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Fizeau interferometer with binary phase Fresnel-zone plate reference for precision measurement of large convex lens
Optics and Lasers in Engineering 110 (2018) 348–355
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Fizeau interferometer with binary phase Fresnel-zone plate reference for precision measurement of large convex lens Xiaohong Wei a, Yuhang He a, Zhengkun Liu b, Kaiyuan Xu a, Bo Gao a, Qiang Li a, Ang Liu a, Liqun Chai a,∗ a b
Research Center of Laser Fusion, China Academy of Engineering Physics, Mianyang 621900, China National Synchrotron Radiation Laboratory, University of Science and Technology of China, Hefei 230029, China
a r t i c l e
i n f o
OCIS codes: Optical testing Interferometry Fresnel zone plate Large optics 120. 2880 120. 3940 120.7000
a b s t r a c t A method of obtaining precision transmission wavefront of large convex lens interferometrically using binary Fresnel zone plate (FZP) was presented. We present the results from a set of experiments that demonstrate the accuracy, flexibility, and the simplicity of performing the FZP test. A direct comparison of the FZP measurement with results from a Fizeau interferometry method shows excellent agreement. Finally, measurement uncertainty of lens due to alignment error and FZP fabrication processes is analyzed. This resulting analysis shows less than 𝜆/10 accuracy for measuring the transmission wavefront of a sphere lens with 31.25 m focal length and Φ410 mm clear aperture.
1. Introduction There is a growing demand for lenses and mirrors in the optics industry, such as information and communication technology or all branches of industrial manufacturing. To ensure the performance of optical systems, the surface shape should be manufactured to an accuracy of a small fraction of a wavelength. Many optical testing methods were developed to guide the fabrication of optical surface accurately [1-5]. Convex spherical lens is commonly measured with matching concave mirrors [6-8], fringes of interference show the shape difference between the two parts. The obvious difficulty with this method is the requirement of making and measuring a concave reference mirror. Moreover, it’s virtually impossible to use this method to measure wavefront of lens with long focal length, such as tens of meters long, for air turbulence induced error and laboratory spatial problem cannot be ignored. However, holographic test could solve this problem simply. In addition, aberrations such as spherical departure of the lens can be compensated by diffraction from a circular computer-generated hologram (CGH) or Fresnel zone plate (FZP) [9-12], which is fabricated onto a flat surface. Fresnel zone plate (FZP), a diffractive equivalent to a reflecting sphere, is a powerful tool for a whole range of alignment and calibration issues in interferometry and diffractive optics in general [13,14]. It is a special case of a CGH. The pattern on a FZP determines whether the wavefront is split into a number of beams, compensates for some aberrations in an optical assembly, or performs other useful optical functions. This high degree of flexibility in creating wavefronts of light with de-
∗
Corresponding author. E-mail address: [email protected] (X. Wei).
sired amplitudes and phases has made FZP extremely useful. A properly designed FZP can perform the functions of a conventional lens or mirror. Diffractive optics has been used for years in optical testing [15–18] and is on its way to become an established technique. With the steady development of fabrication techniques for computer generated hologram (CGH), such as laser direct writing, e-beam writing and ionbeam writing [19-23] that provide both high accuracy and high resolution, this elegant way of controlling light has become widely accepted. In this paper, lens transmission wavefront (TWF) measurement using FZP method is explicitly introduced, and a large zone plate over Φ410 mm aperture was fabricated with very high precision. Both theoretical analysis and experiment result show that the precision of this method is better than 𝜆/10 (𝜆 = 632.8 nm). Compared with Fizeau interferometry method for sphere surface testing method, the FZP method can be effectively applied to the transmission wavefront of lens with long focal length, for the configuration is very simple and compact, which is very easy to be adjusted, thus enable to obtain high precision.
2. Metrology 2.1. System construction Traditionally, the compensation method is used to measure the transmission wavefront of large convex lens with aberration, as shown in Fig. 1. Spherical wave from an interferometer is transmitted through a compensation lens and the lens under test successively, the output parallel light is retroreflected by a reference flat.
https://doi.org/10.1016/j.optlaseng.2018.06.006 Received 15 May 2017; Received in revised form 6 June 2018; Accepted 11 June 2018 0143-8166/© 2018 Elsevier Ltd. All rights reserved.
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Optics and Lasers in Engineering 110 (2018) 348–355
possible with ± 0.5 𝜇m positioning error, offering immediate precision enhancement to the spherical shape this element represents in diffraction. Whereas use of a spherical mirror is more costly and time consuming. The second benefit is the almost arbitrary wavefront generation. The hologram can be designed such that all rays incident on the surface are reflected back via the same path they came from, i.e., the test wave of the interferometer makes a normal incidence angle with the reflective hologram surface and is retroreflected. In this case, the interferogram only shows the deviation of the surface under test from the design prescription. Thus, it can be seen that the FZP can be powerful tools for reducing the number of elements in a setup, thereby enhancing system performance. However, when adopting the traditional compensation method, this null test requires the use of some auxiliary optics to match the f-number and to compensate system aberration, which therefore introduces unwanted error into the test. Thirdly, this method is especially beneficial to the TWF measurement of lens with long focal length, for the optical path length can be less than 1 m with proper optical design. Furthermore, the longer the focal length is, the lower the fabrication difficulty and higher fabrication precision of FZP is. On the other hand, using traditional compensation method, TWF measurement of lens with long focal length creates laboratory space problems, not to mention potential wavefront errors from air turbulence. One in particular is originality respect to previous work done by Zhou and Burge. (1) The reflective wavefront of CGH/FZP is used in the measurement here; Whereas the transmission wavefront of CGH is used to compensate the wavefront of lens under test. (2) The size of FZP could be very large since diffraction pattern on FZP is quite simple; While in the transmission mode, the size of CGH is generally small, and complicated alignment pattern should be designed since the testing system is quite sensitive to adjusting error. (3) Homogeneity of FZP has nothing to do with the measurement accuracy; Conversely, homogeneity is one of the major measurement errors in the transmission mode, and the effect of refractive index deviation should be analyzed in detail.
Fig. 1. Experimental setup for measuring wavefront of lens under test using the compensation method.
The testing optical system of the FZP method is composed of a phaseshifting interferometer (PSI), lens under test and a FZP, as shown in Fig. 2. When performing a lens TWF test, FZP is equivalent to a convex mirror. Part of the collimated light from the PSI is reflected from the transmission flat (TF), forming the reference beam; the rest transmits through the TF and lens, then is retroreflected from the zone plate, and finally into the interferometer, forming the test beam. From the interference fringe, lens TWF can be obtained. A precise characterization of the lens TWF should separate the contribution of the error in FZP substrate flatness. Measuring the substrate before the hologram pattern is generated has two main problems: one is the repositioning issue and the second, worse problem is the unknown substrate figure change introduced by the FZP fabrication process. When we subtract the nonzero-order surface measurement from the zero-order measurement, the FZP substrate error can be totally removed, leaving the residual wavefront errors from the fabrication non-uniformities. Thus, in this paper we adopt the method of measuring the zeroth order diffraction from FZP to account for substrate irregularities. In the zeroth order, the hologram emulates a flat mirror, when it is put behind the TF, the reflection interference wavefront is the reflected zeroth order wavefront of FZP. The measurement procedure of lens TWF is as follow: (1) Measure the reflected zeroth order wavefront of the FZP. (2) Put in lens, leaving a lens-to-FZP spacing 𝜏 the designed value. Keep the position of FZP unchanged and adjust the lens until the interferogram turns into null fringe, the obtained wavefront is the first-order wavefront. Lens TWF due to fabrication error can be obtained through subtracting the zeroth order wavefront from the first-order wavefront. But be warned, pixel matching by superposing edge feature on interferogram should be carried out, since rays propagating to FZP is convergent, as shown in Fig. 2.
2.2. Design and fabrication of FZP The design of FZP is to determine the radii and etching depth of the band structure of concentric rings on FZP substrate. According to diffraction principle, the radial locations for ring edges, i.e., Fresnel-zone boundaries, are the positions that give integral numbers of half-wavelength optical path differences at the center of curvature from the radius of curvature value, R. FZP possesses the lens-like characteristic, and the geometric diagram of FZP is shown in Fig. 3. The diffraction wavefront of FZP should match the transmission wavefront of lens with aberration. Following the aplanatic principle, and via optical design software, the distance between lens and FZP, the radii of FZP, and the coefficients of the various orders phase polynomial could be varied and the system optimized to retroreflect light. The geometric diagram is shown in Fig. 4. All the parallel light coming from the TF of a PSI transmits through the lens under test to the zone plate
Note that the lens-to-FZP spacing 𝜏 should be chosen reasonable. It is desirable to minimize the airspace 𝜏 in order to minimize the amount of atmosphere traversed by the test beam, and at the same time leave enough room for hands and for maneuvering of mounts. The advantages of using a reflective FZP to replace a spherical mirror are threefold. First and foremost, the Fresnel-zone plate consists of nothing more than a circular binary amplitude grating. Hence, it is very easily manufactured with optical lithography. Lithographic writing is
Fig. 2. Experimental setup for measuring wavefront of lens under test using FZP method. 349
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tern. Ion etching is used for fabrication of phase binary FZPs. This thermochemical effect allows the direct generation of patterns with spatial resolution better than 1000 mm−1 onto bare chromium films (Fig. 5). Etching depth concerns diffraction efficiency. According to the previous work done by Zhou and Burge [8], (1) CGHs with 50% duty-cycle have less wavefront error and higher diffraction efficiency; (2) When the etching depth is close to 0.5𝜆, the wavefront error increases dramatically. Thus, in the reflective FZP application, 0.25𝜆 etching depth should be avoided. The fabrication of efficient FZP requires the precise patterning of a large number of binary surface-relief structures. For close-to-ideal performance: duty cycle variation, etching depth variation, and pattern distortion should be controlled as small as possible. Using as a reflective optic, FZP should be thick enough in order to keep the surface shape. In our experiment, the diameter of the FZP is Φ430mm, with a thickness of 70 mm. The designed duty cycle and etching depth are 0.5 and 120 nm, respectively. Since the size of FZP is very large, the band gap of the zones at the edge is only about 10 𝜇m, the fabrication of FZP is very difficult. The measured duty cycle variation is ± 2%, etching depth variation is ± 5%, and pattern distortion is ± 1 𝜇m.
Fig. 3. Geometric diagram of FZP boundary locations.
Fig. 4. Geometric diagram for the design of FZP.
3. Experiment 3.1. Comparison with Fizeau interferometry method
with the same optical path, i.e., 𝑛0 |𝑃 𝑄| + 𝑛1 |𝑄𝑀 | + 𝑛0 |𝑀𝑁 | + 𝜙(𝑟) = 𝑛0 𝑑1 + 𝑛1 𝑑2 + 𝑛0 𝑑3 = 𝐶
(1)
The feasibility and accuracy of the scheme is validated through comparison with Fizeau interferometry method. For a lens with a focal length of 1500 mm and a diameter of 80 mm already in our lab, a FZP with the first-order radius curvature of 1200 mm and a diameter of 80 mm is designed and fabricated. The optical structure of FZP method is shown in Fig. 2. In order to obtain high quality diffraction wavefront, the reflective surface shape of the FZP substrate should be PV < 𝜆/10 (𝜆 = 632.8 nm). The reflective wavefront of the zoneplate substrate is shown in Fig. 6, from which we can see that peak-to-valley (PV) value is 0.065𝜆. The clear aperture is Φ75mm. The transmission wavefront of the lens under test is shown in Fig. 7, the data loss by FZP method originate from the superposition
where C is constant, 𝜑(r) is optical path difference induced by FZP. Thus, 𝜙(𝑟) = 𝑛0 𝑑1 + 𝑛1 𝑑2 + 𝑛0 𝑑3 − (𝑛0 |𝑃 𝑄| + 𝑛1 |𝑄𝑀 | + 𝑛0 |𝑀𝑁 |)
(2)
The computer model may then be used to output the radii aperture coordinates where 𝜋 phase changes occur. FZP with small size and thin substrates can be fabricated by direct laser writing, and ion-beam etching method can be adopted to fabricate FZP with binary or continuousrelief microstructure on to large-diameter and thick optical substrates. The FZP pattern can be created using a direct writing method based on thermochemical effect of laser heating on chromium films. After it is exposed, immersing the substrate in a caustic bath develops the pat-
Fig. 5. Method for fabrication of binary and continuous relief of FZP surface. 350
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Fig. 6. The reflective wavefront of the zoneplate substrate.
Fig. 9. Transmission wavefront of the lens tested by Fizeau interferometry method.
Table 1 The test results of FZP method.
PV RMS
0 order wavefront
Lens wavefront
TWF subtract 0 order
0.078𝜆 6.605nm
0.158𝜆 17.042nm
0.134𝜆 14.583nm
Table 2 The test results of lens in 3 states. (unit: 𝜆).
PV RMS
Original state
FZP rotated
lens rotated
0.134 0.023
0.137 0.019
0. 134 0.016
Fig. 7. The transmission wavefront of a lens tested by FZP method.
was manufactured. The first-order radius of curvature of zone hologram is 31.75 m, with a diameter of Φ430mm. The lens-to-FZP spacing is 0.5 m. The wavelength of the PSI is 632.8 nm. The transmitted wavefront of the lens is represented in Fig. 10. The clear aperture is Φ410 mm, power is removed from the graph. The PV and RMS value are listed in Table 1. As shown in Fig. 9, the substrate error can be effectively reduced by subtracting the zero-order wave front of the FZP, which is carried out by interpolation in order to avoid the lateral difference with direct substracting. On condition that the substrate shape is very smooth, and the machining precision is very high, this background error can be neglected. In addition, astigmatism is obvious in Fig. 9. The following experiment is carried out to judge the cause of this feature. Firstly, keep the status of the lens, and make the FZP anticlockwise turn 90°; Secondly, maintain the FZP the original state, and make the lens anticlockwise turn 90°. The test result is shown in Fig. 11 and Table 2. We can see that when rotating the FZP, the distribution as well as quantitative value is nearly the same, whereas in the situation of rotating the lens, the astigmatism rotates the same angle, with similar PV and RMS values. Thus, we can confirm that the astigmatism is originated from the optical element itself, and error caused by FZP is very small. The repeatability of the testing system is calculated by the following method: The average of 10 consecutive measurements with 6 phase averages without changing tip, tilt or focus will be used as a reference, and the reference will be subtracted from each individual measurement. The mean & standard deviation (𝜎) of the RMS of each of the difference measurements will be computed over a 410-mm diameter aperture. The repeatability equals the mean + 2𝜎. The tested results are shown in Table 3.
Fig. 8. The optical structure for Fizeau interferometry method.
of multiple diffraction orders. The measurement result is PV = 0.266 𝜆, RMS = 36.33 nm. The clear aperture is Ф75 mm. Then this lens is measured by spherical Fizeau interferometry method, the optical structure is shown in Fig. 8. The interferometer used in this experiment is 4D 4020. This is the conventional approach for lens transmission wavefront measurement, and the accuracy is believed to be better than 𝜆/10. Lens TWF distribution is shown in Fig. 9, and the data loss by Fizeau interferometry method comes from reflected light of the lens back surface. The measured PV = 0.226𝜆, RMS = 35.93 nm. The clear aperture isФ75 mm. From the experiment results we can see that both the wavefront distribution and PV, RMS value are consistent, thus confirms the feasibility and accuracy of the FZP method. 3.2. Large-size long focal-length lens TWF measurement The big advantage of FZP method is large-size long focal-length lens TWF measurement, for it can greatly shorten optical path length, meanwhile provide high measurement accuracy. In order to measure the transmission wavefront of lens with the size of Φ430mm and the focal length of 31.25 m, a large size zone hologram 351
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Fig. 10. Test result: (a) 0 order of the FZP; (b) transmission wavefront of the lens under test; (c) transmission wavefront that subtract the substrate error of the FZP.
Fig. 11. Test TWF result: (a) lens in initial state; (b) with zone plate anticlockwise turn 90°; (c) with lens anticlockwise turn 90°. Table 3 The repeatability of FZP method. (unit: 𝜆).
RMS
1
2
3
4
5
6
7
8
9
10
result
0.0031
0.0022
0.0016
0.0021
0.0030
0.0018
0.0020
0.0034
0.0027
0.0034
0.0039
can see that the simulated PV = 0.0000𝜆, RMS = 0.0000𝜆, which reveals that system design error could be ignored. 4.2. Influence of FZP fabrication error FZP fabrication errors may be classified into two basic types: substrate figure errors and pattern errors. Pattern errors may further be classified as fringe position errors, duty-cycle errors and etching depth errors. Typical FZP substrate errors are low spatial frequency surface figure errors that are responsible for the low spatial frequency wavefront aberrations in the diffracted wavefront. In a reflection hologram setup, a surface defect on a FZP substrate with a peak-to-valley deviation of 𝛿 s will produce a phase error in the reflected wavefront that equals 2𝛿s because of the double path configuration. This substrate error is eliminated by subtracting the zeroth order diffraction from FZP, but the wavefront still contains some residual errors due to variations in the duty cycle or etching depth. A binary, linear grating model is used to build the parametric model based on scalar diffraction theory, as shows in Fig. 13, where t and D correspond to the etching depth and duty-cycle, respectively, and D is defined as D = b/p, A1 and A2 correspond to the amplitudes of the output wave fronts from the peaks and valleys of the grating, respectively. The values of A1 and A2 are determined by the amplitude functions of the reflectance coefficients at the grating interface, using Fresnel equations [24]. The phase function 𝜑 represents the phase difference between light from the peaks and valleys of the grating. The wavelength of the incident light is assumed to be much smaller than the grating period, so the scalar diffraction approximations can be applied.
Fig. 12. The designed wavefront map.
4. Error analysis The errors of the FZP null testing system includes wavefront aberrations brought by system design, FZP fabrication and alignment error. In order to assure the validation and accuracy of the measured results, these error sources need to be analyzed carefully. 4.1. System design error FZP used in this experiment is designed by ZEMAX, the simulated result is shown in Fig. 12. By optical design software, the distance between lens and FZP, the radii of FZP, and the coefficients of the various orders phase polynomial could be varied and the system optimized to retroreflect light. Thus, the system design error could be very small. We 352
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Fig. 13. (a) Procedure for the coding of a continuous phase function into the binary CGH pattern. CGH top view on the right. (b) Binary, linear grating profile.
Fig. 14. Fabrication error measured by ZYGO NewView interferometer: (a) oblique plot to show the 3-D distribution and (b) surface/wavefront profile to show the duty-cycle and etching depth error quantitatively.
Table 4 Sensitivity functions.
Table 5 Wavefront error induced by the manufacturing of FZP.
Zero order (m = 0)
Non-zero order (m≠0)
𝜕Ψ 𝜕D
A1 A2 sinφ A22 D2 +A21 (1−D)2 +2A1 A2 D(1−D)cosφ
{
𝜕Ψ 𝜕𝜙 𝜕Ψ 𝜕 A1
A22 D2 +A1 A2 D(1−D)cosφ A22 D2 +A21 (1−D)2 +2A1 A2 D(1−D)cosφ A2 D(1−D)sinφ A22 (1−D)2 +A21 D2 +2A1 A2 D(1−D)cosφ
Error sources Pattern distortions Duty-cycle errors Etching depth
∞ for sinc(mD)= 0 0 otherwise A22 −A1 A2 cosφ
Ψ 2𝜋
deviations in duty-cycle ΔD, phase function Δ𝜑 and amplitude ΔA1 . The error sensitivity functions of the different diffraction orders are shown in Table 4. Duty-cycle non-uniformities have no effect on the wavefront error for the non-zero order diffraction. The amplitude of the light from the etched regions can vary due to limitations in the reactive ion etching process. The surface roughness of the etched area may vary due to etching, so the light incident on the FZP scatters differently. Since the surface roughness is much smaller than the wavelength of incident light, the coupling of the scattered light to the diffracted light is ignored. Variations in duty cycle and etching depth are measured using an interference microscope, which provide the surface relief of the FZP. A set of the sampled points should be measured to obtain the variations in duty cycle and etching depth. One sampled measured result is show in Fig. 14. Based on the measured fabrication errors, wavefront error is computed and listed in Table 5. Nowadays, state of the art photolithography allows the hologram to be generated with high precision and finesse, which greatly reduces the error introduced by FZP manufacture.
(3)
The displacement of the recorded fringe in a hologram from its ideal position is commonly referred to as pattern distortion. The amount of wavefront phase errors produced by the hologram pattern distortions can be expressed as a product of the gradient of the diffracted wavefront function and the pattern distortion vector. For a linear grating, wavefront phase errors produced by grating pattern distortions in the m-th order beam can be calculated as: Δ𝑊 = −𝑚𝜆
𝜀 𝑝
Wavefront errors ±𝜆/34 0 ±𝜆/35
A21 +A22 −2A1 A2 cosφ −A2 sinφ A21 +A22 −2A1 A2 cosφ
Based on Fraunhofer diffraction theory, the far field diffraction wavefront 𝜓 may be related to the original wavefront by a simple Fourier transform relationship. The wavefront phase W in waves can be obtained by 𝑊 =
Fabrication tolerances ±1𝜇m ±2% ±5%
(4)
where 𝜀 is the grating position error in direction perpendicular to the fringes, p is the localized fringe spacing, and m is the diffraction order. Pattern distortions are analyzed from FZP fabrication method. The Fresnel zone is produced by transferring the phase mask to the fused silica substrate. The positioning accuracy of phase mask is ± 0.75 𝜇m, the temperature change during ion beam writing process is less than is 1 °C, the induced position error is ΔL = 210 × 103 × 0.6 × 10−6 = 0.12 𝜇m, where 210 mm is edge-to center distance, 0.6 × 10−6 is the expansion coefficient. Thus, the combined positioning error is less than ± 1 𝜇m. The designed minimum period is 34 𝜇m, according to formula (4), the pattern distortion error is ± 𝜆/34. The wavefront phase sensitivity functions [9] 𝜕 Ψ/𝜕 D, 𝜕 Ψ/𝜕 𝜑, and 𝜕 Ψ/𝜕 A1 are introduced to specify the wavefront error caused by small
4.3. Influence of alignment error The phase function compensates the transmitted wavefront from the lens accurately only when the relative positions of focus of incident TWF from the lens and the FZP surface are identical to the value given in the process of phase function calculation. Any misalignment of lens or FZP surface will lead to additional aberration. In our experiment, when the reflected 0th order wavefront is measured, the position and attitude of FZP remains unchanged, and a 5-D amount is used to adjust lens. In this case, the setup consists only two elements: The lens under test and FZP. Both elements must be aligned 353
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(a)
(b)
Fig. 15. Wavefront error induced by (a) tilt and (b) decenter.
Table 6 Wavefront errors without substrate calibration. Error sources
Wavefront errors
Substrate RMS error Pattern distortions Duty-cycle errors Etching depth Alignment errors Repeatability error Root-sum-square error
0.0127𝜆 ±𝜆/34 0 ±𝜆/35 0.0064𝜆 0.0039𝜆 0.077𝜆/48.7nm
unsolved 3-dimensional fabrication and measurement problem is converted into a 2-dimensional problem. An example of the TWF measurement for lens with 31.25 m focal length and 410 mm clear aperture was given, and a phase FZP with 410 mm clear aperture was fabricated to carry out the experiment. Thanks to the continuous progress in microlithography high precision lateral pattern can be fabricated. The substrate error is the dominant error from the FZP and that it can be removed by subtracting the first-order measurement from the zero-order one. After performing the substrate calibration, the precision of the FZP method is 0.042𝜆 (26.7 nm).
with micron precision or better. Given the correct position of FZP, a rotationally symmetric system has 5 adjustable parameters: 5 degrees of freedom for the lens (z-position, decenter x/y, tilt x/y). However, z-position error mainly induces additional power, which is subtracted when analyzing wavefront, thus can be ignored, and error induced by tilt and decenter is analyzed and computed quantitatively below. Assuming when the bright spot is superimposed on the alignment crosshairs, there is still 2 pixels’ deviation from the ideal position, which can definitely be recognized by human eyes. Combining with the configuration parameters of the ZYGO interferometer used, the adjustment accuracy of the mount, the decenter and tilt error are 45″ and 3 mm, respectively. The TWF of lens under test with tilt angle of 45″ and decenter of 3 mm by ZEMAX simulation are shown in Fig. 15. The induced wavefront error are mainly astigmatism and coma, which can be calculated by the summation of the 5th and 6th terms (astigmatism), 7th and 8th terms of Zernike fringe polynomials (coma). The calculated astigmatism induced by tilt are 0.00046𝜆, 0.00086𝜆; and the calculated coma induced by decenter are 0.00034𝜆, −0.0063𝜆. Thus, the composite alignment error is 0.0064𝜆.
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4.4. Composite error The system measurement error is the combination of all kinds of uncertainties, including FZP design, fabrication, alignment errors and repeatability errors, which is obtained in Section 3. Table 6 lists all the wavefront error sources, notice that FZP design error is not included, that’s because the designed PV and RMS value both are zero. The estimated root-sum-square (rss) wavefront error is 0.077𝜆/48.7 nm, among which, substrate error is dominant. FZP substrate errors can be removed by subtracting the zero-order measurement from the TWF. In this case, the residual rss wavefront error drops from 48.7 to 26.7 nm (0.077 to 0.042𝜆). 5. Conclusion In this paper, the use of a diffractive element, a so called nullFZP, offers a practical solution to generate the reference wavefront for large convex lens TWF measurement with long focal length. Thus, the 354
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[21] Menon R, Patel H, Gil D, Smith HI. Maskless lithography. Mater Today 2005:26–33. [22] William RM, Williams GL, Cowling JJ, Seed NL, Purvis A. High-contrast pattern reconstructions using a phase-seeded point CGH method. Appl Opt 2016;55(7):1703–12.
[23] Dai Xianglu, Xie Huimin. New methods of fabricating gratings for deformation measurements: a review. Opt Laser Eng 2017;92:48–56. [24] Born M, Wolf E. Principles of optics. 6th Ed. N.Y.: Cambridge University Press; 1980.
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Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Fizeau interferometer with binary phase Fresnel-zone plate reference for precision measurement of large convex lens Xiaohong Wei a, Yuhang He a, Zhengkun Liu b, Kaiyuan Xu a, Bo Gao a, Qiang Li a, Ang Liu a, Liqun Chai a,∗ a b
Research Center of Laser Fusion, China Academy of Engineering Physics, Mianyang 621900, China National Synchrotron Radiation Laboratory, University of Science and Technology of China, Hefei 230029, China
a r t i c l e
i n f o
OCIS codes: Optical testing Interferometry Fresnel zone plate Large optics 120. 2880 120. 3940 120.7000
a b s t r a c t A method of obtaining precision transmission wavefront of large convex lens interferometrically using binary Fresnel zone plate (FZP) was presented. We present the results from a set of experiments that demonstrate the accuracy, flexibility, and the simplicity of performing the FZP test. A direct comparison of the FZP measurement with results from a Fizeau interferometry method shows excellent agreement. Finally, measurement uncertainty of lens due to alignment error and FZP fabrication processes is analyzed. This resulting analysis shows less than 𝜆/10 accuracy for measuring the transmission wavefront of a sphere lens with 31.25 m focal length and Φ410 mm clear aperture.
1. Introduction There is a growing demand for lenses and mirrors in the optics industry, such as information and communication technology or all branches of industrial manufacturing. To ensure the performance of optical systems, the surface shape should be manufactured to an accuracy of a small fraction of a wavelength. Many optical testing methods were developed to guide the fabrication of optical surface accurately [1-5]. Convex spherical lens is commonly measured with matching concave mirrors [6-8], fringes of interference show the shape difference between the two parts. The obvious difficulty with this method is the requirement of making and measuring a concave reference mirror. Moreover, it’s virtually impossible to use this method to measure wavefront of lens with long focal length, such as tens of meters long, for air turbulence induced error and laboratory spatial problem cannot be ignored. However, holographic test could solve this problem simply. In addition, aberrations such as spherical departure of the lens can be compensated by diffraction from a circular computer-generated hologram (CGH) or Fresnel zone plate (FZP) [9-12], which is fabricated onto a flat surface. Fresnel zone plate (FZP), a diffractive equivalent to a reflecting sphere, is a powerful tool for a whole range of alignment and calibration issues in interferometry and diffractive optics in general [13,14]. It is a special case of a CGH. The pattern on a FZP determines whether the wavefront is split into a number of beams, compensates for some aberrations in an optical assembly, or performs other useful optical functions. This high degree of flexibility in creating wavefronts of light with de-
∗
Corresponding author. E-mail address: [email protected] (X. Wei).
sired amplitudes and phases has made FZP extremely useful. A properly designed FZP can perform the functions of a conventional lens or mirror. Diffractive optics has been used for years in optical testing [15–18] and is on its way to become an established technique. With the steady development of fabrication techniques for computer generated hologram (CGH), such as laser direct writing, e-beam writing and ionbeam writing [19-23] that provide both high accuracy and high resolution, this elegant way of controlling light has become widely accepted. In this paper, lens transmission wavefront (TWF) measurement using FZP method is explicitly introduced, and a large zone plate over Φ410 mm aperture was fabricated with very high precision. Both theoretical analysis and experiment result show that the precision of this method is better than 𝜆/10 (𝜆 = 632.8 nm). Compared with Fizeau interferometry method for sphere surface testing method, the FZP method can be effectively applied to the transmission wavefront of lens with long focal length, for the configuration is very simple and compact, which is very easy to be adjusted, thus enable to obtain high precision.
2. Metrology 2.1. System construction Traditionally, the compensation method is used to measure the transmission wavefront of large convex lens with aberration, as shown in Fig. 1. Spherical wave from an interferometer is transmitted through a compensation lens and the lens under test successively, the output parallel light is retroreflected by a reference flat.
https://doi.org/10.1016/j.optlaseng.2018.06.006 Received 15 May 2017; Received in revised form 6 June 2018; Accepted 11 June 2018 0143-8166/© 2018 Elsevier Ltd. All rights reserved.
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possible with ± 0.5 𝜇m positioning error, offering immediate precision enhancement to the spherical shape this element represents in diffraction. Whereas use of a spherical mirror is more costly and time consuming. The second benefit is the almost arbitrary wavefront generation. The hologram can be designed such that all rays incident on the surface are reflected back via the same path they came from, i.e., the test wave of the interferometer makes a normal incidence angle with the reflective hologram surface and is retroreflected. In this case, the interferogram only shows the deviation of the surface under test from the design prescription. Thus, it can be seen that the FZP can be powerful tools for reducing the number of elements in a setup, thereby enhancing system performance. However, when adopting the traditional compensation method, this null test requires the use of some auxiliary optics to match the f-number and to compensate system aberration, which therefore introduces unwanted error into the test. Thirdly, this method is especially beneficial to the TWF measurement of lens with long focal length, for the optical path length can be less than 1 m with proper optical design. Furthermore, the longer the focal length is, the lower the fabrication difficulty and higher fabrication precision of FZP is. On the other hand, using traditional compensation method, TWF measurement of lens with long focal length creates laboratory space problems, not to mention potential wavefront errors from air turbulence. One in particular is originality respect to previous work done by Zhou and Burge. (1) The reflective wavefront of CGH/FZP is used in the measurement here; Whereas the transmission wavefront of CGH is used to compensate the wavefront of lens under test. (2) The size of FZP could be very large since diffraction pattern on FZP is quite simple; While in the transmission mode, the size of CGH is generally small, and complicated alignment pattern should be designed since the testing system is quite sensitive to adjusting error. (3) Homogeneity of FZP has nothing to do with the measurement accuracy; Conversely, homogeneity is one of the major measurement errors in the transmission mode, and the effect of refractive index deviation should be analyzed in detail.
Fig. 1. Experimental setup for measuring wavefront of lens under test using the compensation method.
The testing optical system of the FZP method is composed of a phaseshifting interferometer (PSI), lens under test and a FZP, as shown in Fig. 2. When performing a lens TWF test, FZP is equivalent to a convex mirror. Part of the collimated light from the PSI is reflected from the transmission flat (TF), forming the reference beam; the rest transmits through the TF and lens, then is retroreflected from the zone plate, and finally into the interferometer, forming the test beam. From the interference fringe, lens TWF can be obtained. A precise characterization of the lens TWF should separate the contribution of the error in FZP substrate flatness. Measuring the substrate before the hologram pattern is generated has two main problems: one is the repositioning issue and the second, worse problem is the unknown substrate figure change introduced by the FZP fabrication process. When we subtract the nonzero-order surface measurement from the zero-order measurement, the FZP substrate error can be totally removed, leaving the residual wavefront errors from the fabrication non-uniformities. Thus, in this paper we adopt the method of measuring the zeroth order diffraction from FZP to account for substrate irregularities. In the zeroth order, the hologram emulates a flat mirror, when it is put behind the TF, the reflection interference wavefront is the reflected zeroth order wavefront of FZP. The measurement procedure of lens TWF is as follow: (1) Measure the reflected zeroth order wavefront of the FZP. (2) Put in lens, leaving a lens-to-FZP spacing 𝜏 the designed value. Keep the position of FZP unchanged and adjust the lens until the interferogram turns into null fringe, the obtained wavefront is the first-order wavefront. Lens TWF due to fabrication error can be obtained through subtracting the zeroth order wavefront from the first-order wavefront. But be warned, pixel matching by superposing edge feature on interferogram should be carried out, since rays propagating to FZP is convergent, as shown in Fig. 2.
2.2. Design and fabrication of FZP The design of FZP is to determine the radii and etching depth of the band structure of concentric rings on FZP substrate. According to diffraction principle, the radial locations for ring edges, i.e., Fresnel-zone boundaries, are the positions that give integral numbers of half-wavelength optical path differences at the center of curvature from the radius of curvature value, R. FZP possesses the lens-like characteristic, and the geometric diagram of FZP is shown in Fig. 3. The diffraction wavefront of FZP should match the transmission wavefront of lens with aberration. Following the aplanatic principle, and via optical design software, the distance between lens and FZP, the radii of FZP, and the coefficients of the various orders phase polynomial could be varied and the system optimized to retroreflect light. The geometric diagram is shown in Fig. 4. All the parallel light coming from the TF of a PSI transmits through the lens under test to the zone plate
Note that the lens-to-FZP spacing 𝜏 should be chosen reasonable. It is desirable to minimize the airspace 𝜏 in order to minimize the amount of atmosphere traversed by the test beam, and at the same time leave enough room for hands and for maneuvering of mounts. The advantages of using a reflective FZP to replace a spherical mirror are threefold. First and foremost, the Fresnel-zone plate consists of nothing more than a circular binary amplitude grating. Hence, it is very easily manufactured with optical lithography. Lithographic writing is
Fig. 2. Experimental setup for measuring wavefront of lens under test using FZP method. 349
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tern. Ion etching is used for fabrication of phase binary FZPs. This thermochemical effect allows the direct generation of patterns with spatial resolution better than 1000 mm−1 onto bare chromium films (Fig. 5). Etching depth concerns diffraction efficiency. According to the previous work done by Zhou and Burge [8], (1) CGHs with 50% duty-cycle have less wavefront error and higher diffraction efficiency; (2) When the etching depth is close to 0.5𝜆, the wavefront error increases dramatically. Thus, in the reflective FZP application, 0.25𝜆 etching depth should be avoided. The fabrication of efficient FZP requires the precise patterning of a large number of binary surface-relief structures. For close-to-ideal performance: duty cycle variation, etching depth variation, and pattern distortion should be controlled as small as possible. Using as a reflective optic, FZP should be thick enough in order to keep the surface shape. In our experiment, the diameter of the FZP is Φ430mm, with a thickness of 70 mm. The designed duty cycle and etching depth are 0.5 and 120 nm, respectively. Since the size of FZP is very large, the band gap of the zones at the edge is only about 10 𝜇m, the fabrication of FZP is very difficult. The measured duty cycle variation is ± 2%, etching depth variation is ± 5%, and pattern distortion is ± 1 𝜇m.
Fig. 3. Geometric diagram of FZP boundary locations.
Fig. 4. Geometric diagram for the design of FZP.
3. Experiment 3.1. Comparison with Fizeau interferometry method
with the same optical path, i.e., 𝑛0 |𝑃 𝑄| + 𝑛1 |𝑄𝑀 | + 𝑛0 |𝑀𝑁 | + 𝜙(𝑟) = 𝑛0 𝑑1 + 𝑛1 𝑑2 + 𝑛0 𝑑3 = 𝐶
(1)
The feasibility and accuracy of the scheme is validated through comparison with Fizeau interferometry method. For a lens with a focal length of 1500 mm and a diameter of 80 mm already in our lab, a FZP with the first-order radius curvature of 1200 mm and a diameter of 80 mm is designed and fabricated. The optical structure of FZP method is shown in Fig. 2. In order to obtain high quality diffraction wavefront, the reflective surface shape of the FZP substrate should be PV < 𝜆/10 (𝜆 = 632.8 nm). The reflective wavefront of the zoneplate substrate is shown in Fig. 6, from which we can see that peak-to-valley (PV) value is 0.065𝜆. The clear aperture is Φ75mm. The transmission wavefront of the lens under test is shown in Fig. 7, the data loss by FZP method originate from the superposition
where C is constant, 𝜑(r) is optical path difference induced by FZP. Thus, 𝜙(𝑟) = 𝑛0 𝑑1 + 𝑛1 𝑑2 + 𝑛0 𝑑3 − (𝑛0 |𝑃 𝑄| + 𝑛1 |𝑄𝑀 | + 𝑛0 |𝑀𝑁 |)
(2)
The computer model may then be used to output the radii aperture coordinates where 𝜋 phase changes occur. FZP with small size and thin substrates can be fabricated by direct laser writing, and ion-beam etching method can be adopted to fabricate FZP with binary or continuousrelief microstructure on to large-diameter and thick optical substrates. The FZP pattern can be created using a direct writing method based on thermochemical effect of laser heating on chromium films. After it is exposed, immersing the substrate in a caustic bath develops the pat-
Fig. 5. Method for fabrication of binary and continuous relief of FZP surface. 350
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Fig. 6. The reflective wavefront of the zoneplate substrate.
Fig. 9. Transmission wavefront of the lens tested by Fizeau interferometry method.
Table 1 The test results of FZP method.
PV RMS
0 order wavefront
Lens wavefront
TWF subtract 0 order
0.078𝜆 6.605nm
0.158𝜆 17.042nm
0.134𝜆 14.583nm
Table 2 The test results of lens in 3 states. (unit: 𝜆).
PV RMS
Original state
FZP rotated
lens rotated
0.134 0.023
0.137 0.019
0. 134 0.016
Fig. 7. The transmission wavefront of a lens tested by FZP method.
was manufactured. The first-order radius of curvature of zone hologram is 31.75 m, with a diameter of Φ430mm. The lens-to-FZP spacing is 0.5 m. The wavelength of the PSI is 632.8 nm. The transmitted wavefront of the lens is represented in Fig. 10. The clear aperture is Φ410 mm, power is removed from the graph. The PV and RMS value are listed in Table 1. As shown in Fig. 9, the substrate error can be effectively reduced by subtracting the zero-order wave front of the FZP, which is carried out by interpolation in order to avoid the lateral difference with direct substracting. On condition that the substrate shape is very smooth, and the machining precision is very high, this background error can be neglected. In addition, astigmatism is obvious in Fig. 9. The following experiment is carried out to judge the cause of this feature. Firstly, keep the status of the lens, and make the FZP anticlockwise turn 90°; Secondly, maintain the FZP the original state, and make the lens anticlockwise turn 90°. The test result is shown in Fig. 11 and Table 2. We can see that when rotating the FZP, the distribution as well as quantitative value is nearly the same, whereas in the situation of rotating the lens, the astigmatism rotates the same angle, with similar PV and RMS values. Thus, we can confirm that the astigmatism is originated from the optical element itself, and error caused by FZP is very small. The repeatability of the testing system is calculated by the following method: The average of 10 consecutive measurements with 6 phase averages without changing tip, tilt or focus will be used as a reference, and the reference will be subtracted from each individual measurement. The mean & standard deviation (𝜎) of the RMS of each of the difference measurements will be computed over a 410-mm diameter aperture. The repeatability equals the mean + 2𝜎. The tested results are shown in Table 3.
Fig. 8. The optical structure for Fizeau interferometry method.
of multiple diffraction orders. The measurement result is PV = 0.266 𝜆, RMS = 36.33 nm. The clear aperture is Ф75 mm. Then this lens is measured by spherical Fizeau interferometry method, the optical structure is shown in Fig. 8. The interferometer used in this experiment is 4D 4020. This is the conventional approach for lens transmission wavefront measurement, and the accuracy is believed to be better than 𝜆/10. Lens TWF distribution is shown in Fig. 9, and the data loss by Fizeau interferometry method comes from reflected light of the lens back surface. The measured PV = 0.226𝜆, RMS = 35.93 nm. The clear aperture isФ75 mm. From the experiment results we can see that both the wavefront distribution and PV, RMS value are consistent, thus confirms the feasibility and accuracy of the FZP method. 3.2. Large-size long focal-length lens TWF measurement The big advantage of FZP method is large-size long focal-length lens TWF measurement, for it can greatly shorten optical path length, meanwhile provide high measurement accuracy. In order to measure the transmission wavefront of lens with the size of Φ430mm and the focal length of 31.25 m, a large size zone hologram 351
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Fig. 10. Test result: (a) 0 order of the FZP; (b) transmission wavefront of the lens under test; (c) transmission wavefront that subtract the substrate error of the FZP.
Fig. 11. Test TWF result: (a) lens in initial state; (b) with zone plate anticlockwise turn 90°; (c) with lens anticlockwise turn 90°. Table 3 The repeatability of FZP method. (unit: 𝜆).
RMS
1
2
3
4
5
6
7
8
9
10
result
0.0031
0.0022
0.0016
0.0021
0.0030
0.0018
0.0020
0.0034
0.0027
0.0034
0.0039
can see that the simulated PV = 0.0000𝜆, RMS = 0.0000𝜆, which reveals that system design error could be ignored. 4.2. Influence of FZP fabrication error FZP fabrication errors may be classified into two basic types: substrate figure errors and pattern errors. Pattern errors may further be classified as fringe position errors, duty-cycle errors and etching depth errors. Typical FZP substrate errors are low spatial frequency surface figure errors that are responsible for the low spatial frequency wavefront aberrations in the diffracted wavefront. In a reflection hologram setup, a surface defect on a FZP substrate with a peak-to-valley deviation of 𝛿 s will produce a phase error in the reflected wavefront that equals 2𝛿s because of the double path configuration. This substrate error is eliminated by subtracting the zeroth order diffraction from FZP, but the wavefront still contains some residual errors due to variations in the duty cycle or etching depth. A binary, linear grating model is used to build the parametric model based on scalar diffraction theory, as shows in Fig. 13, where t and D correspond to the etching depth and duty-cycle, respectively, and D is defined as D = b/p, A1 and A2 correspond to the amplitudes of the output wave fronts from the peaks and valleys of the grating, respectively. The values of A1 and A2 are determined by the amplitude functions of the reflectance coefficients at the grating interface, using Fresnel equations [24]. The phase function 𝜑 represents the phase difference between light from the peaks and valleys of the grating. The wavelength of the incident light is assumed to be much smaller than the grating period, so the scalar diffraction approximations can be applied.
Fig. 12. The designed wavefront map.
4. Error analysis The errors of the FZP null testing system includes wavefront aberrations brought by system design, FZP fabrication and alignment error. In order to assure the validation and accuracy of the measured results, these error sources need to be analyzed carefully. 4.1. System design error FZP used in this experiment is designed by ZEMAX, the simulated result is shown in Fig. 12. By optical design software, the distance between lens and FZP, the radii of FZP, and the coefficients of the various orders phase polynomial could be varied and the system optimized to retroreflect light. Thus, the system design error could be very small. We 352
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Fig. 13. (a) Procedure for the coding of a continuous phase function into the binary CGH pattern. CGH top view on the right. (b) Binary, linear grating profile.
Fig. 14. Fabrication error measured by ZYGO NewView interferometer: (a) oblique plot to show the 3-D distribution and (b) surface/wavefront profile to show the duty-cycle and etching depth error quantitatively.
Table 4 Sensitivity functions.
Table 5 Wavefront error induced by the manufacturing of FZP.
Zero order (m = 0)
Non-zero order (m≠0)
𝜕Ψ 𝜕D
A1 A2 sinφ A22 D2 +A21 (1−D)2 +2A1 A2 D(1−D)cosφ
{
𝜕Ψ 𝜕𝜙 𝜕Ψ 𝜕 A1
A22 D2 +A1 A2 D(1−D)cosφ A22 D2 +A21 (1−D)2 +2A1 A2 D(1−D)cosφ A2 D(1−D)sinφ A22 (1−D)2 +A21 D2 +2A1 A2 D(1−D)cosφ
Error sources Pattern distortions Duty-cycle errors Etching depth
∞ for sinc(mD)= 0 0 otherwise A22 −A1 A2 cosφ
Ψ 2𝜋
deviations in duty-cycle ΔD, phase function Δ𝜑 and amplitude ΔA1 . The error sensitivity functions of the different diffraction orders are shown in Table 4. Duty-cycle non-uniformities have no effect on the wavefront error for the non-zero order diffraction. The amplitude of the light from the etched regions can vary due to limitations in the reactive ion etching process. The surface roughness of the etched area may vary due to etching, so the light incident on the FZP scatters differently. Since the surface roughness is much smaller than the wavelength of incident light, the coupling of the scattered light to the diffracted light is ignored. Variations in duty cycle and etching depth are measured using an interference microscope, which provide the surface relief of the FZP. A set of the sampled points should be measured to obtain the variations in duty cycle and etching depth. One sampled measured result is show in Fig. 14. Based on the measured fabrication errors, wavefront error is computed and listed in Table 5. Nowadays, state of the art photolithography allows the hologram to be generated with high precision and finesse, which greatly reduces the error introduced by FZP manufacture.
(3)
The displacement of the recorded fringe in a hologram from its ideal position is commonly referred to as pattern distortion. The amount of wavefront phase errors produced by the hologram pattern distortions can be expressed as a product of the gradient of the diffracted wavefront function and the pattern distortion vector. For a linear grating, wavefront phase errors produced by grating pattern distortions in the m-th order beam can be calculated as: Δ𝑊 = −𝑚𝜆
𝜀 𝑝
Wavefront errors ±𝜆/34 0 ±𝜆/35
A21 +A22 −2A1 A2 cosφ −A2 sinφ A21 +A22 −2A1 A2 cosφ
Based on Fraunhofer diffraction theory, the far field diffraction wavefront 𝜓 may be related to the original wavefront by a simple Fourier transform relationship. The wavefront phase W in waves can be obtained by 𝑊 =
Fabrication tolerances ±1𝜇m ±2% ±5%
(4)
where 𝜀 is the grating position error in direction perpendicular to the fringes, p is the localized fringe spacing, and m is the diffraction order. Pattern distortions are analyzed from FZP fabrication method. The Fresnel zone is produced by transferring the phase mask to the fused silica substrate. The positioning accuracy of phase mask is ± 0.75 𝜇m, the temperature change during ion beam writing process is less than is 1 °C, the induced position error is ΔL = 210 × 103 × 0.6 × 10−6 = 0.12 𝜇m, where 210 mm is edge-to center distance, 0.6 × 10−6 is the expansion coefficient. Thus, the combined positioning error is less than ± 1 𝜇m. The designed minimum period is 34 𝜇m, according to formula (4), the pattern distortion error is ± 𝜆/34. The wavefront phase sensitivity functions [9] 𝜕 Ψ/𝜕 D, 𝜕 Ψ/𝜕 𝜑, and 𝜕 Ψ/𝜕 A1 are introduced to specify the wavefront error caused by small
4.3. Influence of alignment error The phase function compensates the transmitted wavefront from the lens accurately only when the relative positions of focus of incident TWF from the lens and the FZP surface are identical to the value given in the process of phase function calculation. Any misalignment of lens or FZP surface will lead to additional aberration. In our experiment, when the reflected 0th order wavefront is measured, the position and attitude of FZP remains unchanged, and a 5-D amount is used to adjust lens. In this case, the setup consists only two elements: The lens under test and FZP. Both elements must be aligned 353
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(a)
(b)
Fig. 15. Wavefront error induced by (a) tilt and (b) decenter.
Table 6 Wavefront errors without substrate calibration. Error sources
Wavefront errors
Substrate RMS error Pattern distortions Duty-cycle errors Etching depth Alignment errors Repeatability error Root-sum-square error
0.0127𝜆 ±𝜆/34 0 ±𝜆/35 0.0064𝜆 0.0039𝜆 0.077𝜆/48.7nm
unsolved 3-dimensional fabrication and measurement problem is converted into a 2-dimensional problem. An example of the TWF measurement for lens with 31.25 m focal length and 410 mm clear aperture was given, and a phase FZP with 410 mm clear aperture was fabricated to carry out the experiment. Thanks to the continuous progress in microlithography high precision lateral pattern can be fabricated. The substrate error is the dominant error from the FZP and that it can be removed by subtracting the first-order measurement from the zero-order one. After performing the substrate calibration, the precision of the FZP method is 0.042𝜆 (26.7 nm).
with micron precision or better. Given the correct position of FZP, a rotationally symmetric system has 5 adjustable parameters: 5 degrees of freedom for the lens (z-position, decenter x/y, tilt x/y). However, z-position error mainly induces additional power, which is subtracted when analyzing wavefront, thus can be ignored, and error induced by tilt and decenter is analyzed and computed quantitatively below. Assuming when the bright spot is superimposed on the alignment crosshairs, there is still 2 pixels’ deviation from the ideal position, which can definitely be recognized by human eyes. Combining with the configuration parameters of the ZYGO interferometer used, the adjustment accuracy of the mount, the decenter and tilt error are 45″ and 3 mm, respectively. The TWF of lens under test with tilt angle of 45″ and decenter of 3 mm by ZEMAX simulation are shown in Fig. 15. The induced wavefront error are mainly astigmatism and coma, which can be calculated by the summation of the 5th and 6th terms (astigmatism), 7th and 8th terms of Zernike fringe polynomials (coma). The calculated astigmatism induced by tilt are 0.00046𝜆, 0.00086𝜆; and the calculated coma induced by decenter are 0.00034𝜆, −0.0063𝜆. Thus, the composite alignment error is 0.0064𝜆.
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4.4. Composite error The system measurement error is the combination of all kinds of uncertainties, including FZP design, fabrication, alignment errors and repeatability errors, which is obtained in Section 3. Table 6 lists all the wavefront error sources, notice that FZP design error is not included, that’s because the designed PV and RMS value both are zero. The estimated root-sum-square (rss) wavefront error is 0.077𝜆/48.7 nm, among which, substrate error is dominant. FZP substrate errors can be removed by subtracting the zero-order measurement from the TWF. In this case, the residual rss wavefront error drops from 48.7 to 26.7 nm (0.077 to 0.042𝜆). 5. Conclusion In this paper, the use of a diffractive element, a so called nullFZP, offers a practical solution to generate the reference wavefront for large convex lens TWF measurement with long focal length. Thus, the 354
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