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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 1 — Jan. 3, 2011
  • pp: 371–379
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Wavefront sensorless adaptive optics: a general model-based approach

Huang Linhai and Changhui Rao  »View Author Affiliations


Optics Express, Vol. 19, Issue 1, pp. 371-379 (2011)
http://dx.doi.org/10.1364/OE.19.000371


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Abstract

Wavefront sensorless adaptive optics (AO) systems have been widely studied in recent years. To reach optimum results, such systems require an efficient correction method. In this paper, a general model-based correction method for a wavefront sensorless AO system is presented. The general model-based approach is set up based on a relationship wherein the second moments (SM) of the wavefront gradients are approximately proportionate to the FWHM of the far-field intensity distribution. The general model-based method is capable of taking various common sets of functions as predetermined bias functions and correcting the aberrations by using fewer photodetector measurements. Numerical simulations of AO corrections of various random aberrations are performed. The results show that the Strehl ratio is improved from 0.07 to about 0.90, with only N + 1 photodetector measurement for the AO correction system using N aberration modes as the predetermined bias functions.

© 2010 OSA

1. Introduction

The model-based approach is a much better choice to reduce the measurements. M. J. Booth has proposed that the model-based approach is capable of correcting aberrations with a minimum of N + 1 photodetector measurements for N aberration modes [4

4. M. J. Booth, “Wavefront sensorless adaptive optics for large aberrations,” Opt. Lett. 32(1), 5–7 (2007). [CrossRef]

,5

5. M. J. Booth, “Wave front sensor-less adaptive optics: a model-based approach using sphere packings,” Opt. Express 14(4), 1339–1352 (2006). [CrossRef] [PubMed]

]. However, the model-based method requires taking different sets of functions as the predetermined bias functions for aberrations of various magnitudes. The approach requires taking Zernike functions as the predetermined bias functions for small aberrations and Lukosz–Zernike (L–Z) functions for large aberrations [4

4. M. J. Booth, “Wavefront sensorless adaptive optics for large aberrations,” Opt. Lett. 32(1), 5–7 (2007). [CrossRef]

,5

5. M. J. Booth, “Wave front sensor-less adaptive optics: a model-based approach using sphere packings,” Opt. Express 14(4), 1339–1352 (2006). [CrossRef] [PubMed]

]. Furthermore, the accuracy of this method relies upon factors such as bias values (the coefficients of the predetermined bias functions).

In this paper we propose a general model-based approach. Unlike the model-based method proposed by M. J. Booth, the general approach is insensitive to the selection of sets of functions as well as the bias values. Besides the L-Z functions, the general model-based approach can also take other kind of modes, such as the Zernike functions, as the predetermined bias functions and correct aberrations effectively, not only for large aberrations but also for small aberrations.

2. The general model-based method

2.1 The wavefront sensorless AO correction system

It is well known that a wavefront sensorless AO can be depicted as shown in Fig. 1
Fig. 1 Schematic diagram of the adaptive system.
, where the input wavefront is incident from the left. After aberration correction by the adaptive element (the DM), the input wavefront is focused onto the photodetector by a positive lens. A mark or a small pinhole is placed on the photodetector to generate a feedback signal, such as the encircled energy signal [4

4. M. J. Booth, “Wavefront sensorless adaptive optics for large aberrations,” Opt. Lett. 32(1), 5–7 (2007). [CrossRef]

]. The encircled energy feedback signal is produced by using a pinhole, which filters energy outside the pinhole and allows energy within the pinhole to be obtained by the photodetector. The feedback signal is used to drive the AO element. The aberration of the input wavefront is described by function Φ(x,y), where x and y are the rectangular coordinates in the pupil plane of the lens. The correcting aberration of the DM is depicted by function Ψ(x,y). The residual aberration of the input wavefront is represented by function R(x,y), and R(x,y)=Φ(x,y)Ψ(x,y). The corresponding far-field intensity distribution I(x',y') is given by [6

6. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1983).

]
I(x',y')=I0|A(x,y)exp[jR(x,y)]ejk2z[(xx')2+(yy')2]dxdy|2,
(1)
where x’ and y’ are the rectangular coordinates in the input plane of the photodetector; I0 is proportional to the incident light power; A is the amplitude of the input wavefront and is set to be uniform in this paper; k=2π/λ, λ is the wavelength of the input wavefront; z is the focal length of the positive lens; and j is the imaginary unit.

We assume that Φ and Ψ can be represented by a series of M orthonormal functions and N orthonormal functions, respectively. In most cases, the number M is greater than N. The orthonormal functions are called modes in the following text, each mode denoted by Fi(x,y):

Φ(x,y)=i=1MνiFi(x,y),Ψ(x,y)=i=1NμiFi(x,y).
(2)

In other words, aberration Φ of the input wavefront can be denoted by vector V, whose elements are the coefficients ofνi. Similarly, the correcting aberration and the residual aberration R can be represented by vectors U and Z, respectively. The elements of U and Z are μi and zi, and moreover, zi=νiμi. Minimizing vector Z is the job of this wavefront sensorless AO system.

In order to minimize vector Z efficiently, the relationship between the information about the far-field intensity distribution and the aberration need to be set up first.

2.2 Relationship between the second moment (SM) of the aberration gradients and the FWHM of far-field intensity distribution

[xΦ(x,y)]2+[yΦ(x,y)]2(x'2+y'2).
(3)

When TL→∞, TLF→∞, Eq. (4) is
xy{[xΦ(x,y)]2+[yΦ(x,y)]2}dxdyx'y'I(x',y')(x'2+y'2)dx'dy',
(5)
where the left-hand side of Eq. (5) is the SM of the aberration gradients, and the right-hand side of Eq. (5) is the sum of the far-field intensity distribution multiplied by a mask [corresponding position(x'2+y'2)]. Since the total sum of far-field intensity distribution I(x',y') is a constant, the right-hand side of Eq. (5) is changed to be

x'y'I(x',y')(x'2+y'2)dx'dy'=R2x'y'I(x',y')(x'2+y'2)R2dx'dy'=R2{x'y'I(x',y')dx'dy'x'y'I(x',y')[1r2R2]dx'dy'}.
(6)

Hence the mask becomes 1r2/R2 for r ≤ R and zero otherwise. r=x'2+y'2, R is a suitable chosen detector radius, and R is weighted by the system’s diffraction limitation (DL). The significant advantage of the new mask is that the new mask is insensitive to the actual detected size. In practice, the new mask could be implemented by the weighting of pixels within the software when the photodetector is replaced by a CCD camera.

Moreover, the whole expression of the right-hand side of Eq. (5) is also normalized by dividing by the sum of I(x',y'). For convenience, Eq. (7) is called the masked detector signal (MDS) in subsequent text.

MDS=x'y'I(x',y')[1r2R2]dx'dy'x'y'I(x',y')dx'dy'.
(7)

Therefore, Eq. (5) can be expressed as
SMc0(1MDS) ,
(8)
where SM is the second moment of the aberration gradients, and c0 is the slop of the trend line, which is determined by the detector radius R.

For comparison, the relationship used in [4

4. M. J. Booth, “Wavefront sensorless adaptive optics for large aberrations,” Opt. Lett. 32(1), 5–7 (2007). [CrossRef]

] is numerically calculated by using random aberrations and are drawn in Fig. 3
Fig. 3 Signal response between aberration magnitude |V| and MDS. Each point in the figure stands for the mean MDS values of 100 random aberrations.
. The relationship between the MDS and aberration magnitude |V| is calculated from 5000 random aberrations, and each circle in Fig. 3 stands for the mean MDS of 100 random aberrations. Note that the MDS is essentially the same as the detected signal used in [4

4. M. J. Booth, “Wavefront sensorless adaptive optics for large aberrations,” Opt. Lett. 32(1), 5–7 (2007). [CrossRef]

], except that the MDS is normalized by the sum of the far-field intensity. Obviously, our method results in a linear relationship between the MDS and the SM.

2.3 Modeling and analysis

So far, the relationship between the MDS and the SM of the aberration gradient has been set up. We will build the general model-based method for a wavefront sensorless AO system according to the relationship.

As we know that the aberration of an input wavefront can be expressed by a series of M orthonormal modesFi(x,y),

Φ(x,y)=i=1MνiFi(x,y).

The input aberration Φ’s gradients in the x and y axes are [9

9. W. J. Hardy, Adaptive Optics for Astronomical Telescopes, (Oxford Univ. Press, 1998).

]
xΦ(x,y)=i=1MνixFi(x,y),yΦ(x,y)=i=1MνiyFi(x,y),
(9)
where Φ(x,y)/x and Φ(x,y)/y are the input aberration gradients in the x and y axes, respectively, and Fi(x,y)/x and Fi(x,y)/y are the orthonormal mode gradients in the x and y axes, respectively.

In order to find out the value of coefficients vi, the N orthonormal mode Fi(x,y) is taken as the predetermined bias function and is added by the DM sequentially with coefficient α to the input aberrations. Then the detector measurements are recorded.

The difference wi,0 between the SM of the gradients of aberration Φ and that after adding a predetermined bias function Fi(x,y) with the coefficient α is declared as
wi,0=SMiSM0=s{[xΦ(x,y)+αxFi(x,y)]2+[yΦ(x,y)+αyFi(x,y)]2}{[xΦ(x,y)]2+[yΦ(x,y)]2}dxdys,
(10)
whereSM0andSMiare the SMs of the gradients of Φ(x,y) and Φ(x,y)+αFi(x,y), respectively. By rearranging, we can get

wi,0=s[αxFi(x,y)2xΦ(x,y)]dxdys+s[αxFi(x,y)]2dxdys+s[αyFi(x,y)2yΦ(x,y)]dxdys+s[αyFi(x,y)]2dxdys=s2α[xFi(x,y)xΦ(x,y)+yFi(x,y)yΦ(x,y)]dxdys+sα2{[xFi(x,y)]2+[yFi(x,y)]2}dxdys.
(11)

After N predetermined bias functions Fi(x, y) are added sequentially by the DM, we will obtain N equations

wi,0=s2α[xFi(x,y)xΦ(x,y)+yFi(x,y)yΦ(x,y)]dxdys+sα2{[xFi(x,y)]2+[yFi(x,y)]2}dxdys,i=1~N.
(12)

Replacing the gradients of aberration Ф by Eq. (9), it is convenient to use the vectors that represent the sampled values of the original functions. Equation (12) becomes
W=2α*S*V+α2*Sm,
(13)
where S=(s1,1s1,2...s1,Ns2,1s2,2.........sN1,NsN,1...sN,N1sN,N............s1,M...sN1,MsN,M), Sm=(s1,1s2,2...sN,N), W=(w1,0w2,0...sN,0), V=(ν1ν2...νN...νM), and

sn,m=s{[xFn(x,y)*xFm(x,y)]+[yFn(x,y)*yFm(x,y)]}dxdys.
(14)

Generally, S is invertible, so the coefficient V of the modes can be calculated by

V=S1(Wα2*Sm)2*α.
(15)

According to Eq. (8) and Eq. (10), we have
wi,0=SMiSM0c0(1MDSi)-c0(1MDS0)-c0(MDSi-MDS0),
where MDS0 and MDSi are the corresponding MDS of input wavefronts Ф(x, y) and Ф(x, y) + αFi(x, y), respectively. That is, Wc0M, thus the vector V is estimated by
VS1(c0*Mα2*Sm)2*α,
(16)
where M=(MDS1MDS0MDS2MDS0...MDSNMDS0).

Note that vector V can be resolved exactly by Eq. (15), no matter what sets of modes are adopted as the predetermined bias functions; hence, it can be concluded that the method is insensitive to the selected sets of modes as well as the choice of the bias value α, and the correction error comes only from Eq. (8). Furthermore, it is known that Eq. (8) is set up based on geometric optics and is satisfied with all kinds of aberrations as long as a suitable detector radius R is selected.

3. Numerical simulations and result analysis

To verify the performance of the general method, 50 random aberrations with a normalized root-meaning-square value of 0.5 λ are produced as the input wavefront aberrations in the paper. The aberrations consist of 18 Zernike modes represented by random coefficients ui. The 18 Zernike modes and 18 L–Z modes are taken sequentially to be the biases of the model-based method and added by the DM to the input wavefront.

The 18 Zernike modes and 18 L–Z modes are drawn in Fig. 4(a)
Fig. 4 Set of aberration modes: (a) 18 Zernike modes (3–20); (b) 18 L–Z modes (3–20).
and Fig. 4(b), respectively. The corresponding inverses S−1 are shown in Fig. 5(a)
Fig. 5 Corresponding inverse S−1 of sets of aberration modes in Fig. 4: The inverse S−1 of (a) 18 Zernike modes (3-20), (b) 18 L–Z modes (3-20).
and Fig. 5(b) respectively.

The biasesFi(x,y)with coefficient α=0.05λ are added sequentially to the aberrations, given the total aberrations U+V,UT=[u1,u2,...,u18],VT=[0,0,.α..,0], and the corresponding MDSi are calculated. The detector radius of far-field intensity R equals 16DL, and the slope of the trend line c0 is −194.1. After the total N biases are added, the correction vector V is calculated by Eq. (16): i = 1,2…,18.

The Strehl Ratio (SR) is adopted to evaluate the correction results, and the SR is defined as
SR=P[I(x,y)]P[I0(x,y)],
(17)
whereP[]is an operation, which calculates the peak intensity. I is the actual intensity distribution, and I0 is the intensity distribution when no aberrations are present.

Since the vector S−1, Sm, and MDS0 are known in advance and do not need recalculation or measurement for each correction, the total detector measurements for one aberration correction would be 19. One aberration correction is considered as one iteration in the following results.

The effect of bias values (the coefficient α) on correction results is under consideration. We take the L–Z modes and Zernike functions as the predetermined bias functions and change the value of coefficient α. The curves of such iterations are show in Fig. 7
Fig. 7 Results of AO correction using (a) L–Z and (b) Zernike modes when coefficientαvary from 0.02 to 0.4 λ. The icons “method 1” and “method 2” in the figure refer to the method proposed by M. J. Booth and the method proposed in this paper.
. As we have expected in the conclusions of Section 2, our method is insensitive to the choice of bias values.

4. Conclusions

A general model-based approach for wavefront sensorless adaptive optics is presented in the paper. The general model-based approach is set up based on the approximate linearity between the MDS and the SM of the wavefront gradient. Because the general method is insensitive to the selected modes, it permits correction of the aberrations by using all kinds of orthogonal aberrations as the predetermined bias functions. Numerical simulations of AO correction to the random aberrations have shown that the general method is a fast and stable wavefront sensorless AO correction method for all kind of modes.

Acknowledgments

The authors thank Wenham Jiang for providing good suggestions. The authors acknowledge helpful suggestions from the reviewers and help from the editors.

References and links

1.

B. Wang and M. J. Booth, “Optimum deformable mirror modes for sensorless adaptive optics,” Opt. Commun. 282(23), 4467–4474 (2009). [CrossRef]

2.

M. A. Vorontsov, G. W. Carhart, M. Cohen, and G. Cauwenberghs, “Adaptive optics based on analog parallel stochastic optimization: analysis and experimental demonstration,” J. Opt. Soc. Am. A 17(8), 1440–1453 (2000). [CrossRef]

3.

P. Piatrou and M. Roggemann, “Beaconless stochastic parallel gradient descent laser beam control: numerical experiments,” Appl. Opt. 46(27), 6831–6842 (2007). [CrossRef] [PubMed]

4.

M. J. Booth, “Wavefront sensorless adaptive optics for large aberrations,” Opt. Lett. 32(1), 5–7 (2007). [CrossRef]

5.

M. J. Booth, “Wave front sensor-less adaptive optics: a model-based approach using sphere packings,” Opt. Express 14(4), 1339–1352 (2006). [CrossRef] [PubMed]

6.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1983).

7.

J. Braat, “Polynomial expansion of severely aberrated wave fronts,” J. Opt. Soc. Am. A 4(4), 643–650 (1987). [CrossRef]

8.

R. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66(3), 207–211 (1976). [CrossRef]

9.

W. J. Hardy, Adaptive Optics for Astronomical Telescopes, (Oxford Univ. Press, 1998).

OCIS Codes
(010.1080) Atmospheric and oceanic optics : Active or adaptive optics
(010.7350) Atmospheric and oceanic optics : Wave-front sensing

ToC Category:
Atmospheric and Oceanic Optics

History
Original Manuscript: August 17, 2010
Revised Manuscript: October 11, 2010
Manuscript Accepted: November 24, 2010
Published: December 24, 2010

Citation
Huang Linhai and Changhui Rao, "Wavefront sensorless adaptive optics: a general model-based approach," Opt. Express 19, 371-379 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-1-371


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