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Computer-aided aberration compensation in optical system assembly
Optik 124 (2013) 1930–1935
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Computer-aided aberration compensation in optical system assembly Chao Tian, Yongying Yang ∗ , Yongmo Zhuo State Key Laboratory of Modern Optical Instrumentation, Zhejiang University, 38 Zheda Road, Hangzhou 310027, China
a r t i c l e
i n f o
Article history: Received 7 January 2012 Accepted 29 May 2012
Keywords: Lens assembly Aberration compensation Interferometry Testing
a b s t r a c t We propose a computer-aided approach for residual aberration compensation of optical systems after initial assembly. The approach considers that intrinsic variables of an optical system such as index of refraction, radius of curvature and thickness are approximately equivalent in small ranges and the residual aberrations may be attributed to the error of one of them (e.g., index). In this way, if experimental wavefront aberration seriously deviates from theoretical one, it may be easy to estimate the exact amount of axial movements of one or more lenses needed to compensate for the aberrations. Normally, two iterative attempts (i.e., disassembly, spacer customization, reassembly and testing) are enough to give an acceptable level of performance, which may save much time and labor compared to the traditional trial-and-error method. Two examples from practice have demonstrated its validity. © 2012 Elsevier GmbH. All rights reserved.
1. Introduction A designed optical system generally needs to go through four phases before it could be delivered to users, that is, optical and mechanical design, elements fabrication, lens assembly and system testing. At the initial phase, designers should give both optical and mechanical solutions to the system according to technical specifications, perform tolerance analysis and prepare lens as well as mechanical drawings. Then these drawings are sent to factories and the elements are fabricated according to requirements. After that, the system is assembled in optical shops and finally, tested to evaluate its performance. Each of the first three steps is important and critical to the final performance of the optical system. If no fatal errors exist in the designs, all elements are fabricated within tolerances and the system is properly aligned and mounted, the actual performance of the system after initial assembly should be quite close to the theoretical one. However, serious deviation between them may frequently occur in practice. For these situations, we may axially adjust one or more lenses (i.e., aberration compensators) to fix the problem (i.e., compensate for residual aberrations). One well-known example is that adjustments of the central air space of a double-Gauss lens may be used to control the astigmatism of the system without affecting other aberrations much. Several other successful applications can also be found in the literature [1]. Vukobratovich [2] and Bacich [3] presented some successful solutions to adjust one or more lenses to compensate for residual aberrations of optical systems with adjustable screws built into the design. Williamson [4,5] described the selection rule
of aberration compensators, which indicates that compensators should be selected based on the sensitivity of each element for primary aberrations such as spherical aberration, coma, astigmatism and distortion so that reasonable movement produces the desired effect. Although built-in adjusting mechanisms bring convenience for further system optimization because sequential movements of compensators are easy, they increase the mechanical complexity and are generally applied in high-quality lens assembly. For most optical systems that do not have such mechanisms, we have to iteratively disassemble the system, change air spaces, assemble and test it if better performance is to be expected. This is a time-consuming and laborious work because exact amount of movements of compensators is difficult to determine and several iterative attempts are normally required before an optimal state is reached. This situation becomes worse when more than expected fabrication and/or assembly errors are present (see Section 3) or the system is very complex. To improve this situation, we propose a computer-aided method for residual aberration compensation during the stage of lens assembly. The method can exactly estimate the amount of axial movements of aberration compensators that is needed to correct the residual aberrations. According to the estimate, the performance of an optical system may be improved to an acceptable level within two attempts.
2. Basic idea ∗ Corresponding author. Tel.: +86 571 87951514; fax: +86 571 87951514. E-mail addresses: [email protected] (C. Tian), [email protected] (Y. Yang). 0030-4026/$ – see front matter © 2012 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2012.05.033
The performance of an optical system can be measured by many indicators, such as ray aberration, wavefront aberration, point
C. Tian et al. / Optik 124 (2013) 1930–1935
tair . Therefore, the system performance may be enhanced by axially adjusting certain lenses to change the air space tair and the amount of adjustments is also determinable. According to the approximately equivalent idea, the deviation between the real aberration W and the nominal one W could be mainly attributed to be the error of one (e.g., refractive index of one glass) instead of several variables of the system. As thus, the analysis can be simplified and we get
Reference wavefront
Exit pupil
Aberrated wavefront
Spherical wavefronts from ojbect point
Q´
Q Optical system
Paraxial focus n
O´
O Q Q´ W = n.QQ´
Fig. 1. Wavefront aberration of an optical system. The small diagram shows the details.
spread function (PSF) and modulation transfer function (MTF). Among them, the wavefront aberration, which unifies the geometrical and diffraction theories of image formation, is the most essential measure. Suppose that an axial object point O is imaged by an optical system (Fig. 1). If the system is perfect, the wavefront emerging from the exit pupil should be spherical and centered at the paraxial image point O , which is geometrically perfect. When aberrations are present, however, the emerging wavefront will be aberrated and the image is not perfect any more. The deviation QQ between the aberrated wavefront Waber and the reference wavefront Wref is the wavefront aberration W or optical path difference (OPD), which reflects the degradation of the image quality. Mathematically, it can be written as W = Wref − Waber = n · QQ , where n is the refractive index of the medium. From the point of wavefront aberration, it is easy to explain the approximately equivalent idea of the method. For an optical system after initial assembly, the nominal wavefront aberration W at the exit pupil can be expressed as the function of its intrinsic variables, that is, W = f (n, r, tlen , tair ),
(1) T
n , r , tlen
tair
W = f (n , r , tlen , tair ) ≈ f (n , r, tlen , tair ),
(3)
where n is the equivalently predicted refractive index (which may not actually be the case). With the fictitious refractive index n , we may first select one or more lenses as aberration compensators [4] and then estimate the amount of axial movements tair needed to compensate for the residual aberrations. If we disassemble the system, axially adjust the compensators by tair , the residual aberrations should almost disappear, that is, W = f (n , r, tlen , tair ) ≈ W,
(4)
where tair = tair + tair is the corrected thickness vector of air spaces. It should be noted that since optical systems after assembly are usually tested by interferometers and the interferogram I is the even function of the wavefront aberration W (i.e., I = cos(±2 · 2W /)), two fictitious values n1 and n2 of the refractive index n may simultaneously exist for one interferogram and satisfy n1 < n < n2 . For these situations, we may use some prior knowledge to help determine which value is more reasonable (see Section 3.1). If no prior knowledge exists, we need to try one by one. The proposed computer-aided method for aberration compensation in optical system assembly may be outlined as follows:
T
where n = [n1 , n2 , n3 , . . .] , r = [r1 , r2 , r3 , . . .] , tlen = [tlen1 , tlen2 , tlen3 , . . .]T and tair = [tair1 , tair2 , tair3 , . . .]T are the nominal index vector, radius vector, thickness vectors of lenses and air spaces, respectively; ni , ri , tleni and tairi are the nominal refractive index of each glass, radius of each surface, thickness of each lens and air space, separately, where i = 1, 2, 3, . . . Although the nominal wavefront aberration W is usually small and quite close to zero for well designed and optimized system, it only represents their theoretical performance. To see their real performance after assembly, it is necessary to measure the actual wavefront aberration W using some techniques such as interferometers [6], which can be written as W = f (n , r , tlen , tair ),
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(2)
where and are the real index vector, radius vector, thickness vectors of lenses and air spaces, respectively. If the real wavefront aberration W is also close to zero (as the theoretical one W), we may consider that the optical system meets the requirements and can be delivered to users. However, if it deviates from the theoretical one W too much, it is certain that some errors exist and the optical system should be fixed before applications. Considering that all the intrinsic variables (i.e., n , r , tlen and tair ) are related to rays deflection ability of the system, their variations in small ranges will affect the wavefront aberration linearly. In other words, the wavefront aberration induced by the errors of one or more variables may be corrected by slightly adjusting other variables and therefore, these intrinsic variables may be approximately regarded equivalent in small ranges. Based on this idea, if we want to fix an optical system with residual wavefront aberration after initial assembly, we may simply modify the values of certain variables. For a specified optical system after assembly [Eq. (2)], since the index n , radius r and thickness of the lenses tlen are all determined and cannot be changed, the only freedom is the air space
(1) Compute the nominal wavefront aberration W with nominal values of the variables n, r, tlen , tair and compare it to the physical one W . (2) Modify the nominal value of one of the variables (e.g., refractive index n) until the wavefront aberration at the exit pupil is close to W and obtain two fictitious indices n1 and n2 . (3) Determine which value is more reasonable. If difficult to judge, try n1 . (4) Select aberration compensators, adjust air spaces tair until the wavefront aberration approaches W and record the amount of movements tair . (5) Disassemble the system, move the selected compensators axially by tair , assemble and test the system again. (6) If the physical wavefront aberration W deviates from the nominal one W more seriously, try n2 and repeat steps 4–5. Otherwise, stop. After steps 1–6, the experimental wavefront aberration W should be much closer to the theoretical one W and the performance of the system should be greatly enhanced. However, if the performance criteria are still not achieved, these steps may be repeated by replacing the air spaces tair with tair . Notice that since intrinsic variables of the system are approximately equivalent, other parameters such as radius of curvature, thickness may also be used to interpret the deviation between W and W . 3. Experimental verifications and discussions The validity of the computer-aided method was tested with two practical experiments. In the first, it predicted the change of wavefront aberration of an aplanatic system when one of the lenses was axially adjusted by known amounts, which agrees well
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C. Tian et al. / Optik 124 (2013) 1930–1935 Table 1 Lens data of the aplanatic system.a
b
1 2 3 4 5 6 IMG
Radius
Thickness
Glassc
Semidiameter
Conic
450.000 −80.962 −52.333 −101.614 39.028 86.990 Infinity
7.690 16.018 5.000 2.500 7.040 86.990 –
(1.51682, 64.36)
25.0 25.0 25.0 25.0 25.0 25.0
0 0 0 0 0 0 0
(1.61618, 36.87) Same as surface 1
a
The radius, thickness and semidiameter are all in millimeters. The surface is the stop of the system. c The first and second data in parenthesis are the index nd and Abbe number vd of d-light (at 587.5618 nm), respectively. b
Beam splitter
Reference Transmission sphere flat L1 L2 L3
Reference surface
C
Imaging lens
tair 1
Detector
Fig. 2. Opto-mechanical structure (a) and nominal wavefront aberration (0.30) from Zemax (b) of the aplanatic system.
with experimental result. In the second, it successfully guided the aberration compensation of an afocal system by just one attempt. 3.1. Aberration compensation of an aplanatic system (transmission sphere) Transmission sphere (TS) plays an important part in interferometric testing of spherical surfaces [6]. It has two basic functions, that is, transforming the incident plane wave into a spherical wave and returning a reference wave from its reference surface (usually the last surface). Since the shape of the transformed wavefront is expected to be perfectly spherical, the TS is required to be free of spherical aberration. Also because the reference and the test wavefronts should follow the same path through the interferometer to meet the null condition [7], the focus of the TS should coincide with the curvature center (aplanatic point) of the reference surface. In the current work, we designed and assembled an F/4.3 TS for a homemade Fizeau interferometer [8] with its last surface as the reference. The working wavelength and entrance pupil diameter is 632.8 nm and 25.0 mm, respectively. The opto-mechanical structure and nominal lens data after initial assembly are shown in Fig. 2(a) and Table 1, separately. Theoretically, the peak to valley (PV) value of the wavefront aberration W at the exit pupil and diameter of the spot diagram at the image surface (0-degree field of view) is about 0.03 [Fig. 2(b)] and 0.55 m [Fig. 6(a)], respectively, which indicates that the system has excellent performance. However, when the assembled system was tested in a Fizeau interferometer (Fig. 3), we found that the residual wavefront aberration (main defocus) on the detector actually has a magnitude of about 9.0 [Fig. 4(a)], which is contrary to our expectations. To see whether the aberration is caused by incorrect assembly, we carefully examined the air space tair between lenses and found no abnormality. At this point, the most reasonable explanation is that there is something wrong with the lenses and real values of them deviate too much from their nominal ones. The problem was fixed by axially adjusting one of the lenses.
Fig. 3. Measurement of the aplanatic system by a Fizeau interferometer.
We first employed the traditional trial-and-error strategy to correct the system. The first lens L1 was chosen as the aberration compensator. To see which direction of the axial movement of L1 was correct, we disassembled the system, increased the air space tair1 (between lenses L1 and L2) for 0.10 mm by adding a 0.10 mm spacer and retested it in the Fizeau interferometer. Unfortunately, fringes of the interferograms (not shown) became denser than before [Fig. 4(a)], which means that the air space tair1 should be decreased. As thus, we iteratively reduced the air space tair1 by 0.20 mm (tair1 = 15.818 mm), 0.50 mm (tair1 = 15.518 mm), and 0.70 mm (tair1 = 15.318 mm), repeated the previous four steps (i.e., disassembly, spacer customization, reassembly and testing) and showed the corresponding testing results in Fig. 4(b)–(d). As we can see, most of the residual aberrations have been compensated by decreasing the air space tair1 for 0.70 mm and therefore, the performance of the TS is greatly improved [the magnitude of the residual wavefront aberration in Fig. 4(d) is smaller than 0.5].
Fig. 4. (a–h) Aberration compensation results by the traditional method and correspondingly predicted results by the proposed method. The first and second row, experimental and predicted wavefronts for tair1 = 0, −0.20, −0.50, −0.70 mm, respectively.
C. Tian et al. / Optik 124 (2013) 1930–1935
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Fig. 5. Numerical relation between PV value of the wavefront aberration and error of the radius rref . We can see that rref = ±0.28 mm both correspond to 9.0 wavefront aberration.
Although the traditional trial-and-error method may be utilized to enhance the system performance, it is time-consuming and laborious because each iterative attempt is discrete and requires four tedious steps mentioned above. This situation may be eased by use of the proposed computer-aided method. Reconsider the physical wavefront aberration of the system [Fig. 4(a)] in the Fizeau interferometer. We assumed that all other variables of the system are correct except the radius rref (i.e., r6 ) of the reference surface. By modeling the interferometer (Fig. 3) in Zemax [7,9], the relation between the OPD on the detector and error of the radius rref was easily obtained, as shown in Fig. 5. We can see that there are = 87.270 mm and two fictitious values of the radius rref (i.e., rref = 86.710 mm), which both can produce interferograms with rref 9.0 circular fringes [Fig. 4(e)] as those in the interferometric testing [Fig. 4(a)]. Taking into account that the returned beam from the reference surface was convergent and its diameter was obviously smaller than 25 mm (an experimental phenomenon and with no use for the traditional method), the fictitious value of the radius rref should be greater than its nominal value 86.990 mm and therefore, = 87.270 mm was a more reasonable option. This conclusion rref can be easily obtained if we trace rays through the TS, as shown in Fig. 7(a). Based on the result, we numerically reduced the air space tair1 by 0.20, 0.50, 0.70 mm as before and found that the correspondingly predicted interferograms in Zemax [Fig. 4(f)–(h)] agree well with those in experiment [Fig. 4(b)–(d)]. In other words, the residual aberrations of the system could be compensated to an acceptable level by just one trial if the computer-aided method is utilized. Since all variables are equivalently determined in the proposed method, it can also be used to estimate other measures of the system besides the wavefront aberration. In this example, we also predicted the spot diagram of the system before [Fig. 6(b)] and
Fig. 7. Ray tracing of the TS with parallel incident beam (black solid lines): (a) ray paths and (b) change of the OPD when the mirror M is in different positions (pay attention to the three critical points, F, C and F2 ). See the text for details.
after [Fig. 6(c)] the correction. The evaluation surfaces were both located in the curvature center C (i.e., 87.270 mm from the reference surface). As we can see, the spot diameter after correction is much smaller than that before correction, which also indicates improvement of the system performance. As the TS is ultimately used in an interferometer, we here briefly discuss how the residual aberrations if not corrected may influence the interferometric testing results. Supposing that parallel beam (W0 = 0) is incident to the simplified model [Fig. 7(a)] of the TS, we can obtain that its back focal length AF must be shorter than the radius AC (i.e., rref ) because the wavefront W1 reflected by the reference surface Sref was convergent. This conclusion holds, no matter which variables of the system may be the problem. If we mount the TS on a Fizeau interferometer and place a flat mirror M in the vicinity of its focus F, the OPD on the detector can be written as OPD = W1 − W2 , where W1 and W2 are the wavefronts returned from the reference surface Sref and the mirror M, respectively. How the OPD will change if the mirror M is continuously moved from the negative direction F− of the focus F to its positive direction F+? To answer this question, we need to investigate three critical positions, i.e., the focus F, the curvature center C and the point F2 (image point of F when the mirror M is in the position C). For the position F (i.e., cat’s eye position), the incident parallel beam W0 will be symmetrically reflected by MF and the returned beam W2F will also be parallel (i.e., W2F = 0). Therefore, the OPDF on
Fig. 6. Spot diagrams of the aplanatic system from Zemax: (a) designed spot, (b) and (c) predicted spot before and after aberration compensation, respectively. Note that all figures were plotted under the same scale.
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C. Tian et al. / Optik 124 (2013) 1930–1935
of view), respectively, from which we can see that the system is near-diffraction-limited in principle. However, large wavefront aberration appeared [Fig. 10(a)] when the system after initial assembly was tested in the CRSI (Fig. 9). Problems occurred again and we used the proposed method to correct the residual aberrations. Before the correction, one thing, i.e., conversion relationship between a radial shearing interferogram and a Fizeau or TwymanGreen interferogram should be clear. This is mainly due to the fact that modeling of the telescope in a CRSI is much more difficult than that in a Fizeau or Twyman-Green interferometer. Therefore, we simply simulated the telescope in a Fizeau interferometer by adding a perfect lens [Fig. 8(a)] and compared the synthetic interferograms to those in the CRSI through conversion. In the CRSI, the contracted wavefront Wc [11] can be written as Wc = W0 2 0 2 + W0 4 0 4 + W1 3 1 3 cos + W2 2 2 2 cos2 , Fig. 8. Optical structure (a) and nominal wavefront aberration (0.04) from Zemax (b) of the afocal system.
the detector can be given as OPDF = W1 which means that interferograms with the same shape as Fig. 4(a) can be obtained if the mirror M is in the position F. For the position C, the returned wavefront W2C may be viewed as emitted from the point F2 . Since FC = CF2 , we have ˛ ≈ ˇ, W2C ≈ W1 and OPDC ≈ 0 which means that zero-fringe interferogram will appear if the mirror M is in the position C. For the position F2 , we may use the same analysis method and obtain W2F2 ≈ 2W1 . In this way, the OPDF2 on the detector has the same magnitude OPD
where and are radial and angular coordinates over a unit circle defining the pupil, Wl m n are aberration coefficients and l, m and n are integers [12]. If we define the effective radial shear R = c /e , the expanded wavefront We can be written as We = W0 2 0 (R)2 + W0 4 0 (R)4 + W1 3 1 (R)3 cos + W2 2 2 (R)2 cos2 ,
(6)
where c and e are the radii of the contracted and expanded beams, respectively. Subtracting Eq. (6) from Eq. (5), the OPD between the two wavefronts is
= Wc − We = W0 2 0 2 (1 − R2 ) + W0 4 0 4 (1 − R4 ) + W1 3 1 3 (1 − R3 ) cos + W2 2 2 2 (1 − R2 ) cos2
as W1 but with opposite sign, that is, OPDF2 ≈ −W1 , which indicates that interferograms at the positions F and F2 are the same in shape. The three positions divide the image space into four regions, that is, A − F, F − C, C − F2 and F2 − +∞ and the relationship between the OPD and the position of the mirror M can be qualitatively obtained, as illustrated in Fig. 7(b). From the analysis above, we know that if the aberrations of the TS are not corrected, its focus F will not coincide with the curvature center C of the reference surface Sref . When the flat mirror M is in the cat’s eye position, the reference and the test wavefronts (i.e., W1 and W2F ) will not follow the same path through the interferometer and retrace errors exist [7]. This may introduce significant errors to interferometric measurements, such as absolute calibration of spheres [6] and radius measurement [10]. On the other hand, if the aberrations are compensated, the corrected focus F will be quite close to the curvature center C and errors can be largely reduced. 3.2. Aberration compensation of an afocal system (Galilean telescope) In the previous example, we show that the computer-aided method successfully predicts the performance changes of the aplanatic system when one compensator is axially moved. Here we give another example, in which the method is employed to guide aberration compensation of an afocal system, i.e., a Galilean telescope used to contract and expand beams in a cyclic radial shearing interferometer (CRSI) [6]. The optical structure and lens data (after optimization during initial assembly) of the system are shown in Fig. 8(a) and Table 2, separately, and its entrance pupil diameter, working wavelength and magnification (ratio of the focal length of the objective lens to that of the eye piece lens) are 86.0 mm, 52.7.0 nm and 2.0, respectively. Fig. 8(b) and 12 show its theoretical wavefront aberration at the exit pupil and MTF (dash-dot curve, 0-degree field
(5)
.
(7)
Comparing Eq. (7) with Eq. (5), we may find that the resulting interferograms in a CRSI is the same as those in a Fizeau interferometer if each aberration coefficient is divided by (1 − Rm ). Considering that defocus W0 2 0 is the dominant aberration and the effective radial shear is 1/2, the magnitude of wavefront aberration of the afocal system should be 11.0 × (4/3) = 14.7 [Fig. 10(a)] if tested in a Fizeau interferometer. In this case, we attributed the serious deviation to the error of the index of dlight nd1 of lens L1 (also L3) and obtained the relationship (Fig. 11) between PV value of the wavefront aberration W and error of the index nd1 by tracing rays through the Fizeau interferometer in Zemax. As we can see, there are also two fictitious values of nd1 , i.e., nd1 = 1.51585 and nd1 = 1.51785, which can yield interferograms with a magnitude of 14.7 [Fig. 10(b)]. Since no prior knowledge helped us to determine which value is more reasonable, we were ready to try them one by one with the computer-aided method. Here the first lens L1 was chosen as the compensator and could be
M1
BS L1 L2 L3 M2
Imaging lens We Wc Detector
Fig. 9. Measurement of the afocal system by a CRSI. BS, beam splitter; M1 and M2, mirrors.
C. Tian et al. / Optik 124 (2013) 1930–1935
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Table 2 Lens data of the afocal system.a
1b 2 3 4 5 6 7c IMG a b c
Radius
Thickness
Glass
Semidiameter
Conic
Remarks
108.683 −334.190 −347.265 417.384 −72.599 −1219.07
24.992 6.288 10.990 103.264 5.988 50.000 100.000 –
(1.51685, 64.59)
52.0 52.0 52.0 52.0 28.0 28.0 28.0
0 0 0 0 0 0 0
f = 100.000
Infinity
(1.61307, 34.94) Same as surface 1
The radius, thickness, semidiameter and focal length are all in millimeters. The surface is the stop of the system. The surface is a perfect lens with its focal length f = 100.000 mm.
Fig. 10. Aberration correction results by the proposed method: (a) experimental wavefront (about 11.0) after initial assembly in the CRSI, (b) synthetic wavefront (about 14.7) in the simulated Fizeau interferometer when nd = 1.51585, (c) predicted wavefront when t = 0.20 mm, (d) experimental wavefront after reassembly in the CRSI when t = 0.20 mm.
the telescope in the CRSI after reassembly. Fortunately, most of the aberrations were corrected and the physical wavefront aberration [Fig. 10(d)] agrees with the predicted one [Fig. 10(c)]. Therefore, the system performance has been greatly improved by just one trial using the proposed method. Using the fictitious index nd1 = 1.51585, we also predicted the MTF of the telescope before (dotted curve) and after (dashed curve) the correction, which are shown in Fig. 12. As we can see, the corrected MTF also has been significantly enhanced and is close to the designed one. 4. Conclusion We present a simple and effective computer-aided method that is capable of compensating for residual aberrations of optical systems in lens assembly. The method utilizes the proposed equivalent idea and can predict the exact amount of axial movements of aberration compensators needed to correct the residual aberrations. This improvement may overcome the trial-and-error shortcoming of the traditional method and greatly improve the working efficiency when applied to practical lens assembly. Acknowledgement
Fig. 11. Numerical relation between PV value of the wavefront aberration and error of the refractive index nd1 . We can see that nd1 = ±0.001 both correspond to 14.7 wavefront aberration.
This work was supported by the Key Programs of the State Key Laboratory of Modern Optical Instrumentation, Zhejiang University, and the Scholarship Award for Excellent Doctoral Student granted by the Ministry of Education. References
Fig. 12. MTF of the afocal system from Zemax.
axially moved to modify the air space tair1 between L1 and L2. We first tried nd1 = 1.51585 in Zemax and found that the residual aberrations could be almost completely compensated [Fig. 10(c)] if the = 6.488 mm). According air space tair1 is increased by 0.20 mm (tair1 to this result, we increased the air space tair1 by 0.20 mm and tested
[1] R. Tomlinson, R. Alcock, J. Petzing, J. Coupland, Computer-aided lens assembly, Appl. Opt. 43 (2004) 579–584. [2] D. Vukobratovich, Optomechanical systems design, in: M.C. Dudzik (Ed.), The Infrared and Electro-Optical Systems Handbook, Ann Arbor and SPIE, Bellingham, WA, 1993. [3] J.J. Bacich, Precision lens mounting, U.S. Patent 4,733,945 (1988). [4] D.M. Williamson, Compensator selection in the tolerancing of a microlithographic lens, Proc. SPIE 1049 (1989) 178–186. [5] H.N. Chapman, D.W. Sweeney, Rigorous method for compensation selection and alignment of microlithographic optical systems, SPIE (1998) 102–113. [6] D. Malacara, Optical Shop Testing, 3rd ed., Wiley, 2007. [7] C. Tian, Y. Yang, T. Wei, Y. Zhuo, Nonnull interferometer simulation for aspheric testing based on ray tracing, Appl. Opt. 50 (2011) 3559–3569. [8] Y. Yang, D. Liu, G. Xin, C. Tian, Y. Luo, Y. Shen, Y. Zhuo, Research of precision interference locating method for a partial null compensator at aspheric testing, Proc. SPIE 7426 (2009) 74260R–74266R. [9] Zemax, Optical Design Program User’s Guide, Zemax Development Corporation, 2005. [10] L.A. Selberg, Radius measurement by interferometry, Opt. Eng. 31 (1992) 1961–1966. [11] E.P. Goodwin, J.C. Wyant, Field Guide to Interferometric Optical Testing, SPIE, Bellingham, WA, 2006. [12] J.C. Wyant, K. Creath, Basic wavefront aberration theory for optical metrology, in: R.R. Shannon, J.C. Wyant (Eds.), Applied Optics and Optical Engineering, Academic, 1992, pp. 15–17.
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Optik journal homepage: www.elsevier.de/ijleo
Computer-aided aberration compensation in optical system assembly Chao Tian, Yongying Yang ∗ , Yongmo Zhuo State Key Laboratory of Modern Optical Instrumentation, Zhejiang University, 38 Zheda Road, Hangzhou 310027, China
a r t i c l e
i n f o
Article history: Received 7 January 2012 Accepted 29 May 2012
Keywords: Lens assembly Aberration compensation Interferometry Testing
a b s t r a c t We propose a computer-aided approach for residual aberration compensation of optical systems after initial assembly. The approach considers that intrinsic variables of an optical system such as index of refraction, radius of curvature and thickness are approximately equivalent in small ranges and the residual aberrations may be attributed to the error of one of them (e.g., index). In this way, if experimental wavefront aberration seriously deviates from theoretical one, it may be easy to estimate the exact amount of axial movements of one or more lenses needed to compensate for the aberrations. Normally, two iterative attempts (i.e., disassembly, spacer customization, reassembly and testing) are enough to give an acceptable level of performance, which may save much time and labor compared to the traditional trial-and-error method. Two examples from practice have demonstrated its validity. © 2012 Elsevier GmbH. All rights reserved.
1. Introduction A designed optical system generally needs to go through four phases before it could be delivered to users, that is, optical and mechanical design, elements fabrication, lens assembly and system testing. At the initial phase, designers should give both optical and mechanical solutions to the system according to technical specifications, perform tolerance analysis and prepare lens as well as mechanical drawings. Then these drawings are sent to factories and the elements are fabricated according to requirements. After that, the system is assembled in optical shops and finally, tested to evaluate its performance. Each of the first three steps is important and critical to the final performance of the optical system. If no fatal errors exist in the designs, all elements are fabricated within tolerances and the system is properly aligned and mounted, the actual performance of the system after initial assembly should be quite close to the theoretical one. However, serious deviation between them may frequently occur in practice. For these situations, we may axially adjust one or more lenses (i.e., aberration compensators) to fix the problem (i.e., compensate for residual aberrations). One well-known example is that adjustments of the central air space of a double-Gauss lens may be used to control the astigmatism of the system without affecting other aberrations much. Several other successful applications can also be found in the literature [1]. Vukobratovich [2] and Bacich [3] presented some successful solutions to adjust one or more lenses to compensate for residual aberrations of optical systems with adjustable screws built into the design. Williamson [4,5] described the selection rule
of aberration compensators, which indicates that compensators should be selected based on the sensitivity of each element for primary aberrations such as spherical aberration, coma, astigmatism and distortion so that reasonable movement produces the desired effect. Although built-in adjusting mechanisms bring convenience for further system optimization because sequential movements of compensators are easy, they increase the mechanical complexity and are generally applied in high-quality lens assembly. For most optical systems that do not have such mechanisms, we have to iteratively disassemble the system, change air spaces, assemble and test it if better performance is to be expected. This is a time-consuming and laborious work because exact amount of movements of compensators is difficult to determine and several iterative attempts are normally required before an optimal state is reached. This situation becomes worse when more than expected fabrication and/or assembly errors are present (see Section 3) or the system is very complex. To improve this situation, we propose a computer-aided method for residual aberration compensation during the stage of lens assembly. The method can exactly estimate the amount of axial movements of aberration compensators that is needed to correct the residual aberrations. According to the estimate, the performance of an optical system may be improved to an acceptable level within two attempts.
2. Basic idea ∗ Corresponding author. Tel.: +86 571 87951514; fax: +86 571 87951514. E-mail addresses: [email protected] (C. Tian), [email protected] (Y. Yang). 0030-4026/$ – see front matter © 2012 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2012.05.033
The performance of an optical system can be measured by many indicators, such as ray aberration, wavefront aberration, point
C. Tian et al. / Optik 124 (2013) 1930–1935
tair . Therefore, the system performance may be enhanced by axially adjusting certain lenses to change the air space tair and the amount of adjustments is also determinable. According to the approximately equivalent idea, the deviation between the real aberration W and the nominal one W could be mainly attributed to be the error of one (e.g., refractive index of one glass) instead of several variables of the system. As thus, the analysis can be simplified and we get
Reference wavefront
Exit pupil
Aberrated wavefront
Spherical wavefronts from ojbect point
Q´
Q Optical system
Paraxial focus n
O´
O Q Q´ W = n.QQ´
Fig. 1. Wavefront aberration of an optical system. The small diagram shows the details.
spread function (PSF) and modulation transfer function (MTF). Among them, the wavefront aberration, which unifies the geometrical and diffraction theories of image formation, is the most essential measure. Suppose that an axial object point O is imaged by an optical system (Fig. 1). If the system is perfect, the wavefront emerging from the exit pupil should be spherical and centered at the paraxial image point O , which is geometrically perfect. When aberrations are present, however, the emerging wavefront will be aberrated and the image is not perfect any more. The deviation QQ between the aberrated wavefront Waber and the reference wavefront Wref is the wavefront aberration W or optical path difference (OPD), which reflects the degradation of the image quality. Mathematically, it can be written as W = Wref − Waber = n · QQ , where n is the refractive index of the medium. From the point of wavefront aberration, it is easy to explain the approximately equivalent idea of the method. For an optical system after initial assembly, the nominal wavefront aberration W at the exit pupil can be expressed as the function of its intrinsic variables, that is, W = f (n, r, tlen , tair ),
(1) T
n , r , tlen
tair
W = f (n , r , tlen , tair ) ≈ f (n , r, tlen , tair ),
(3)
where n is the equivalently predicted refractive index (which may not actually be the case). With the fictitious refractive index n , we may first select one or more lenses as aberration compensators [4] and then estimate the amount of axial movements tair needed to compensate for the residual aberrations. If we disassemble the system, axially adjust the compensators by tair , the residual aberrations should almost disappear, that is, W = f (n , r, tlen , tair ) ≈ W,
(4)
where tair = tair + tair is the corrected thickness vector of air spaces. It should be noted that since optical systems after assembly are usually tested by interferometers and the interferogram I is the even function of the wavefront aberration W (i.e., I = cos(±2 · 2W /)), two fictitious values n1 and n2 of the refractive index n may simultaneously exist for one interferogram and satisfy n1 < n < n2 . For these situations, we may use some prior knowledge to help determine which value is more reasonable (see Section 3.1). If no prior knowledge exists, we need to try one by one. The proposed computer-aided method for aberration compensation in optical system assembly may be outlined as follows:
T
where n = [n1 , n2 , n3 , . . .] , r = [r1 , r2 , r3 , . . .] , tlen = [tlen1 , tlen2 , tlen3 , . . .]T and tair = [tair1 , tair2 , tair3 , . . .]T are the nominal index vector, radius vector, thickness vectors of lenses and air spaces, respectively; ni , ri , tleni and tairi are the nominal refractive index of each glass, radius of each surface, thickness of each lens and air space, separately, where i = 1, 2, 3, . . . Although the nominal wavefront aberration W is usually small and quite close to zero for well designed and optimized system, it only represents their theoretical performance. To see their real performance after assembly, it is necessary to measure the actual wavefront aberration W using some techniques such as interferometers [6], which can be written as W = f (n , r , tlen , tair ),
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(2)
where and are the real index vector, radius vector, thickness vectors of lenses and air spaces, respectively. If the real wavefront aberration W is also close to zero (as the theoretical one W), we may consider that the optical system meets the requirements and can be delivered to users. However, if it deviates from the theoretical one W too much, it is certain that some errors exist and the optical system should be fixed before applications. Considering that all the intrinsic variables (i.e., n , r , tlen and tair ) are related to rays deflection ability of the system, their variations in small ranges will affect the wavefront aberration linearly. In other words, the wavefront aberration induced by the errors of one or more variables may be corrected by slightly adjusting other variables and therefore, these intrinsic variables may be approximately regarded equivalent in small ranges. Based on this idea, if we want to fix an optical system with residual wavefront aberration after initial assembly, we may simply modify the values of certain variables. For a specified optical system after assembly [Eq. (2)], since the index n , radius r and thickness of the lenses tlen are all determined and cannot be changed, the only freedom is the air space
(1) Compute the nominal wavefront aberration W with nominal values of the variables n, r, tlen , tair and compare it to the physical one W . (2) Modify the nominal value of one of the variables (e.g., refractive index n) until the wavefront aberration at the exit pupil is close to W and obtain two fictitious indices n1 and n2 . (3) Determine which value is more reasonable. If difficult to judge, try n1 . (4) Select aberration compensators, adjust air spaces tair until the wavefront aberration approaches W and record the amount of movements tair . (5) Disassemble the system, move the selected compensators axially by tair , assemble and test the system again. (6) If the physical wavefront aberration W deviates from the nominal one W more seriously, try n2 and repeat steps 4–5. Otherwise, stop. After steps 1–6, the experimental wavefront aberration W should be much closer to the theoretical one W and the performance of the system should be greatly enhanced. However, if the performance criteria are still not achieved, these steps may be repeated by replacing the air spaces tair with tair . Notice that since intrinsic variables of the system are approximately equivalent, other parameters such as radius of curvature, thickness may also be used to interpret the deviation between W and W . 3. Experimental verifications and discussions The validity of the computer-aided method was tested with two practical experiments. In the first, it predicted the change of wavefront aberration of an aplanatic system when one of the lenses was axially adjusted by known amounts, which agrees well
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C. Tian et al. / Optik 124 (2013) 1930–1935 Table 1 Lens data of the aplanatic system.a
b
1 2 3 4 5 6 IMG
Radius
Thickness
Glassc
Semidiameter
Conic
450.000 −80.962 −52.333 −101.614 39.028 86.990 Infinity
7.690 16.018 5.000 2.500 7.040 86.990 –
(1.51682, 64.36)
25.0 25.0 25.0 25.0 25.0 25.0
0 0 0 0 0 0 0
(1.61618, 36.87) Same as surface 1
a
The radius, thickness and semidiameter are all in millimeters. The surface is the stop of the system. c The first and second data in parenthesis are the index nd and Abbe number vd of d-light (at 587.5618 nm), respectively. b
Beam splitter
Reference Transmission sphere flat L1 L2 L3
Reference surface
C
Imaging lens
tair 1
Detector
Fig. 2. Opto-mechanical structure (a) and nominal wavefront aberration (0.30) from Zemax (b) of the aplanatic system.
with experimental result. In the second, it successfully guided the aberration compensation of an afocal system by just one attempt. 3.1. Aberration compensation of an aplanatic system (transmission sphere) Transmission sphere (TS) plays an important part in interferometric testing of spherical surfaces [6]. It has two basic functions, that is, transforming the incident plane wave into a spherical wave and returning a reference wave from its reference surface (usually the last surface). Since the shape of the transformed wavefront is expected to be perfectly spherical, the TS is required to be free of spherical aberration. Also because the reference and the test wavefronts should follow the same path through the interferometer to meet the null condition [7], the focus of the TS should coincide with the curvature center (aplanatic point) of the reference surface. In the current work, we designed and assembled an F/4.3 TS for a homemade Fizeau interferometer [8] with its last surface as the reference. The working wavelength and entrance pupil diameter is 632.8 nm and 25.0 mm, respectively. The opto-mechanical structure and nominal lens data after initial assembly are shown in Fig. 2(a) and Table 1, separately. Theoretically, the peak to valley (PV) value of the wavefront aberration W at the exit pupil and diameter of the spot diagram at the image surface (0-degree field of view) is about 0.03 [Fig. 2(b)] and 0.55 m [Fig. 6(a)], respectively, which indicates that the system has excellent performance. However, when the assembled system was tested in a Fizeau interferometer (Fig. 3), we found that the residual wavefront aberration (main defocus) on the detector actually has a magnitude of about 9.0 [Fig. 4(a)], which is contrary to our expectations. To see whether the aberration is caused by incorrect assembly, we carefully examined the air space tair between lenses and found no abnormality. At this point, the most reasonable explanation is that there is something wrong with the lenses and real values of them deviate too much from their nominal ones. The problem was fixed by axially adjusting one of the lenses.
Fig. 3. Measurement of the aplanatic system by a Fizeau interferometer.
We first employed the traditional trial-and-error strategy to correct the system. The first lens L1 was chosen as the aberration compensator. To see which direction of the axial movement of L1 was correct, we disassembled the system, increased the air space tair1 (between lenses L1 and L2) for 0.10 mm by adding a 0.10 mm spacer and retested it in the Fizeau interferometer. Unfortunately, fringes of the interferograms (not shown) became denser than before [Fig. 4(a)], which means that the air space tair1 should be decreased. As thus, we iteratively reduced the air space tair1 by 0.20 mm (tair1 = 15.818 mm), 0.50 mm (tair1 = 15.518 mm), and 0.70 mm (tair1 = 15.318 mm), repeated the previous four steps (i.e., disassembly, spacer customization, reassembly and testing) and showed the corresponding testing results in Fig. 4(b)–(d). As we can see, most of the residual aberrations have been compensated by decreasing the air space tair1 for 0.70 mm and therefore, the performance of the TS is greatly improved [the magnitude of the residual wavefront aberration in Fig. 4(d) is smaller than 0.5].
Fig. 4. (a–h) Aberration compensation results by the traditional method and correspondingly predicted results by the proposed method. The first and second row, experimental and predicted wavefronts for tair1 = 0, −0.20, −0.50, −0.70 mm, respectively.
C. Tian et al. / Optik 124 (2013) 1930–1935
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Fig. 5. Numerical relation between PV value of the wavefront aberration and error of the radius rref . We can see that rref = ±0.28 mm both correspond to 9.0 wavefront aberration.
Although the traditional trial-and-error method may be utilized to enhance the system performance, it is time-consuming and laborious because each iterative attempt is discrete and requires four tedious steps mentioned above. This situation may be eased by use of the proposed computer-aided method. Reconsider the physical wavefront aberration of the system [Fig. 4(a)] in the Fizeau interferometer. We assumed that all other variables of the system are correct except the radius rref (i.e., r6 ) of the reference surface. By modeling the interferometer (Fig. 3) in Zemax [7,9], the relation between the OPD on the detector and error of the radius rref was easily obtained, as shown in Fig. 5. We can see that there are = 87.270 mm and two fictitious values of the radius rref (i.e., rref = 86.710 mm), which both can produce interferograms with rref 9.0 circular fringes [Fig. 4(e)] as those in the interferometric testing [Fig. 4(a)]. Taking into account that the returned beam from the reference surface was convergent and its diameter was obviously smaller than 25 mm (an experimental phenomenon and with no use for the traditional method), the fictitious value of the radius rref should be greater than its nominal value 86.990 mm and therefore, = 87.270 mm was a more reasonable option. This conclusion rref can be easily obtained if we trace rays through the TS, as shown in Fig. 7(a). Based on the result, we numerically reduced the air space tair1 by 0.20, 0.50, 0.70 mm as before and found that the correspondingly predicted interferograms in Zemax [Fig. 4(f)–(h)] agree well with those in experiment [Fig. 4(b)–(d)]. In other words, the residual aberrations of the system could be compensated to an acceptable level by just one trial if the computer-aided method is utilized. Since all variables are equivalently determined in the proposed method, it can also be used to estimate other measures of the system besides the wavefront aberration. In this example, we also predicted the spot diagram of the system before [Fig. 6(b)] and
Fig. 7. Ray tracing of the TS with parallel incident beam (black solid lines): (a) ray paths and (b) change of the OPD when the mirror M is in different positions (pay attention to the three critical points, F, C and F2 ). See the text for details.
after [Fig. 6(c)] the correction. The evaluation surfaces were both located in the curvature center C (i.e., 87.270 mm from the reference surface). As we can see, the spot diameter after correction is much smaller than that before correction, which also indicates improvement of the system performance. As the TS is ultimately used in an interferometer, we here briefly discuss how the residual aberrations if not corrected may influence the interferometric testing results. Supposing that parallel beam (W0 = 0) is incident to the simplified model [Fig. 7(a)] of the TS, we can obtain that its back focal length AF must be shorter than the radius AC (i.e., rref ) because the wavefront W1 reflected by the reference surface Sref was convergent. This conclusion holds, no matter which variables of the system may be the problem. If we mount the TS on a Fizeau interferometer and place a flat mirror M in the vicinity of its focus F, the OPD on the detector can be written as OPD = W1 − W2 , where W1 and W2 are the wavefronts returned from the reference surface Sref and the mirror M, respectively. How the OPD will change if the mirror M is continuously moved from the negative direction F− of the focus F to its positive direction F+? To answer this question, we need to investigate three critical positions, i.e., the focus F, the curvature center C and the point F2 (image point of F when the mirror M is in the position C). For the position F (i.e., cat’s eye position), the incident parallel beam W0 will be symmetrically reflected by MF and the returned beam W2F will also be parallel (i.e., W2F = 0). Therefore, the OPDF on
Fig. 6. Spot diagrams of the aplanatic system from Zemax: (a) designed spot, (b) and (c) predicted spot before and after aberration compensation, respectively. Note that all figures were plotted under the same scale.
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C. Tian et al. / Optik 124 (2013) 1930–1935
of view), respectively, from which we can see that the system is near-diffraction-limited in principle. However, large wavefront aberration appeared [Fig. 10(a)] when the system after initial assembly was tested in the CRSI (Fig. 9). Problems occurred again and we used the proposed method to correct the residual aberrations. Before the correction, one thing, i.e., conversion relationship between a radial shearing interferogram and a Fizeau or TwymanGreen interferogram should be clear. This is mainly due to the fact that modeling of the telescope in a CRSI is much more difficult than that in a Fizeau or Twyman-Green interferometer. Therefore, we simply simulated the telescope in a Fizeau interferometer by adding a perfect lens [Fig. 8(a)] and compared the synthetic interferograms to those in the CRSI through conversion. In the CRSI, the contracted wavefront Wc [11] can be written as Wc = W0 2 0 2 + W0 4 0 4 + W1 3 1 3 cos + W2 2 2 2 cos2 , Fig. 8. Optical structure (a) and nominal wavefront aberration (0.04) from Zemax (b) of the afocal system.
the detector can be given as OPDF = W1 which means that interferograms with the same shape as Fig. 4(a) can be obtained if the mirror M is in the position F. For the position C, the returned wavefront W2C may be viewed as emitted from the point F2 . Since FC = CF2 , we have ˛ ≈ ˇ, W2C ≈ W1 and OPDC ≈ 0 which means that zero-fringe interferogram will appear if the mirror M is in the position C. For the position F2 , we may use the same analysis method and obtain W2F2 ≈ 2W1 . In this way, the OPDF2 on the detector has the same magnitude OPD
where and are radial and angular coordinates over a unit circle defining the pupil, Wl m n are aberration coefficients and l, m and n are integers [12]. If we define the effective radial shear R = c /e , the expanded wavefront We can be written as We = W0 2 0 (R)2 + W0 4 0 (R)4 + W1 3 1 (R)3 cos + W2 2 2 (R)2 cos2 ,
(6)
where c and e are the radii of the contracted and expanded beams, respectively. Subtracting Eq. (6) from Eq. (5), the OPD between the two wavefronts is
= Wc − We = W0 2 0 2 (1 − R2 ) + W0 4 0 4 (1 − R4 ) + W1 3 1 3 (1 − R3 ) cos + W2 2 2 2 (1 − R2 ) cos2
as W1 but with opposite sign, that is, OPDF2 ≈ −W1 , which indicates that interferograms at the positions F and F2 are the same in shape. The three positions divide the image space into four regions, that is, A − F, F − C, C − F2 and F2 − +∞ and the relationship between the OPD and the position of the mirror M can be qualitatively obtained, as illustrated in Fig. 7(b). From the analysis above, we know that if the aberrations of the TS are not corrected, its focus F will not coincide with the curvature center C of the reference surface Sref . When the flat mirror M is in the cat’s eye position, the reference and the test wavefronts (i.e., W1 and W2F ) will not follow the same path through the interferometer and retrace errors exist [7]. This may introduce significant errors to interferometric measurements, such as absolute calibration of spheres [6] and radius measurement [10]. On the other hand, if the aberrations are compensated, the corrected focus F will be quite close to the curvature center C and errors can be largely reduced. 3.2. Aberration compensation of an afocal system (Galilean telescope) In the previous example, we show that the computer-aided method successfully predicts the performance changes of the aplanatic system when one compensator is axially moved. Here we give another example, in which the method is employed to guide aberration compensation of an afocal system, i.e., a Galilean telescope used to contract and expand beams in a cyclic radial shearing interferometer (CRSI) [6]. The optical structure and lens data (after optimization during initial assembly) of the system are shown in Fig. 8(a) and Table 2, separately, and its entrance pupil diameter, working wavelength and magnification (ratio of the focal length of the objective lens to that of the eye piece lens) are 86.0 mm, 52.7.0 nm and 2.0, respectively. Fig. 8(b) and 12 show its theoretical wavefront aberration at the exit pupil and MTF (dash-dot curve, 0-degree field
(5)
.
(7)
Comparing Eq. (7) with Eq. (5), we may find that the resulting interferograms in a CRSI is the same as those in a Fizeau interferometer if each aberration coefficient is divided by (1 − Rm ). Considering that defocus W0 2 0 is the dominant aberration and the effective radial shear is 1/2, the magnitude of wavefront aberration of the afocal system should be 11.0 × (4/3) = 14.7 [Fig. 10(a)] if tested in a Fizeau interferometer. In this case, we attributed the serious deviation to the error of the index of dlight nd1 of lens L1 (also L3) and obtained the relationship (Fig. 11) between PV value of the wavefront aberration W and error of the index nd1 by tracing rays through the Fizeau interferometer in Zemax. As we can see, there are also two fictitious values of nd1 , i.e., nd1 = 1.51585 and nd1 = 1.51785, which can yield interferograms with a magnitude of 14.7 [Fig. 10(b)]. Since no prior knowledge helped us to determine which value is more reasonable, we were ready to try them one by one with the computer-aided method. Here the first lens L1 was chosen as the compensator and could be
M1
BS L1 L2 L3 M2
Imaging lens We Wc Detector
Fig. 9. Measurement of the afocal system by a CRSI. BS, beam splitter; M1 and M2, mirrors.
C. Tian et al. / Optik 124 (2013) 1930–1935
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Table 2 Lens data of the afocal system.a
1b 2 3 4 5 6 7c IMG a b c
Radius
Thickness
Glass
Semidiameter
Conic
Remarks
108.683 −334.190 −347.265 417.384 −72.599 −1219.07
24.992 6.288 10.990 103.264 5.988 50.000 100.000 –
(1.51685, 64.59)
52.0 52.0 52.0 52.0 28.0 28.0 28.0
0 0 0 0 0 0 0
f = 100.000
Infinity
(1.61307, 34.94) Same as surface 1
The radius, thickness, semidiameter and focal length are all in millimeters. The surface is the stop of the system. The surface is a perfect lens with its focal length f = 100.000 mm.
Fig. 10. Aberration correction results by the proposed method: (a) experimental wavefront (about 11.0) after initial assembly in the CRSI, (b) synthetic wavefront (about 14.7) in the simulated Fizeau interferometer when nd = 1.51585, (c) predicted wavefront when t = 0.20 mm, (d) experimental wavefront after reassembly in the CRSI when t = 0.20 mm.
the telescope in the CRSI after reassembly. Fortunately, most of the aberrations were corrected and the physical wavefront aberration [Fig. 10(d)] agrees with the predicted one [Fig. 10(c)]. Therefore, the system performance has been greatly improved by just one trial using the proposed method. Using the fictitious index nd1 = 1.51585, we also predicted the MTF of the telescope before (dotted curve) and after (dashed curve) the correction, which are shown in Fig. 12. As we can see, the corrected MTF also has been significantly enhanced and is close to the designed one. 4. Conclusion We present a simple and effective computer-aided method that is capable of compensating for residual aberrations of optical systems in lens assembly. The method utilizes the proposed equivalent idea and can predict the exact amount of axial movements of aberration compensators needed to correct the residual aberrations. This improvement may overcome the trial-and-error shortcoming of the traditional method and greatly improve the working efficiency when applied to practical lens assembly. Acknowledgement
Fig. 11. Numerical relation between PV value of the wavefront aberration and error of the refractive index nd1 . We can see that nd1 = ±0.001 both correspond to 14.7 wavefront aberration.
This work was supported by the Key Programs of the State Key Laboratory of Modern Optical Instrumentation, Zhejiang University, and the Scholarship Award for Excellent Doctoral Student granted by the Ministry of Education. References
Fig. 12. MTF of the afocal system from Zemax.
axially moved to modify the air space tair1 between L1 and L2. We first tried nd1 = 1.51585 in Zemax and found that the residual aberrations could be almost completely compensated [Fig. 10(c)] if the = 6.488 mm). According air space tair1 is increased by 0.20 mm (tair1 to this result, we increased the air space tair1 by 0.20 mm and tested
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