In this paper we show an ability to use a multi-order (multi-channel) diffractive optical element for wavefront relief function expansion in terms of Zernike polynomials. This approach can be successfully used for small meanings of wavefront aberrations when the wavefront relief can be represented as a linear superposition of Zernike polynomials. Unfortunately, linear approximation is becoming unworkable with increasing of aberration meanings. In this work we study an applicability of this Zernike expansion method.
–2 0 2 –3 –1 1 3 –4 –2 2r cos () 6r2 sin (2) 3 (2r2 1) 6r2 cos (2) 2 2r3 sin (3) 2 2(3r3 2r)sin () 2 2 (3r3 2r)cos() 2 2r3 cos(3)
10r4 sin (4) 10 (4r4 3r2)sin (2) 5 (6r4 6r2 1) 10 (4r4 3r2)cos(2) 10r4 cos(4)
(Trefoil)
In this work we assume that Zernike polynomials appear as follows: nm r,
Computer Optics and Nanophotonics / P.A. Khorin, S.A. Degtyarev here Rnm (r) is radial Zernike polynomials:
Rnm r nm 2 p0
1 p n p ! n m p! 2
n m p ! 2 r R p ! W r, exp iwr, ,
N n wr, cnmnm r, .
n0 mn
(
There is so-called “Zernike pyramid” in figure 1. 2D patterns of few Zernike polynomials form this pyramid. In a vertical direction radial number n varies from 0 to 4 (n = 0 to 4) and in a horizontal direction azimuthal number m varies from -n to n (m = -n to n).
Optical elements work together and create an image. Unfortunately, created with an optical system image can not be ideal. Lenses and mirrors have their own aberrations; in addition, aberrations can arise due to inaccuracies of fabrication, misalignments, and so on. Partial error compensation can be provided after precise measurements of wavefront aberrations.
In our simulations we introduce aberrations into the optical system through the adding to the relief of the first surface of the second lens. These approach and algorithm are discussed in detail in paper [13]. 2D patterns of point-spread function are shown in figure 3. This patterns are arranged as a pyramid corresponding to Zernike pyramid in figure 1.
In the paper [14] authors investigate an ability of using multi-order diffractive optical element for analysis of human eye optical system aberrations [15, 16]. Note that, this way may be used for other optical systems, including telescopes [17-19].
It should be noticed that Zernike bases can differ not only in indexing and normalization but in the angular dependence shape. In particular, in [20] exponential and trigonometric angular dependences are used for designing multi-order diffractive optical elements for wavefront analysis.
Optical Zernike analyzer is a combination of diffractive optical element and lens; DOE is matched with basis function and lens does Fourier transform. Achieved Fourier spectrum at the center has the value which means scalar product of input light function and the basis function. Experimental testing of the same devise is described in [20, 21].
Certain basis Zernike function is coded in each order of multi-order diffractive optical element. DOE transmission function looks as follows:
P Q
(x, y) p0 q0 pq x, y exp i pq x pq y (
There is an illustration of 8-order diffractive optical filter working in figure 5. Plane wave (wavefront is flat w( x, y) 1 ) illuminates the element and diffracts on it. As can be seen, all orders are holed. It means that analyzed wavefront does not have any aberrations.
In this work we propose a physical model of optical device (figure 6) for wavefront aberration analysis. This device is based on multi-order Zernike-matched diffractive optical element. Analyzed wavefront w(x,y) passes through the element. The lens O with focal distance f makes Fourier transform. Sensor matrix should be posed at the focal distance from the lens. Sensor plane has u and v coordinate axes.
DOE
Another testing simulation of proposed Zernike analyzer model is provided for initial beam wavefront w(x,y)=ψ2,0(x,y)+ ψ2,2(x,y). Resulting pattern is shown in fig. 7. Simulation shows that 8-order filter detects defocusing and astigmatism aberrations as it is set for initial beam. Detected coefficient meanings are C20=0.995, С22=0.996 but in initial beam they are equal to 1.
Fig. 8. Resulting amplitude patterns which is formed with 8-order analyzer after filtering the initial wavefront w(x,y)=exp[iα c31ψ3,1(x,y)].
Once more example of working of proposed model is considered if the incident wavefront is w(x,y)= c31ψ3,1(x,y), c31=2√2 ≈ 2.8284. The analyzer detects com-a aberration with coefficient cdetected 2.7489 . Resulting 2D amplitude patterns are 31 shown in fig. 8.
It should be noticed that aberrations detection can be successful only for weak aberrations of the wavefront. With increasing of aberrations linear approximation of the wavefront as a sum Zernike polynomials is becoming nonapplicable and false detection happens.
In the table 2 a dependence of detection quality is shown. Incident wavefront is constant and equal w(x,y)=exp[iαc31ψ3,1(x,y)], c31=2√2 ≈ 2.8284. Parameter α is varied from 0 to 2.0; we observe a varying of detected value cdetected 31 . From the table 2 it is seen that for too small and too high meanings of parameter α the detection is mistaken.
Every conventional aberration can be expressed in terms of decomposition into Zernike basis. Evidently, to compensate the most prominent aberration it is enough to modify the incident field by adding a phase which is complex-conjugated to revealed aberration. It can be done with diffractive optics methods including etching relief to the lenses surfaces or adding DOE to t he optical system.
It is worth to notice that the size of proposed DOE from Zernike analyzer is equal about 5 mm. 8-channel filter sensor has 512x512 pixels. Therefore it is easy to calculate resolution power of the device, it means 12.4 um.
However, resolution power of a diffractive optical element nowadays is limited by the 1 um meaning. Resolution of proposed 8-order DOE is far from technical limit. Thus, it says that the diffractive optical element can be easily produced with modern etching machines.
In this work provided simulation has shown that proposed Zernike analyzer can successfully detect aberration types and the model also reveals limitation of incident field aberration to be detected with the device. Defined tolerance range of aberrat ion coefficient α is {0.5;1.75}. Optimal meaning is 1. If α goes out of tolerance range, aberration can not be detected or false detection can happens.
This work was supported by the Ministry of Education and Science of the Russian Federation and Russian Foundation for Basic Research grant No. 15-29-03823ofi_m.