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PHASE RECONSTRUCTION USING A ZERNIKE DECOMPOSITION FILTER
S. N. Khonina, V. V. Kotlyar, VA. Soifer Image Processing Systems Institute, Russian Academy of Sciences Y. Hang and D. Zhao Beijing Institute of Technology, Beijing 100081. China e-mail :ytwang@sun. ihep. ac. cn
Abstract
Coherent wavefronts are analysed using a Zernike filter that decomposes the analyzed light field into a set of diffraction orders with amplitudes proportional to the circular Zernike polynomials. We also apply the algorithm to the calculation of the light field phase from measurements of the modules of decomposition coefficients. Operation of several filter is simulated.
/. Introduction
Reconstruction of the coherent light field phase is a topical problem in digital data processing. We cannot directly measure the light field phase but have to determine it indirectly, via light intensity measurements. For example, the wavefront of the light field can be reconstructed from an interferogram [1], from measurements of the intensity distribution of the spatial Fourier-spectrum [2). A Shack-Hartmann wavefront sensor consisting of an array of equivalent pinholes or a microlens matrix [3] lias also found use for phase retrieval. One can also reconstruct the phase using amplitude and phase filters capable of decomposing the light field in an orthogonal basis [4,5]. Paper [6] deals with the theory of spatial filters, called modans. intended for analyzing (selecting) the spatial modes of laser light. The number of basis terms that can provide the effective analysis of light fields is minimized using a basis matched to the field under analysis.
Paper [7] suggests that wavefront aberrations should be analyzed using Zernike polynomials |8j. Note, however, that it turns out to be impossible to design a spatial filter that would be able to produce an intensity distribution proportional to the coefficients expansion of the sought light field phase in terms of Zernike polynomials. In other words, we cannot solve the formulated problem using a purely optical approach We will have to do additional calculations.
In Ref. [7] an iterative algorithm for calculating the phase of the light field complex amplitude was developed.
In the present paper we compare the results of phase retrieval obtained by the iterative algorithm proposed in Ref. f7] and by the noniterativc method considered in Ref. [5].
2. ZERNIKE decomposition coefficients
There is complete set of orthogonal functions with angular harmonics in the circle of radius r0. These are the circular Zernike polynomials [8J:
^nm (r,<P) = Awi^n (r ) exp(itfl, (1)
where
A.... =
ln + l
W
R'.
(n-m)'2
'(')= I M Y
p. 0
n+m
n - m
r
V'o y
(2)
(3)
/¿"(r)are the radial Zernike polynomials:
K'L <r) = (XR^1 (r) = 0\m\ < njil (r) = 1,
(r, (pj are polar coordinates.
The decomposition of the light complex amplitude K(r, <p) into a series in terms of (1) is given by
¡R:<r)R;(r)rdr =
2(« + l)
r<i 2x
C^llEM^iwyrdrdv.
(5)
(6)
In the plane of a spatial Fourier-spectrum which can be generated using a spherical lens with focal length / the light field complex amplitude F(p,0) will take the form
H) 2x
xexp
2 */
' * 1 -1 —i-pcos(<2?-H9) \rdrd(p
. f j
(7)
where is the wavenumber of light, X is the
wavelength of light and (p,0) are polar coordinates Based on Eq. (4), we can represent the decomposilion of the light field in (7) into a series in terms of Zernike polynomials in Eq. (I) as:
J /1-0 m- n
\KWm{-rp)rdr
o J
<»)
In deriving Eq. (8) we employed an integral representation of the Bessel function of first kind and of m-th order:
r 2.T
Jm (x) = — f expi- ixcos t + imtpt.
In I
We can find the integral in Eq. (8) in an explicit form [8]
o '
=(-1)
(fl m) 2 2
0w\p)
(9)
From Eq.(9) it is seen that at n >0 the complc.x amplitude is equal to zero at the central point p 0
()m > 0
0) = < 1 2 n (2 r° ^ =
(10)
Therefore, at n>0 tlie intensity distribution of basis diffraction orders in the Fourier plane will have a like-ring structure.
In Fig. 1 is shown an optical setup (Zernike's analyzer) demonstrating the use of a phase Zernike filter for analysis of the wavefront with complex amplitude FJr.pt Analogous to the wavefront Shack-Hartmann sensor [3). the Zernike filter (ZI) is placed immediate]) in the plane of the wavefront under study A spherical lens L of focal length / is place beside it. In the rear focal plane of the lens L we place an array of photore-ceivers PA coupled to the computer PC.
t-'M
tig. I Optical setup
The transmission function of the ZF should be the phase one
i tp) — cxp[/'S'(r. <p)\ (11)
and is sought-for in the form
«r,<p) = I
rt-0m n »
x exp[/*/ 1 rPnm cos( <p-0nm) + vnm]
(12)
where (pm, (p„j arc the vectors of carrier spatial frequencies in polar coordinates and vnm ore the task's free parameters tliat should be fitted in such a manner that Eq. (12) be the correct equality.
Provided the effective spatial separauon in the Fourier plane of separate basis diffraction orders, wc can consider the functions
Xm(r,<p)exp[/*/ lrPnm cos(<p- Onm) +] as being nearly orthogonal and compute the parameters in Eq (12) by
(t. 2.T
0 0
< exp[- ikf 1rpm cos[<p- 0nm )]rdrd(p}
(13)
The Zernike filter of Eq. (12) produces the spatial separation of certain coefficients Cm from the expansion of the field E(r,<p, in Eq (4). If such a filter r'x.yi is placed near a sphencal lens and is illuminated by a light wave of amplitude Ffx.y/, the light intensity at the focal plane points u~arm and will be ap-
proximately proportional to the squared modulus of the expansion coefficients (',„,.
kf ' f J /•.'(x. i')r(x.,i')cxp[ - tkf luf - u )}/vr/v * / t
V n
cxp(/ vn„)6{u aKm. v - (inm)
n C rn n
(14)
where (am.fi!-jr.) arc the sectors of spatial carrier frequencies in Cartesian coordinates The more ciTecti\el\ is performed the spatial separation of basis beams propagating at different angles, the rnore accurate \sill be the approximate equality (14)
3. Iterative algorithm of the fight field phase reconstruction
After the light intensity proportional to the squared modules of coefficients of the expansion (4)
A»=K'«r (15)
has been measured at discrete points of the Fourier-plane (Fig. 1), we will have to do additional calculations to find the light field phase
Q{r,<p) = mgE</,<p). (16)
For this purpose, we may use an iterative algorithm similar to the algorithm of Eqs.(13) and (14). In this case, the estimate of the light field phase in the
(k+l)-\h iteration will be given by
= tl7j
where are the free parameters in the k -th iteration obtained from the equation
= arg J Jexpfe (r, <pydrd<p , (18)
0 0
(19)
where Qk (r, q>) is the desired phase estimation in the k -th iteration.
4. Method of coefficients phase definition
Thus, the intensity at points of spatial frequencies (Pnm, 9«m) is proportional to the squared amplitude of expansion coefficients, Inm «¡C^j . The phase <f>nm =arg(Cnm)may be reconstructed if one adds into the filter (7) terms in the form of linear combination of neighboring basis functions [5]:
^nm ^»iml
xexp^/V/'lcos^-C)^!
In this case, the additional channels corresponding to the Fourier-spectrum points with spatial frequency' (p^, O'wi) and (p"^, the intensity will be given by:
=\c4 +|CnV|2 +2|C,w,|CnW|cos(^m, -<fi„m\ m Pnm =\Cj( +|CnW|2+2|Cnm|Q.m.|sin(^v-+m\ thus making it possible to find the phase <{w assuming
The recursive relationship for the sought-for phases may be written in various forms. In view of assumed smallness of certain coefficients, the following function mav be used:
e _ / _ / , ,
nm nm rim' J
(21)
field from an orthogonal basis, and reconstructing this field.
5. Numerals examples The simulation parameters are: 256 pixels on radius r and 256 pixels on angle 0, rD= 1 mm, k=l(f mm"1, f=100 mm.
Using the iterative algorithm described in section 3 we designed a 25-channel Zernike filter forming the basis diffraction orders (n,m) at m<8 and n <& that propagate at some angles to the optical axis. Shown in Fig. 2 are: a half-tonic amplitude (a) and phase (b) of the Zernike filter (black colour - phase 0 and white colour - phase 2ri) and the configuration of the numbers (n,m) and linear combinations distribution between the orders (c).
a)
(7,1) (6,6) (6,4) « (6,0)
(73) (3,1) <Z2) (2,0) (5,5)
(7,5) (33) m (U) (53)
(7,7) (4,0) (4,2) {4,4) (5,1)
(8,0) (8,2) (8,4) (8.6) m
lttus, an optical method provides a sufficient accuracy in finding the complex coefficients of the light
c)
Fig. 2 Amplitude ondphase of a 2 Srchannel Zernike filter for 25 polynomials
Figure 3 shows the result of operation of the phase-only 25-channel Zernike filter illuminated by a beam:
E(x, y) = exp[/ cos(ax+by)],
the beam parameters are chosen to be a=0.5, ¿=1.
In Fig.3 are shown: the illuminating beam phase (a) with cross-sections along the x- and .y-axis (b), the diffraction pattern in the Fourier-plane for a phase-only filter (c), the reconstructed light field using the iterative algorithm (d) with cross-sections on axis x and y (e).
In Fig.3 are shown: the illuminating beam phase (a) with cross-sections along the x- (top) and .y-axis (bottom) (b), the diffraction pattern in the Fourier-plane for a phase-only filter (c), the light field reconstructed using the iterative algorithm (d) with cross-sections along the x- (top) andy-axis (bottom) (e).
Fig. 3. Result ofphase reconstruction using the iterative algorithm
Figure 4 depicts how the r.m.s. error of reconstruction, 5, depends on the number of iterations k, Eq. (17). One can see that if one uses a phase-only filter the process steadily converges during 100 iterations, although any acceptable reconstruction error is not achieved (5>0.2). Note that the use of an amplitude-phase filter (Fig. 2a,b) resulted in the fast convergence (after 5 iterations) of the process of reconstructing the phase function E[x,y\.
0 5 to 1S 20 25 30 35 40 «5 50 55 Ed 65 /0 75 80 S5 30 95
Fig. 4. R. nts error of reconstruction 5 vs the number of iterations
Using the method described in section 4 we designed a 25-channel Zernike filter forming the basis diffraction orders (n,m) at «<4 and m<4 (total 9) and their linear combination (total 8+8) that propagate at some angles to the optical axis. Shown in Fig. 5 are: a half-tonic amplitude (a) and phase (b) of the Zemike filter and the configuration of the numbers (n,m) and linear combinations distribution between the orders (c).
In Fig.6 are shown: the illuminating beam phase (a) with cross-sections along the x- (top) and y-axis (bottom) (b), the diffraction pattern in the Fourier-plane
for a amplitude-phase filter (fl), the reconstructed light field using method (21) (d) with cross-sections along the x- (top) andy-axis (bottom) (e).
i if ' h)
SR S7 S6 SS S4
PI '(3,1) (2,2) (2,0) S3
P2 (3,3) (0,0) (U) S2
PS (4,0) M (4,4) SI
P4 P5 PG P7 P8
Fig. 5. Amplitude and phase of a 25-channel Zernike filter for 9 polynomials
In this case, a tolerable reconstruction of the phase took place only with the use of an amplitude-pliasc filter, the r.m.s. error being 6=0.14. From Fig. 6, the phase is seen to be reconstructed up to a turn. A subsequent application of the iterative algorithm (17)-<18) did not result in the decreasing error. In all cases the process diverged at once.
b
Fig. 6. Result ofphase reconstruction using method (21)
e
6. Conclusions
The results obtained allow us to infer that the light field phase is possible to reconstruct using a filter matched to Zernike polynomials. In the present context, the method (21) has been shown to be better than the iterative algorithm (17)-(18).
Acknowledgments
This work was supported by Russian Foundation of Fundamental Research (№№ 96-01-10021, 98-01-00894).
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