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Abstract
We present the use of a computer-generated hologram (CGH) to test the mid-spatial frequency error of a large aperture long-focal-length lens. In order to verify this test approach, a 450 mm × 450 mm reflective CGH is designed and fabricated for testing the 440 mm × 440 mm spatial filter lens. Both 0th and 1st order diffraction wavefront of CGH are measured, and the 0th order diffraction wavefront is used to calibrate the substrate error. The mid-spatial frequency wavefront error caused by the CGH fabrication errors are evaluated using the binary linear grating model and power spectral density (PSD) theory. Experimental results and error analysis show that the CGH test approach is also feasible for the measurement of mid-spatial frequency error, and the measurement accuracy of PSD1 can reach 0.8832 nm RMS.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Large aperture long-focal-length lens has been widely used as spatial filter lens in high power laser system [1]. Transmitted wavefront error is an important specification for long-focal-length lens, and it will directly affect the filtering effect and beam quality. According to the performance requirements of laser system, wavefront error is usually specified over a continuous range of spatial frequencies from 2.5 × 10−3 to 100 mm-1. During the development of National Ignition Facility (NIF) in the United States, Lawrence Livermore National Laboratory (LLNL) divided wavefront errors into three frequency bands: low-spatial frequency error (2.5 × 10−3 to 3 × 10−2 mm-1), mid-spatial frequency error (3 × 10−2 to 8.3 mm-1), and high-spatial frequency error (8.3 to100 mm-1) [1]. These bands, from low to high spatial frequencies are called figure, waviness, and roughness. LLNL defined the mid-spatial frequency error as any wavefront error whose gradient results in light scattered into angles which are not accepted by the target, but are subject to non-linear growth [2]. Error of this type is difficult to control using the root-mean-square (RMS) and peak-to-valley (PV) specifications, because the non-linear gain experienced by these errors varies by orders of magnitude depending on the spatial frequency. Therefore, LLNL introduced the power spectral density (PSD) to characterize the mid-spatial frequency errors and developed the PSD specification for the optics used in NIF [2]. Since there is no instrument can cover the entire mid-spatial frequency region, they further subdivided this region into two bands: waviness1 or PSD1 (3 × 10−2 to 0.4 mm-1) and waviness2 or PSD2 (0.4 to 8.3 mm-1). For the specifications of PSD2, only some small regions of the surfaces are sampled, which can be tested by small-aperture interferometers. However, for the specification of PSD1, the transmitted wavefront measurement of entire optic is required. Due to large aperture and long-focal-length, the transmitted wavefront measurement of this lens is usually difficult. There are two conventional methods for testing the transmitted wavefront of long-focal-length lens. One method involves a spherical interferometer, a concave reference mirror and a compensating lens [3]. In this method, the concave mirror and compensating lens are used for shortening the optical path and compensating the aberration, respectively. Although this method has a strong compatibility, one set of concave reference mirror and compensating lens can measure several different spatial filter lenses, the alignment of optical elements is difficult and the test optical path is still relatively long. Another method involves a large aperture interferometer and a convex mirror [4]. This method has a simple test configuration and the optical path is very short. However, measurements of large aperture convex mirrors are difficult because they require auxiliary optics and flat interferometer that are larger than the surfaces being tested.
Recent years, with the developments in diffractive optics and lithography technology, the use of large aperture computer-generated hologram (CGH) has become an attractive alternative [5–8]. Therefore, we proposed to measure the transmitted wavefront of long-focal-length lens using a reflective CGH [9]. In this method, a CGH with circular lines is used to emulate the reflective properties of the convex mirror, which can avoid the fabrication of large aperture convex mirror and keep the advantage of simple test configuration, as shown in Fig. 1. Moreover, the design of CGH is flexible, one CGH with multiple zones can provide the wavefront correction for null testing and alignment information for optical elements simultaneously. In previous work [9], we have studied the low-spatial frequency errors of large aperture long focal-length lens and the experimental results indicated that the CGH test scheme is feasible and valid. In this paper, we will further study the mid-spatial frequency errors of large aperture long-focal-length lens using CGH and focus on the measurement of PSD1.
Fig. 1. Schematic drawing of test configuration for large aperture long-focal-length lens. (a) Convex mirror method; (b) CGH method.
2. Calculation of PSD from 2D wavefront map
The power spectral density (PSD) is the power spectrum per unit frequency, which describes the wavefront error in terms of spatial frequency. In this section, we will briefly describe the mathematical calculations of obtaining PSD from a two-dimensional (2D) wavefront map.
Considering a wavefront map $u(x,y)$ of transmissive optic (or, surface height map of reflective optic) over an area ${L_x} \times {L_y}$, the finite-length Fourier transform and 2D PSD are defined as [10]
3. Experiments and results
In order to verify the feasiblity of the proposed test approach, we designed a CGH for testing the 440 mm × 440 mm spatial filter lens. This lens is made of fused silica (n = 1.450@λ = 1053 nm) and has an effective focal length of 32500 mm (@λ = 1053 nm). The center thickness of lens is 48 mm, and the vertex radii of curvature are R1 = 17842 mm (CX) and R2 = 80810 mm (CX), respectively. In the clear aperture of 400 mm × 400 mm, the transmitted wavefront requirements of lens are PV <λ/3 (@λ = 632.8 nm), PSD1 band-passed RMS <1.8 nm and 1D collapse PSD <1.01${\nu ^{\textrm{ - }1.55}}$. Here, 1.01${\nu ^{\textrm{ - }1.55}}$ is the “Not-To-Exceed” line and $\nu$ represents the spatial frequency in arbitrary direction. The CGH is binary and works in reflection mode. The 1st order diffraction light is used for CGH design. In our design, the distance of CGH from long-focal-length lens is 500 mm, and the minimum CGH fringe spacing is 32.9 μm in the testing zone of 430 mm × 430 mm. The CGH fringe pattern is fabrticated on a 450 mm × 450 mm × 70 mm fused silica substrate. To get a high diffraction efficiency, phase CGH is employed, and the duty-cycle and etching depth are set to 0.5 and 110 nm for the consideration of obtaining a relatively small wavefront error [12]. The CGH substrate was made in our optical shop, which has a figure of PV = 0.059λ and PSD1 band-passed RMS = 0.8018 nm within 400 mm × 400 mm aperture, as shown in Fig. 2. The CGH fringe pattern was fabricated in National Synchrotron Radiation Laboratory, University of Science and Technology of China. First, the CGH fringe pattern was fabricated on a mask by laser writer, then copied to the photo-resist layer on the fused silica substrate by ultraviolet exposure. Finally, the fringe pattern was transferred into the fused silica substrate through reactive ion etching.
Fig. 2. CGH substrate figure before etching (a) CGH substrate figure map: PV = 0.059λ and RMS = 0.0062λ; (b) PSD1 band-passed figure map: RMS = 0.8018nm.
In our experiments, the large aperture interferometer is a commercialized interferometer produced by Zygo, which can measure up to 32-inch diameter. Note that the camera of this interferometer has 1000 × 1000 pixels and the effective pixel size is 0.827 mm, in order to ensure the validity of PSD data, we require at least 4 samples per cycle for the high frequency end, which means that the pixel size must be smaller than 0.625 mm for the PSD1 band from 3 × 10−2 to 0.4 mm-1. Therefore, a magnification of 1.4× is used to make this interferometer cover all of the PSD1 range, in which case the equivalent pixel size becomes 0.580 mm.
The CGH test process includes two steps [9]: (1) put the CGH substrate in optical path and measure the 0th order diffraction wavefront; (2) put the long-focal-length lens in optical path and measure the 1st order diffraction wavefront. Through measuring the 0th order diffraction wavefront of the CGH, we can obtain the substrate figure error and back it out from 1st order one. Figures 3 and 4 are the data maps of two measurements. In the center of maps, we can see that there is a small area of invalid data caused by the spurious order diffraction of CGH and reflection of lens surface [13,14]. For PSD calculations, the Fourier transform requires real numbers in the entire matrix, therefore these invalid data are removed and pathed by interpolation. After data patching, the measured CGH substrate figure error is PV = 0.118λ and PSD1 band-passed RMS = 0.6924 nm. The transmitted wavefront of long-focal-length lens measured by 1st order wavefront is PV = 0.225λ and PSD1 band-passed RMS = 3.0232 nm.
Fig. 3. 0th order measurement. (a) CGH substrate figure map before data patching: PV = 0.161λ and RMS = 0.0145λ; (b) CGH substrate figure map after data patching: PV = 0.118λ and RMS = 0.0145λ; (c) PSD1 band-passed figure map: RMS = 0.6924 nm.
Fig. 4. 1st order measurement. (a) Wavefront map before data patching: PV = 0.225λ and RMS = 0.0181λ; (b) Wavefront map after data patching: PV = 0.225λ and RMS = 0.0181λ; (c) PSD1 band-passed wavefront map: RMS = 3.0232 nm.
For the CGH substrate calibration, data mapping must be done to make two measurements have a correct map registration. Figure 5 shows the mapping function between the radius positions of CGH and the measured lens, which also represents the mapping relationship between 0th order map and 1st order map. According to this mapping function, the 0th order map is transformed to match the 1st order map. After data mapping, the CGH substrate figure map is PV = 0.113λ and PSD1 band-passed RMS = 0.6894 nm. The transmitted wavefront of long-focal-length lens after CGH substrate calibration is PV = 0.201λ and PSD1 band-passed RMS = 2.8908 nm, as shown in Fig. 6.
Fig. 5. Data mapping between 0th order and 1st order. (a) Mapping function; (b) CGH substrate figure map after data mapping: PV = 0.113λ and RMS = 0.0134λ; (c) PSD1 band-passed substrate figure map: RMS = 0.6894 nm.
Fig. 6. Transmitted wavefront of large aperture long-focal-length lens after CGH substrate calibration. (a) Wavefront map: PV = 0.201λ and RMS = 0.0112λ; (b) PSD1 band-passed wavefront map: RMS = 2.8908 nm.
The specification RMS in the PSD1 band is obtained by performing a band-pass filtering, and then calculating the RMS of the filtered wavefront error. For specification purposes, the PSD1 band-passed RMS of the entire clear aperture is used. However, local finishing marks or defects can heavily influence the overall RMS, therefore the RMS in 16 sub-apertures of 100 mm × 100 mm size are also calculated and reported, as shown in Fig. 7. While not a specification, this information is very useful in identifying the poorly finished areas. From Fig. 7, we can find that the sub-apertures at four corners still have a worse RMS, which leads to the PSD1 band-passed RMS of this measured lens not meeting the requirements yet.
At high intensities, the phase noise caused by wavefront error can grow nonlinearly for some spatial frequencies, and the effect on the laser performance will be much worse if the wavefront error is concentrated into one narrow band rather than spread across the entire spectrum. Therefore, the 1D collapse PSD curves are used to identify the strong frequency components in the spectrum, and a “Not-To-Exceed” line is specified to bound the peak value of the curves, wherein none of the 1D collapse PSD curves can exceed this line over the PSD1 band. Figure 8 shows the PSD1 band-passed 1D collapse PSD curves. Based on the Radon transform, one can construct the projections of 2D PSD from different angles and calculate the 1D collapse PSD along these projection angles. In our case, six different angles of 0, 30, 60, 90, 120 and 150 degrees are calculated, and the 1D collapse PSD curves are expressed in log-log form. From Fig. 8, we can see that there are no obvious peaks in the 1D collapse PSD curves and all of the curves are under the “Not-To-Exceed” line in PSD1 band.
Fig. 8. The PSD1 band-passed 1D collapse PSD curves of large aperture long-focal-length lens (a) with and (b) without the CGH substrate calibration.
4. Error analysis
The CGH test errors includes design errors, fabrication errors and aligment errors. All of these errors will influence the optical performance of test setup. In this section, we will evaluate the wavefront errors for the CGH test in PSD1 frequency band.
4.1 Binary linear grating model
The performance of a CGH is directly related to its diffraction characteristics, so we use the binary linear grating model [15,16] to estimate the PSD1 band-passed wavefront errors that result from the CGH fabrication errors.
Figure 9 shows the schematic drawing of binary linear grating model, which is described by line spacing S and etching depth t. The duty-cycle is defined as D = b/S, where b is width of the unetched area, and A0 and A1 represent the amplitudes of the output wavefront from the etched and unetched areas of the grating, respectively. The phase step ϕ corresponds to the phase difference between two areas, which equals 4πt/λ for the grating used in reflection mode. According to Fraunhofer diffraction theory, the diffraction wavefront phase Ψ and the wavefront phase sensitivity functions can be derived, as summarized in Table 1.
Table 1. Diffraction wavefront and sensitivity functions for error analysis of CGH
The wavefront errors due to the variation in the duty cycle and phase step can be given by
where ΔWD and ΔWϕ are the wavefront errors in waves produced by small variations in duty cycle ΔD and in phase step Δϕ. As for the CGH pattern distortion, the introduced wavefront error is proportional to the diffraction order ${m_o}$ and inversely to local fringe spacing S, that is where ε is the grating position error in the direction perpendicular to the pattern.Combined wavefront phase sensitivity functions and the PSD theory mentioned above, the PSD1 band-passed wavefront errors caused by the CGH fabrication nonuniformities can be evaluated. Assuming that the CGH fabrication errors are zero mean normal random distributed and have a variance of $\sigma _{}^2$, the corresponding variance of the introduced wavefront errors is $\sigma _{\Delta W}^2$, then the PSD1 band-passed wavefront errors $\sigma _{PSD1}^{}$ can be calculated by
4.2 Evaluation of each error source
The wavefront errors from each error source for the CGH test in PSD1 frequency band are given in Table 2. All of these errors are given up to 4 digits behind the decimal point. Note that the test beam passes through the measured lens twice, a 0.5 factor has been counted in the analysis.
Table 2. Errors analysis for the CGH test in PSD1 frequency band
The design residual of CGH is 0.0000 nm RMS, which can be ignored. In the CGH manufacturing process, the substrate figure error is a primary error source. After substrate calibration, the substrate figure calibration residual error is 0.4150 nm RMS. For a phase type CGH used in refelction, A0 and A1 are equal, if the duty-cycle is 0.5 with a 5% (3σ) variation, the introduced wavefront error is 0.6500 nm RMS. Moreover, for a 0.5 duty-cycle, the wavefront sensitivitis of etching depth are equal for both 0th and 1st order. After substracting the 0th order measurement from the 1st order one, the wavefront error caused by etching depth is cancelled. The line spacing of fringe pattern is non-uniform and has a minimum value of 32.9 μm, therefore the wavefront error caused by 0.5 μm (3σ) pattern distortion is 0.3470 nm RMS. For the alignment error of CGH, we only need to consider the tilt because of the collimated test light. If we adjust the CGH to one fringe in both x and y directions, the introduced wavefront error is 0.0000 nm RMS, which can be neglected.
When we design the CGH, parameters of the measured lens are assumed to be ideal. Howerver, in practice, the measured lens has a torlerance range and its fabrication errors will also introduce corresponding wavefront errors. The deviation of radius of curvature will cause an offset to the focal length. The focal length of this lens is 32500mm, then the wavefront error caused by 0.1% offset is 0.0083 nm RMS, wherein the power term has been removed by using a misalignment correction of defocus. For the deviation of thickness, the wavefront error caused by 0.2 mm offset is 0.0006 nm RMS. As for the alignment of measured lens, we use the “spurious fringe spot” and “ghost fringe spot” to align the lens with CGH. In x and y direction, the wavefront error caused by 0.2 mm decenter is 0.0022 nm RMS, and the wavefront error caused by 0.04′ tilt is 0.0009 nm RMS. In z direction, the wavefront error caused by 1 mm displacement is 0.0006 nm RMS. During substrate calibration, the mapping error between two maps will be also an error source. The wavefront error due to average 1-pixel mapping error in x and y directions is 0.1800 nm RMS.
Assuming all the errors are un-related and independent, the total PSD1 band-passed wavefront error of this CGH test can then be estimated as root-sum-square (RSS) of these errors, which is approximately 0.8832 nm RMS.
5. Conclusion
The method of using CGH to test the mid-spatial frequency error of large aperture long-focal-length lens is presented. In this paper, we focused on the measurement of PSD1 and designed a 450 mm × 450 mm CGH to measure the 440 mm × 440 mm spatial filter lens. Both 0th and1st order diffraction wavefront of CGH are measured for the substrate calibration, and the binary linear grating model and PSD theory are used to estimate the mid-spatial frequency wavefront error introduced by the CGH fabrication errors. Experimental results and error analysis indicate that the measurement accuracy of PSD1 can reach 0.8832 nm RMS, which means the CGH test approach can also be used to measure the mid-spatial frequency error of large aperture long-focal-length lens.
Disclosures
The authors declare no conflicts of interest.
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