship between the PDFs of the residual wavefront variance and the control loop bandwidths of the tilt and deformable mirrors can be obtained. Furthermore, a novel availability index can be introduced for evaluating the AO system performance and for determining to what extent a certain AO system satisfies the requirement of its “parent” optical system.
#256902 © 2016 OSA
Received 5 Jan 2016; revised 1 Feb 2016; accepted 1 Feb 2016; published 3 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.002818 | OPTICS EXPRESS 2826
4.1 Determining the control loop bandwidth based on the probability analysis For a tilt-control system with the bandwidth (3 dB cut-off frequency) of ft-3dB, the angular tilt variance after compensation is [9]: 2
f λ Var (α ) = TG f t − 3dB d
2
(29)
where fTG is the tilt Greenwood frequency [9,19] and d and λ denote the beam diameter and the wavelength, respectively. From Eq. (16) and Eq. (29), the angular tilt variance related to the AO system parameters is: 1 λ fTG σα = 0.429 d ft − 3dB In addition, Eq. (19) can be rewritten as:
2
2
2
(30)
2
πd πd 2 2 E (σ ) = E (α ) = Var (α ) + E (α ) 2λ 2λ 2 t
2
fTG π 2 πd 2 = 0.429σ α + σ α = 11.5 2 2λ ft − 3dB On the other hand, from Eq. (2) we have: E (σ t2 ) = 2σ zt2
2
(31)
(32)
Finally, the parameter σ can be determined directly from the tilt Greenwood frequency and the bandwidth of the tilt mirror control loop, and is: 2 zt
2
f σ = 5.75 TG (33) f t −3dB In the same manner, the expected value of the higher-order wavefront variance limited by the insufficient bandwidth of the deformable mirror control loop, E (σ h2 ) , is [9]: 2 zt
f E (σ ) = G f d −3dB
53
2 h
(34)
where fd-3dB is the bandwidth of the deformable mirror control loop, and fG is the Greenwood frequency [15]. According to Eq. (5), we have: E (σ h2 ) = nσ zh2
(35)
The parameter σ zh2 can be determined directly from the Greenwood frequency and the bandwidth of the deformable mirror control loop, and is: 53
1 f σ = G (36) n f d −3dB The relationship between the PDF of the residual wavefront variance and the AO system bandwidth could be obtained by combining Eq. (9) with Eq. (33) and Eq. (36). This allows obtaining the PDF pattern of the AO wavefront variance, rather than the average variance. 2 zh
#256902 © 2016 OSA
Received 5 Jan 2016; revised 1 Feb 2016; accepted 1 Feb 2016; published 3 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.002818 | OPTICS EXPRESS 2827
Furthermore, we can design the PDF pattern by controlling the control loop bandwidth of the tilt mirror and deformable mirror, which yields a better analysis of the AO system performance. 4.2 Estimating the availability of an AO system Usually, the extent to which an AO system is in an operable state of satisfying the requirement of its “parent” optical system is implicit in its design stage. Using the probability distribution of the residual wavefront variance, we can introduce the concept of availability for more precisely characterizing this issue; this would help to evaluate the price-performance ratio of an AO system. Simply put, the availability is the fraction of time a system spends in a functioning condition, and its value is usually computed as the ratio of the up time over the total time of a system [20]. The result is often referred to as a “mission capable” rate. Thus, the availability of an AO system could be calculated by integrating the corresponding PDF in the interval between zero and a given wavefront variance threshold, C, determined by the “parent” optical system. It could be defined as: C
A = f ( x)dx
(37)
0
Here, we only discuss the estimation of the system availability by using the residual wavefront variance variable. Estimations using the Strehl ratio can be transformed into the form of Eq. (37). Based on the aforementioned definition of availability, the common imaging AO system maybe a low-availability system, with the availability of 1%, implying that 1% of images are effective; this availability may be sufficient for satisfying the requirement for getting clear details of stars. For a laser power transmitting system, 90% availability may be desired. For a high-speed laser communication system, the availability should reach an extremely high level, i.e., “five nines” high availability standard. If the variable x in Eq. (37) denotes the instantaneous wavefront variance, the instantaneous availability of an AO system can be calculated by substituting Eq. (9) into Eq. (37). From the M = 2.9 and M = 1.9 data, the availability values owing to the insufficient bandwidth for different wavefront variance thresholds are calculated from Eq. (9) and Eq. (37). The 1-A curves are shown in Fig. 6. 0
10
M = 1.9 M = 2.9
-2
1-A
10
-4
10
-6
10
-8
10
0
0.2
0.4
0.6
0.8
1
1.2
residual wavefront variance threshold C (λ 2)
Fig. 6. The estimated availability of the 127 elements AO system, for different wavefront variance thresholds. at the 0.55μm wavelength. The availability A was calculated from Eq. (9) and Eq. (37).
#256902 © 2016 OSA
Received 5 Jan 2016; revised 1 Feb 2016; accepted 1 Feb 2016; published 3 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.002818 | OPTICS EXPRESS 2828
Only by considering the effect of the insufficient bandwidth on the residual wavefront error, the value of 1-A is log-linear with respect to the variance threshold, with the slope −1 ( 2σ zt2 ) for large wavefront variances, in accord with Eq. (11). From the results in Fig. 6, the availability of the AO system dramatically decreases with decreasing the wavefront variance threshold. By averaging the M = 2.9 and M = 1.9 availability, the instantaneous availability values for the 127 elements AO system are 36% and 1.35% for C = 0.1λ 2 and 0.05λ 2 , respectively. On the other hand, the requirement of high availability may impose significant challenges on the AO system. For example, the mean wavefront variance of 1 rad2 suits state-of-the-art imaging AO systems, but building an AO system for optical communication, which requires the wavefront variance probability to be under 10−6 for variances above 1 rad2 may be technically and financially challenging. If the variable x in Eq. (37) denotes the wavefront variance average, by normalizing the PDF of σ 2 , the N frames long-term exposure availability of an AO system can be calculated as: 1 A = P N
N
σ i =1
2 i
C−μ < C ≈ Φ N δ
(38)
where Φ ( ⋅) is the cumulative distribution function of a standard normal distribution. Finally, by combining Eqs. (24)–(26), Eq. (31), Eq. (34), and Eq. (38), the long-term exposure availability of an AO system can be estimated directly by using the AO system parameters. 5. Conclusions and discussion
The probability analysis of an AO system can contribute to a better analysis of the AO system performance, compared with the power spectrum method. We have derived analytical formulas for the PDFs of the residual wavefront variance and the Strehl ratio, measured instantaneously and in the long term, subject to the insufficient control loop bandwidth. The formulas are quite useful for designing and probabilistically evaluating AO systems. For many applications, the availability of an AO system can be calculated by using these PDFs. The results derived in this paper should be considered as a conservative estimate, based on the assumptions of isotropy and uniformity of the Kolmogorov atmosphere that limit applicability to practical scenarios. More comprehensive probability-based studies of AO systems should be performed in future. The error sources, such as the fitting error of a deformable mirror, the Hartmann sensor noise, the angular anisoplanatic aberration, and the non-Kolmogorov turbulence should be investigated together. The availability provides a better link between the AO system and its “parent” optical system. To improve the availability of an AO system, decreasing the residual wavefront error and relaxing the budget of the “parent” optical system can help. The tradeoff among all factors, such as technical parameters, system requirements, time and money constraints, should be carefully considered for different AO applications because of the complexity of system engineering. Acknowledgments
We gratefully acknowledge the suggestions of Professor Wenhan Jian, Institute of Optics and Electronics, Chinese Academy of Sciences. The work is supported by National Natural Science Foundation of China (NSFC) (No.61405023).
#256902 © 2016 OSA
Received 5 Jan 2016; revised 1 Feb 2016; accepted 1 Feb 2016; published 3 Feb 2016 8 Feb 2016 | Vol. 24, No. 3 | DOI:10.1364/OE.24.002818 | OPTICS EXPRESS 2829
