a spherical mirror testing using a cgh with small errors

8/3/2019 A Spherical Mirror Testing Using a CGH With Small Errors

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Aspherical mirror testing using a CGH withsmall errorsAkira Ono and James C. Wyant

A method for reducing errors in aspherical mirror testing using a computer-generated hologram (CGH) isdescribed. By using a modified filtering method the carrier frequency in the CGH can be reduced by two-thirds, and the resulting error due to distortion is only one-half of that of a conventional CGH. By adoptinga Fizeau-type optical setup, only the surfacequality of the reference affects the measured results.

1. IntroductionComputer-generated holograms (CGH) are veryuseful for testing aspherical lenses or mirrors. CGHscan be in-line1 or off-axis.2 '3 The in-line CGH has anadvantage of allowing the compensation of strongeraspheric contributions, but filtering of the spuriousdiffraction beams and optimizing the system are moredifficult than for the off-axis CGH.4 For these reasons,the off-axis CGH is more commonly used than the on-axis CGH.In using the off-axis CGH for testing, a distortion in

the CGH is one of the most significant error sources.58The maximum distortion is almost proportional to themaximum spatial frequency in the CGH. To eliminatespurious diffraction orders, in the commonly usedtesting setup, the CGH must have three times highercarrier frequency than the maximum spatial frequencyf of the object wave.5 If the carrier frequency can belower, testing errors would be reduced.Optical element aberration or distortion is anothersignificant error source. Figure 1 shows a typical opticalsetup using the off-axis CGH for testing an asphericalmirror. If a beam splitter, a reference mirror, and adiverger lens have wave front distortions, AWb,AWr,and A Wd, respectively, a total wave distortion can bewritten statistically as

AW = (2AWb)2 + AW2 + AW2. (1)

Akiro Ono is with Toshiba Corporation, Manufacturing Engi-neering Laboratory, Shinsugita, Isogo, Yokohama, Japan, and J. C.Wyant is with University of Arizona, Optical Sciences Center, Tucson,Arizona95721.Received 23 July 1984.0004-6935/85/040560-04$02.00/0.1985 Optical Society of America.

From Eq. (1) it is seen that each optical element (espe-ciallythe beam splitter) has to have better quality thanthat of a mirror under test.This paper is concerned with improving those prob-lems by changing filtering means and adapting a Fizeau-type interferometric optical setup.11. Optical Setup

In the conventional CGH setup shown in Fig. 1, theOth-order diffracted beam from the object wave, whichcontains the aberrations from the object under test, andthe 1st-order diffracted beam from the reference waveare passed through the spatial filter. Figure 2 showsspatial frequency distributions of the diffracted beamsfrom the reference wave by the CGH. Since the widthof the sidelobe of the second-order diffracted beam istwice that of the 1st-order diffracted beam, for the filterto block the second-order beam the carrier frequencyf, has to be at least three times larger than the sidelobef of the st-order beam.5 7 The maximum frequencyin the CGH willbe f + f = 4f.If the 1st-order diffracted beam of the reference wavehas a wave front of W(x,y), the Nth-order diffractedbeam has the wavefront of NW(xy), and the asphericalmirror also can be tested using interference between theNth-order beam of the reference wave and the N -1st-order beam of the object beam. Figure 3 shows aspatial frequency distribution of the diffracted beamsfrom the objective wave. When a Oth-order beam of thereference wave and a -1st-order beam of the objectwave are considered, the sidelobe width of both beamsis almost zero, and adjacent diffracted beams-have awidth of f. In this case f is sufficient for the carrierfrequency fCin the CGH. The carrier frequency can beone-third, and the maximum frequency (f, + f) in theCGH can be one-half compared to the filteringmethod shown in Fig. 1. Consequently, a distortion inthe CGH can be one-half of the former method.

560 APPLIED OPTICS / Vol. 24, No. 4 / 15 February 1985


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FILM(ixtdo, est) RROR

Fig. 1. Typical optical setup using an off-axis CGH for testing anaspherical mirror.

- /2_-,AFig. 4. Optical setup.

I = C-Ro- (T) 2 -D-1 I, = CR, Do,'O +1 +2

. fpA.,-f,4f~

Fig. 2. Spatial frequency distribution of the diffracted beams fromthe reference wave produced by a CGH.

(2)where C is a coefficient, Ro and Rr are reflective ratiosof an aspherical mirror under test and a reference sur-face, respectively, Tr is a transparent ratio, and Do andD-1 are CGH diffraction ratios of Oth-order and-1st-order, respectively. If there is no absorption inthe reference surface, Tr is written as

T. 1 - Rr.

-2 -I 0 +1 +2f f f \A

Fig. 3. Spatial frequency distribution of the diffracted beams fromthe object beam.

Figure 4 shows the optical setup. A spherical surfaceis used for a reference surface instead of a plane mirror.A filter passes the -1st-order diffracted beam from theobject wave and the Oth-order diffracted beam from thespherical reference surface.The optical setup shown in Fig. 4 is of the Fizeau in-terferometer type. Since in the Fizeau interferometeran object wave and a reference wave go through almostthe same position at each optical element, aberrationsof the beam splitter and a diverger lens do not have tobe considered. Only the reference surface is requiredto have good quality. In this paper, a spherical surfacehaving quality of better than /10 was used for thereference. Ichioka and Lohmannl also suggested theFizeau-type optical setup for the CGH. The differencebetween their optical setup and the optical setup shownin Fig. 4 is not only that the latter uses an off-axisCGH but also that the CGH is located in the collimatedbeams. If the CGH is located in diverging beams likeIchioka's optical setup, the alignment error of the CGHpositioning affects the testing result seriously.By adopting the Fizeau-type optical setup, an addi-tional advantage of taking a high visibility interfero-gram is obtained. In the optical setup shown in Fig. 4,the intensity of the -1st-order diffracted beam of theobject wave 1o and intensity of the Oth-order diffractedbeam of the reference wave Ir are written as

(3)Since the best visibility in the interferogram can beobtained under the condition of I, = Ir, from Eqs. (2)and (3) an optimum value of Rr can be written as

D, _(D.2 iDol\/I1VJ +4RoRr = 1 + (42RoDo and D- 1 are defined by the procedure of making theCGH and usually Ro 1.For the results in this paper, the CGH was drawnusing a plotter (Hewlett Packard 7225A) connected toa desk-top computer (Hewlett Packard model 85) andphotoreduced using a conventional camera with a clo-se-up lens. This is one of the easiest procedures formaking the CGH, but diffraction efficiencyis very lowbecause a zero level in binary data is represented by agray tone instead of black (see Sec. V). For example,the value of Do/D- 1 was 24 in our CGH. SubstitutingDoID-1 = 24 and Ro = 1 into Eq. (4) yields

Rr = 0.038. (5)Since the reflective ratio of glass for a normal incidentbeam is almost the same value as Eq. (5), good visibilityinterference fringes could be obtained by using a non-coated spherical glass surface for the reference sur-face.111. Experiment

For an application of the Fizeau-type CGH opticalsetup with modified filtering method, a f/13204-mmdiam parabolic mirror was tested. Figure 5 showstheCGH pattern. The object wave from the parabolicmirror has an optical phase difference of W4 = 2 X23.44 from the reference wave at the edge of the mirror.When adding a defocusing of W2 = -1.5W4 = -27r X35.16, he maximumspatial frequency (the same as thesidelobe f) in the object wave is written as

15 February 1985 / Vol. 24, No. 4 / APPLIEDOPTICS 561

o - Z s l x - -

I


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Fig. 5. CGH pattern used for aspherical mirror testing.

f = 23.44/r, (6)where r is the radius of the CGH.6 In ideal cases, thecarrier frequency f can be equal to f, as mentionedabove. Considering that the sidelobe might be broad-ened by distortions of diverger and image lens and thephotofilm, f = 36/r (1.5f) was selected for filtering thespurious diffracted beam successfully. The maximumfrequency is then F + fc = 59.44/r. In the usual CGHsystem shown in Fig. 1, the value of c is bigger than 3f,and the maximum frequency is bigger than 4f(=93.76/r). This means that the CGH distortion shownin Fig. 4 is -3/5 of the CGH distortion shown in Fig.1 The CGH distortion was measured using the inter-ferometric method9"1 in advance of testing the para-bolic mirror. In this method a grating was drawn andreduced to a photofilm with the same plotter and cam-era used to make the CGH; then the distortion wasmeasured from an interferogram between +Nth- and-Nth-order diffracted beam by the grating. A wavedistortion AW in the -1st-order diffracted beam by theCGH can be estimated as

A W AP.fm N * , (7)where /r is the maximum spatial frequency in the CGH,fg is a spatial frequency in the grating, AP is the maxi-mum fringe distortion in the interferogram, and X s thewavelength of the laser. Figure 6 shows a resultantinterferogram between +3rd- and -3rd-order diffractedbeam by a grating with a frequency of 75/r. Since Fig.5 has the maximum fringe distortion of -- /4 fringe, fromEq. (7) the -1st-order diffracted beam by the CGH wasconsidered to have a maximum wave distortion of0.1X.The CGH radius r is determined as

r=d rm (8)21mwhere d and 1mare focal lengths of the diverger lenses

and the aspherical mirror, respectively, and r is a ra-

Fig. 6. Interferogram obtained measuring CGH distortion using theinterferometric method.

dius of the mirror. A radius error Ar of the CGH causesa wave distortion of A Wr written asAWr Ar-f- X.

From Eqs. (6), (8), and (9), A Wr is rewritten asAW 23442Ar-l.- Xld - rm

(9)

(10)Equation (10) shows that not onlythe CGH radius errorAr but also focal ength d affect the wave distortion. Itis difficult to know precisely the value of d for a com-mercial lens. A CGH with suitable radius can be se-lected from among many CGHs where each has a dif-ferent radius from others, so that spherical aberrationin an interferogram may be smallest. Here ten CGHswith an average radius of 8 mm and a radius differenceof 0.05 mm between each were made. From Eq. (9) AWwas estimated as 0.073X. Consequently, in this ex-periment, mirror testing is considered to have an ac-curacy of 0.17X even including a distortion of 0.05Xcaused by a CGH alignment error. The accuracy of0.17X s fairly good taking account of the procedure formaking CGHs and the quality of the optical elementsused in this experiment.Figure 7 shows an interferogram of testing a parabolicmirror mentioned above using the optical setup shownin Fig. 4. The maximum fringe distortion in the in-terferogram is 0.2 fringe. Considering testing accuracy,a wave reflected by this mirror has a distortion of


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From Eq. (11) intensities of the Oth-and 1st-order dif-fracted beams are obtained asI(0) = (a + b) 2 = a 2 + b2 + 2ab, (12)

Fig. 7. Resultant interferogram for testing a parabolic mirror usingthe optical setup shown in Fig. 4.

0ILCLJ

A 2

tXLJN2a~~~~~~~~b1 _ _POSITION

Fig. 8. Amplitude distribution for a gray tone rectangular grating.about half of the usual CGH distortion. For an appli-cation, a parabolic mirror with a 204-mm diameter and612-mm focal length was tested. Testing results with anerror of

a spherical mirror testing using a cgh with small errors

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