Role of boundary measurements in curvature sensing

1 October 1999

Optics Communications 169 Ž1999. 59–62 www.elsevier.comrlocateroptcom

Role of boundary measurements in curvature sensing E. Acosta a

a,)

, S. Rıos ´ a, M. Soto a, V.V. Voitsekhovich

b

´ de Optica, ´ Area Departamento de Fısica Aplicada, Facultade de Fısica, UniÕersidade de Santiago de Compostela, ´ ´ 15706 Santiago de Compostela, Galicia, Spain b Instituto de Astronomıa, ´ UNAM, AP 70-264 Cd. UniÕersitaria, 04510 Mexico D.F., Mexico Received 17 May 1999; received in revised form 3 August 1999; accepted 6 August 1999

Abstract The role of boundary measurements in curvature sensing is analyzed in basis of the method of weighting functions. It is shown that depending on the orthogonal basis chosen to expand the wavefront some modes can be exactly restored from the curvature data only, without the boundary information. This fact implies that curvature sensing technique can be used without contour measurements to evaluate some given modes. Therefore, if the contribution to the total phase distortion of the non-estimated modes is relatively small compared to the estimated ones, the method can be successfully applied avoiding edge measurements. q 1999 Published by Elsevier Science B.V. All rights reserved. Keywords: Curvature sensing; Phase retrieval; Weighting polynomials; Modal approach

1. Introduction Curvature sensing w1x is a promising method of phase measurements which is becoming more and more popular in adaptive optics, mirror testing and atmospheric research. The main advantage of the curvature sensing is that it allows a more simple experimental set-up than other techniques. In curvature sensing the normalized intensity difference between corresponding points in two defocused images of the pupil of an optical system is evaluated, giving

)

Corresponding author. E-mail: [email protected]

the sensor signal, I Ž r ., which is related to the phase S Ž r . at the pupil as follows

E D S Ž r . y dŽ r y R . l rs r , f

hs

l fŽ f y l .

,

h

Er

SŽ r . s

IŽ r . s

R2

IŽ r . ,

I1 Ž r . y I2 Ž r . I1 Ž r . q I2 Ž r .

,

Ž 1. where D is the Laplace operator, d is the Dirac delta-function, f is the focal length, l is the distance from the focus to the defocused images, and I1 and I2 are the intensities of two symmetrically defocused images. From the mathematical point of view, the phase can be restored by solving the corresponding Poisson

0030-4018r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 4 2 8 - 9

E. Acosta et al.r Optics Communications 169 (1999) 59–62

60

equation with the Neumann boundary condition given by the edge signal. In this context, the obtained solution is unique apart from an additive constant Žpiston.. In previous works w2,3x, the authors proposed a direct least square fit of modal expansions of the phase to the sensor signal. If the basis functions chosen to expand the phase are orthogonal, the orthogonality is preserved, so that each mode coefficient can be independently evaluated as sum of weighted integrals over the Laplacian of the phase inside the pupil domain and over the outward normal derivative on the domain contour. In this paper we will show what part of phase aberrations can be restored without boundary measurements. To do this we will apply the method of weighting functions w2,3x which allows us to analyze separately the role of curvature and boundary measurements in the reconstruction depending on the basis chosen to expand the wavefront.

j i Ž r . are the so-called weighting functions that allow the estimation of the mode coefficients in terms of the Laplacian of the phase, its normal derivative at the domain contour and its value at the boundary, C ŽDirichlet condition for unique solution.. For a given orthogonal basis, the  j i 4 set can be easily calculated by solving the Poisson equation D j i Ž r . s Fi Ž r .

with the requirement that j i has to be analytic in s q C. If the weighting functions also obey the following condition in their normal derivative

Ej i Ž r . En

s 0 in C

a i s cy1 i

2

Hsd sj Ž r . D S Ž r . yEC dl j Ž r . =S Ž r . i

Q Ž r . s Ý a i Fi Ž r . ,

Ž 2.

i

an approximating function defined as a linear combination of a set of basis functions, orthogonal in a domain s with contour C. The expansion coefficients a i have to be determined from the Laplacian of S Ž r . in order to minimize the square error 2

d 2s

Ž 3.

where s usually represents the input pupil of the optical system. It was shown w 2 x that the mode coefficients a i which minimize Eq. Ž2. can be obtained as a i s cy1 i

2

Hsd sj Ž r . D S Ž r . qEC dl =j Ž r . S Ž r . i

y dl j i Ž r . =S Ž r .

EC

i

Ž 4.

dl being a differential contour vector normal to C pointing outwards the domain and c i normalization constants given by c i d i iX s d 2s Fi Ž r . FiX Ž r .

Hs

i

Ž 8.

Let SŽ r . be a given phase and

Hs Ž S y Q .

Ž 7.

we obtain the desired final result in terms of the information given by the sensor

2. Method of weighting functions

SyQ s

Ž 6.

Ž 5.

Thus, Eq. Ž8. can be used to perform a direct fit of the mode coefficients to the measurements provided by curvature sensors through Eq. Ž1.. The possibility of restoring some coefficients with no contour measurements arise in those basis functions such that their corresponding weighting functions satisfy the following additional contour condition: j Ž r . s 0 in C, and therefore a i s cy1 i

2

Hsd sj Ž r . D S Ž r . i

Ž 9.

The possibility of obtaining such weighting functions depends strongly on the chosen basis and allow us to perform a deep analysis of the role of boundary conditions in curvature sensing. In the next Section we will apply the method to three different sets of basis functions, all of them orthogonal in the unit circle. The first ones, Zernike polynomials w4,5x and pseudoanalytic approximation of Karhunen–Loeve ` functions w6x, are presented due to their interest in optical testing and adaptive optics. The last one, the Appell and Kampe´ de Feriet polynomials w7x, are ´ included in order to completely illustrate the role of boundary conditions.

E. Acosta et al.r Optics Communications 169 (1999) 59–62

3. Weighting functions. Role of boundary conditions 3.1. Zernike polynomials (Zi ( r ,u ) s Zi ( r )) It was shown w2,3x that when these polynomials are used as a basis set of orthogonal functions in the unit circle, the associated weighting polynomials j i verifying the Poisson equation D j i Ž r . s Zi Ž r . with the contour condition Ej i Ž r . rE n s 0 in the circumference, can be straightforward evaluated. Now, we will try to find those polynomials j i that also obey j i Ž r s 1. s 0. ŽThe normalization convection used in Ref. w5x, will be adopted here.. a. Rotational symmetric modes, Zn,0 , n is even. It can be shown that a solution in the form r 1 rX j n ,0 ( r ) s R n ,0 Ž r XX . r XX d r XX d r X q B X 0 r 0 Ž 10 .

žH

H

/

satisfies Eq. Ž6. and Ej n ,0 Ž r s 1 . 1 s R n ,0 Ž r . r d r s 0 Ž 11 . Er 0 and if r 11 Bsy R n ,0 Ž r X . r X d r X Ž 12 . 0 r 0 then j n,0 Ž r s 1 ) s 0, showing that rotational symmetric modes can be calculated with only curvature measurements of the phase. b. Non-symmetric modes, Zn,l , l / 0 It can be shown that a solution in the form r l r X Ž 1y l . l X j n ,l ( r , u ) s A r l q r R n Ž r . d rX 2 l 0

H

H H

ž

H

ry l

r

2 l

H0 r

y

X Ž 1q

l.

R nl Ž r X . d r X

sin Ž lu . if l ) 0 = cos Ž lu . if l - 0

½

5

/ Ž 13 .

is solution of the corresponding Poisson equation D j n,l Ž r . s Zn,l Ž r .. We will distinguish here two cases: b.1. Modes whose azimutal frequency, l , is different than the radial one, n. If 1 1 Asy R Ž r . r 1y l d r Ž 14 . 2 l 0 n ,l

H

61

then not only Ej n,l Ž r s 1.rE n s 0 but also j n,l Ž r s 1 ) s 0, showing that these modes can be also calculated with only curvature measurements of the phase. b.2. Modes whose azimutal frequency, l , is equal to the radial one, n ŽHarmonic modes., Z l ,l . Here the weighting polynomials expression reduces to

ž

j l ,l ( r , u ) s A r l q

=

½

r l q2 4 Ž l q 1.

sin Ž lu . if l ) 0 cos Ž lu . if l - 0

/ 5

Ž 15 .

with Asy

Ž l q 2. 4 l Ž l q 1.

Ž 16 .

in order to obtain Ej n,l Ž r s 1.rE n s 0. Nevertheless, in this case j n,l Ž r s 1 ) s y2r4 l Ž l q 1. / 0 and therefore, these modes need the boundary measurements to be evaluated. This fact is expected due to their Harmonic character. In conclusion, all the expansion coefficients for the Zernike polynomials but those with azimutal frequency equal to the radial one can be exactly restored from curvature data only.

3.2. Pseudoanalytic Karhunen–LoeÕe ` (K–L) func( ( )) tions K i r ,u An accurate approximation of K–L functions in terms of Zernike polynomial was given in Ref. w6x. The approximation is built in such a way that the radial part of a given K–L function with azimutal frequency l can be obtained by a linear combination of Zernike polynomials with the same azimutal frequency. The evaluation in this case of the weighting functions is straightforward since they can be built as linear combination of the weighting functions evaluated above. The approximation for every K–L functions with the exception of the rotational symmetric ones always contain a harmonic Zernike

E. Acosta et al.r Optics Communications 169 (1999) 59–62

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Table 1 Polynomials of Apell and Kampe´ de Feriet ´ U0,0 s1 U1,0 s x U0,1 s y U2,0 s 3 x 2 q y 2 y1 U0,2 s x 2 q3 y 2 y1 U1,1 s xy U3,0 s 5 x 3 q3 xy 2 y3 x U0,3 s 5 y 3 q3 x 2 y y3 y

polynomial in its expansion and therefore they can not be calculated without the edge signal. Thus, only l s 0 modes can be restored from only the Laplacian of the phase. 3.3. Appell and Kampe´ de Feriet polynomials ´ (Ui (x,y)) These polynomials represent an orthogonal basis in the unit circle not all of them separable in radial and angular coordinates. The general expression for them is given by w 7 x

E kq m Uk m s D

k

ExEy

m

Ž x 2 q y 2 y 1.

kqm

to expand the phase there is the possibility of restoring some modes coefficients with no edge information of the wavefront derivative. As the information about Harmonic functions lies exclusively on the boundary, the maximum information about the wavefront without boundary conditions can be obtained from a modal approach by the properly choice of an orthogonal basis to expand the wavefront. Thus, for instance, when Zernike polynomials are used as an orthogonal basis in the unit circle it has been demonstrated that only the modes with equal radial and azimutal frequencies need boundary condition to be restored. For Karhunen– Loeve ` pseudoanalytic functions, only the rotational symmetric modes can be evaluated in this way. Since depending on the chosen basis some modes can be obtained with no contour measurement, it can be concluded that, in one hand that some aberrations can be retrieved and, in the other hand, if the contribution to the total phase of the remaining modes to the total distortion is small enough, an accurate reconstruction can be done with curvature sensing method only using the Laplacian of the phase information.

Ž 17 .

where D is a normalization constant. The first polynomials are listed in Table 1. For the listed polynomials it can be shown that in order to calculate the weighting functions, not only the normal derivative of the phase has to be known but also a Dirichlet condition of phase in the circumference. Thus, the method of weighting functions for these polynomials does not lead to a solution of the problem for the information obtained from curvature sensing technique. 4. Conclusion The reconstruction of a modal approach of a wavefront through curvature sensing measurements without contour information has been analyzed in basis of the weighting functions method. It has been shown that depending on the chosen orthogonal basis

Acknowledgements This work was supported by Ministerio de Educacion ´ y Cultura ŽSpain., Ref.: SAB1995-0729, by Sistema Nacional de Investigadores ŽMexico. and by Secretarıa ´ Xeral de Investigacion ´ e Desenvolvemento, XUGA20605B98. References w1x F. Roddier, Appl. Opt. 27 Ž1988. 1223. w2x S. Bara, ´ S. Rıos, ´ E. Acosta, J. Opt. Soc. Am. A 13 Ž1996. 1467. w3x V.V. Voitsekhovich, J. Opt. Soc. Am. A 12 Ž1995. 2194. w4x R.J. Noll, J. Opt. Soc. Am. 66 Ž1976. 207. w5x D. Malacara, Optical Shop Testing, Wiley, New York, 1992, p. 465. w6x G. Dai, J. Opt. Soc. Am. A 12 Ž1995. 2182. w7x A.H. Stroud, Approximate Calculation of Multiple Integrals, Prentice-Hall, New Jersey, 1971, Chap.3.