The point diffraction interferometer (PDI) employs a point-diffraction spherical wavefront as ideal measurement reference, and it overcomes the accuracy limitation of reference optics in traditional interferometers. To overcome the limitation of measurement range either with pinhole (low light transmission) or with single-mode fiber (low NA), a single-mode fiber with narrowed exit aperture has been proposed to obtain the point-diffraction wavefront with both high NA and high power. It is a key issue to analyze the point-diffraction wavefront error in PDI, which determines the achievable accuracy of the system. The FDTD method based on the vector diffraction theory provides a powerful tool for the design and optimization of the PDI system. In addition, a high-precision method based on shearing interferometry can be applied to measure point-diffraction wavefront with high NA, in which a double-step calibration including three-dimensional coordinate reconstruction and symmetric lateral displacement compensation is used to calibrate the geometric aberration. The PDI is expected to be a powerful tool for high-precision optical testing. With the PDI method, a high accuracy with RMS value better than subnanometer can be obtained in the optical surface testing and submicron in the absolute three-dimensional coordinate measurement, demonstrating the feasibility and wide application foreground of PDI.
- point diffraction
- optical testing
- optical fiber
- wavefront sphericity
The development of optical design and fabrication such as projection optics for extreme ultraviolet lithography (EUVL) and laser fusion, etc. places ultrahigh requirement on the optical testing precision and accuracy. In the EUVL operating at a wavelength of 13.5 nm [1–4], the projection optics is composed of 4–6 aspheric mirrors and each mirror needs to be finished with the figure error less than 0.2–0.3 nm (RMS), requiring a higher testing accuracy (e.g., 0.1 nm RMS). The optical interferometers, including the Twyman-Green  and Fizeau interferometers , have been widely used in the testing lenses and mirrors for figure metrology, in which standard lenses are applied to produce the necessary reference wavefront. Due to the fabricating error of the reference optics, the achievable measurement accuracy of the traditional interferometers is generally limited within
2. PDI for wavefront and surface testing
2.1. Wavefront testing with PDI
The PDI was first proposed by Smartt and Strong in 1972 . The early version of the PDI was an interferometer using a PDI plate with partial transmission. Figure 1 shows the principle of the PDI with partially transparent PDI plate. The PDI plate consists of an absorbing metal coating mask on a clear substrate and a tiny pinhole placed near focus to divide the wave into two parts, namely the testing wave and reference wave. The pinhole picks off part of the incident focused light wave and generates the diffracted spherical wavefront as reference wavefront. The test wave passes through the PDI plate and then interferes with diffracted spherical wave. The PDI can be used to measure the wavefront error from imaging optics and flow field, etc. In the design of the PDI shown in Figure 1, it is a key issue to determine the pinhole size and plate transmittance . Due to the reduction in light intensity incident on the tiny pinhole, the controllable reduction in the test wave intensity by mask attenuation is required to get maximum fringe contrast.
Another PDI plate with double apertures has been applied in the PDI for the evaluation of wavefront error in imaging optics , as shown in Figure 2. A beam splitter such as the transmission grating is used to divide the test wave into two waves with a small angular separation. The PDI plate consists of one tiny pinhole and one large window on an opaque mask, and both of them are placed at the respect focal points of two beams. The pinhole is applied to generate the spherical reference wavefront by diffraction and the window transmits the test wave. The intensity of the diffracted wave relative to that of the conventional PDI is increased by several orders of magnitude. With the application of double-aperture PDI plate, there is no need to further attenuation in the test wave to match the intensities of interfering waves. Besides, the beam division enables the potential to introduce the various phase shifts between the interfering waves.
2.2. Optical surface testing with PDI
Figure 3 shows the typical configuration of PDI for optical surface testing. Either the pinhole (Figure 3a) [13–15] or the single-mode optical fiber (Figure 3b) [16–18] can be used as the point-diffraction source to generate the required spherical wavefront. The diffracted wavefront is separated to two parts, i.e., the test and reference wavefronts. The test wave travels toward the spherical surface under test, and the pinhole (exit end of fiber) is positioned at the curvature center of the test surface, so that the reflected wave from the test surface converges at the pinhole mirror (semitransparent metallic film on the output end of fiber) and then is reflected at the pinhole mirror (semitransparent metallic film on the output end of fiber). The test wave combines and interferes with the reference wave after reflection. By translating the test surface with a precise PZT scanner, the test surface error can be measured with the phase-shifting method. The selection of beam polarization state in the system can significantly influence the measurement. The polarization would affect the sphericity of the diffracted wavefront, and the reflection at mirror over a high NA can also introduce polarization-dependent phase shifts. In the PDI system, the polarization state of diffracted spherical wave is generally adjusted to be circularly polarized , in which the effect of polarization on the measurement precision is negligible.
In the PDI with pinhole method, the circular pinhole of submicron (or even smaller) diameter is adopted to obtain high measurable NA. Figure 4(a) shows a scanning-electron microscope (SEM) picture of the pinhole fabricated by etching the chromium film with the focused ion-beam etching (FIBE) method , in which the metallic layer is sputtered onto the silica substrate. A nearly perfect circular pinhole can be obtained with the FIBE method. The high measurable NA can be achieved with the pinhole method; however, the nonadjustable fringe contrast would limit the measurement accuracy in the testing of low-reflectivity surface, due to the poor fringe contrast and difficulty in fringe processing.
A pinhole PDI with adjustable fringe contrast can be adopted for the testing of high-NA spherical surfaces with low reflectivity . The polarizing elements are applied to transform the polarization states and adjust the relative intensities of the interfering beams, by which the adjustable fringe contrast can be achieved. Figure 5 shows the optical configuration of the pinhole PDI with adjustable fringe contrast. A quarter-wave plate (QWP2) with special structure (consisting of a thin waveplate film and a plano-convex substrate) is placed at the test path, with the fast axis oriented at −45° to horizontal. The diffracted wave is adjusted to be circularly polarized. The test wave
Figure 6 shows the testing results about a spherical surface with the reflectivity 4%, NA 0.40 and aperture diameter 137.7 mm. Figure 6(a) is the surface error measured with the adjustable-contrast pinhole PDI system shown in Figure 5, and Figure 6(b) is that obtained with the ZYGO interferometer. According to Figure 6, a good agreement between the PDI result and that from the ZYGO interferometer is achieved, and the PV and RMS differences between the testing results are about and , respectively. Thus, high-accuracy testing of the surface with low reflectivity is realized with the adjustable-contrast PDI system.
The light transmission through the pinhole is quite low (<0.1‰), and it significantly limits the achievable measurement range of the pinhole PDI. In the PDI with single-mode fiber, the adjustable fringe contrast is easy to realize and high light transmittance (>10%) can be obtained; however, the measurable NA in the fiber method is limited by the NA of the fiber, which is commonly less than 0.2. Besides, the measurable aperture of the test surface is approximately half that of the diffraction wave, both for the pinhole PDI and single-mode fiber PDI shown in Figure 3. A PDI system with two optical fibers as point-diffraction sources was developed for full use of the diffracted wavefront , in which the diffracted wave from one fiber serves as reference wavefront and that from the other fiber as test wave. Besides, a novel submicron-aperture (SMA) fiber with cone-shaped exit end, as shown in Figure 4(b), has been proposed to obtain both the high diffraction light power and high-NA spherical wavefront [19–21]. The SMA fiber taper surface is coated with metallic film and the exit aperture is formed from the polished tip, it is formed with the same processing technology as manufacturing of fiber-based probes for the scanning near-field optical microscopy. With the SMA fiber, the measurable range can be almost extended to a half space, and the corresponding light transmittance over 50% is obtained . Thus, it is considered as a feasible way to extend the measurement range of the system.
3. PDI for 3D coordinate measurement
In this section, a fiber PDI with submicron aperture , which is based on the single-mode fiber with a narrowed exit aperture, for the absolute 3D coordinate measurement within large angle range is described, and the system configuration is shown in Figure 7. After passing through the half-wave plate (HWP1), the frequency-stabilized laser beam is separate into p-polarized and s-polarized beams by the polarized beam splitter (PBS), and they are coupled into the single-mode fibers SF1 and SF2, respectively. The exit ends of the fibers SF1 and SF2 are integrated into a target with certain lateral offset, and the output point-diffraction waves
To increase the NA of diffracted wavefront with optical fiber, a single-mode fiber with submicron aperture (as shown in Figure 4b) is applied as point-diffraction source. Both the exit ends of two single-mode fibers are integrated in a target with certain lateral offset. According to the PDI system shown in Figure 7, the absolute 3D coordinates of target can be measured from the phase distribution in interference field corresponding to the optical path difference (OPD), as shown in Figure 8.
Denoting the plane of CCD detector as the
where , is the known 3D coordinate of the point , and are those of two fiber apertures in the target under measurement. According to the one-to-one correspondence relationship of the phase at position (that is the pixel on CCD) and 3D coordinates of two fiber apertures in the target, the phase difference can be written as follows:
where the vector indicates the coordinates of two fiber apertures under measurement that is ; is the measured phase difference. The phase differences
where is the 3D coordinate of the pixel on CCD. From Eq. (2), various equations can be established corresponding to the pixel positions on the CCD, and we have the matrix equations,
where the subscript
Thus, the space coordinates of two fiber apertures can be determined from the global minimum of the function (that is the least-square solution of Eq. (4)), with a single true solution set.
An experimental SMA-fiber PDI system with aperture size about 0.5 μm, has been set up for the measurement of 3D coordinates. For comparison, the target is installed on the probe of a CMM with the positioning accuracy about 1.0 μm, and the 3D coordinates measured with CMM is taken as the nominal value. Figure 9 shows the measurement results about the 3D coordinate deviations in
4. Numerical analysis of point-diffraction wavefront
As one of the most important elements in the PDI, the point source for the diffracted reference wavefront determines the achievable accuracy in the measurement. Purely empirical design of point source parameters is both time consuming and costly. The numerical method based on diffraction theory is a feasible way for the analyzing point-diffraction wavefront. The scalar diffraction theory is valid only when the pinhole size is several times larger than the operating wavelength. For the high-NA spherical wavefront emerging from a tiny aperture with the size comparable with or less than operating wavelength, the vector diffraction theory (that is a nonapproximate method) is required to realize the accurate estimation of diffracted wavefront error. In this section, numerical analysis based on finite difference time domain (FDTD) method (that is a vector diffraction theory)  is presented. Figure 10 shows the flow diagram for the simulation of point-diffraction wavefront based on FDTD method. Due to limitations of computer memory capacity and runtime, FDTD cannot be directly applied to calculate the far-field distribution of pinhole diffraction. In the first step, the near-field distribution of point diffraction is analyzed with the FDTD method, and then the near-to-far field transition based on Huygens’ principle is performed to obtain the far-field distribution of point-diffraction wavefront at the position under study. Finally, the sphericity evaluation is carried out to get the departure of point-diffraction wavefront from an ideal sphere.
4.1. Simulation of pinhole diffraction
Figure 11 shows the amplitude and phase distribution in the near-field diffraction, respectively, corresponding to the pinholes of 0.5 μm and 1 μm diameters, respectively. According to Figure 11, the phase distribution within Airy disk range is close to an ideal sphere, and the angle of Airy disk increases with the decrease of pinhole diameter.
To analyze the wavefront error where the test surface with 500 mm curvature radius is placed, far field is positioned at 500 mm away from pinhole. The wavefront errors over various NAs of diffracted wavefronts corresponding to different pinhole diameters are shown in Figure 12(a). According to Figure 12(a), the point-diffraction wavefront error grows both with the pinhole size and NA range, and the wavefront error RMS for the 1 μm pinhole diameter over 0.35 NA is smaller than 0.53 nm. Thus, it can be taken as ideal reference wavefront and applied to realize the measurement accuracy reaching to RMS (
4.2. Simulation of SMA fiber diffraction
Figure 13(a) shows the simulation results about the full wavefront aperture angle for the SMA fibers with various exit apertures . According to Figure 13(a), the full aperture angle of diffracted wave obviously increases with the decreasing of fiber aperture, and that corresponding to the 0.5 μm fiber aperture are about 160°, providing the necessary conic boundary to extend the measurement range almost within a half space. Figure 13(b) shows the wavefront errors over various NAs of diffracted wavefronts corresponding to different SMA fiber apertures . From Figure 13(b), the increase in NA range and fiber aperture could lead to the growth of diffracted wavefront error. The wavefront error over 0.70 NA range for the 0.5 μm fiber aperture is better than RMS, and it can be applied as ideal spherical reference wave in the PDI system.
Figure 14 shows the analyzing result about light transmittance in the point diffraction both with the SMA fiber method and pinhole method. According to Figure 14, the light transmittance in both the SMA fiber diffraction and pinhole diffraction grows with the increase in aperture size. It can be seen from Figures 12(a) and 13(a) that the similar aperture angles can be obtained with the same diffraction aperture size, however, the light transmittance in the SMA fiber diffraction is far larger than that in pinhole diffraction according to Figure 14. The light transmittance corresponding to the 0.5 μm aperture in the SMA fiber method and pinhole method is about 67% and 6%, respectively. Thus, the SMA fiber provides a feasible way to obtain the high light intensity required in the optical testing, enabling the extension of measurement range with PDI system.
5. Experimental measurement of point-diffraction wavefront
The numerical simulation based on the vector diffraction theory such as FDTD method provides an easy and efficient way to estimate the point-diffraction wavefront error. However, both the computational accuracy and complex practical factors (such as environmental disturbance and performances of various optical parts) can introduce significant deviation from an ideal case. Due to the accuracy limitation of standard optics, the traditional interferometers fail to measure the point-diffraction wavefront error, which is expected to be in the order of subnanometer or even smaller. Various experimental testing methods have been proposed to measure the point-diffraction wavefront error, the majority of which are based on the hybrid method  and null test . Typically, the hybrid method requires several measurements with the rotation and displacement of the optics under test, it is sensitive to environmental disturbance and cannot completely separate the systematic error. The null test is a self-reference method, and it is widely applied to reconstruct the point-diffraction wavefront and calibrate the PDI. However, it requires the foreknowledge about the system configuration to remove the high-order aberrations, which are introduced by the point-source separation and cannot be negligible in the case of high-NA wavefront. In this section, a high-precision measurement method, which is based on shearing interferometry, is presented to analyze the point-diffraction wavefront from SMA fiber .
Figure 15 shows the schematic diagram of SMA fiber projector used to evaluate the sphericity of point-diffraction wavefront. The SMA fiber can be applied to obtain both high diffraction light power and high-NA spherical wavefront. Four parallel SMA single-mode fibers S1, S2, S3, and S4 with coplanar exit ends are integrated in a metal tube. To obtain the identical wavefront parameters, the four SMA fibers with same aperture size and cone angle are carefully chosen in the experiment to minimize the measurement error. The coherent beams are coupled into the SMA fibers, and the point-diffraction waves from the fibers interfere on a detection plane, as shown in Figure 15(b). By alternatively switching on the waves from the fiber pairs G1 (S1 and S2) and G2 (S3 and S4), the shearing interferograms in
5.1. Point-diffraction wavefront retrieval method
With the shearing wavefronts obtained from the projector shown in Figure 15, the differential Zernike polynomials fitting method can be applied to retrieve the SMA fiber point-diffraction wavefront. Denoting the point-diffraction wavefront under test as , we have the shearing wavefronts and in
where is the shearing amount as shown in Figure 15(b). The wavefront can be described with Zernike polynomials and can be decomposed into a series ( terms) of orthonormal polynomials with the corresponding coefficients ,
According to Eqs. (6) and (7), the shearing wavefronts and can be expressed as follows:
where the differential Zernike polynomials and can be written as follows:
Denoting the shearing wavefronts, differential Zernike polynomials and the coefficients in Eq. (8) as , and , respectively, Eq. (8) can be transformed into a matrix form,
Thus, the coefficients of Zernike polynomials in Eq. (7) can be obtained from the least-squares solution of Eq. (10),
According to Eq. (11), the retrieval of point-diffraction wavefront depends on the measurement precision of shearing wavefronts and . Traditionally, the systematic error introduced by lateral displacement between SMA fibers can be calibrated by removing Zernike tilt and power terms. However, the residual high-order aberrations can significantly influence the measurement precision, especially in the case of high NA and large lateral displacement. Thus, a general and rigorous method for geometric error removal is required to realize the high-precision measurement of point-diffraction wavefront.
5.2. High-precision method for systematic error calibration
In the null test of pinhole diffraction wavefront and single-mode fiber diffraction wavefront, the high wavefront NA in a pinhole PDI and large lateral displacement between fibers in fiber PDI could introduce some high-order geometric aberrations, respectively. Different from traditional pinhole PDI and single-mode fiber PDI, the null test of SMA fiber diffraction wavefront involves both high NA and large lateral displacement, placing much higher requirement on the calibration of the systematic error introduced by lateral displacement between SMA fibers. A double-step calibration method based on three-dimensional coordinate reconstruction and symmetric lateral displacement compensation can be applied to completely remove systematic error. It should be noted that the possible longitudinal displacement between SMA fibers may also introduce certain systematic error; however, it can be well minimized with the fine adjusting mechanism. Besides, the error introduced by the longitudinal displacement is low-order aberration, and it can be well calibrated with traditional misalignment calibration method by subtracting the Zernike piston, tilt and power terms.
5.2.1. First-step calibration based on three-dimensional coordinate reconstruction
Without loss of generality, we take the displacement in
Supposing that the distance between exit apertures of fiber pairs G1 (S1 and S2) and an arbitrary point on the detection plane are and , we have the corresponding optical path difference OPD,
where and are the three-dimensional coordinates of S1 and S2, is that of the arbitrary known point . To simplify the analysis, the origin of the coordinate system is located at S1, the distance between the SMA fiber projector and CCD detector is . Thus, the OPD in Eq. (12) can be simplified as follows:
According to the one-to-one correspondence of the OPD distribution on the detection plane and the coordinate of fiber apertures, the 3D coordinate measurement method with PDI introduced in Section 3 can be applied to reconstruct the 3D coordinates in Eq. (12) that is the global minimum of the residual function ,
where the vector is the coordinates of SMA fibers under measurement, the subscript
However, the reconstruction accuracy of fiber coordinates can only reach the order of submicron in the practical case, resulting in obvious residual error in the calibration result. The root mean square (RMS) value of the residual error in the preliminary calibration is corresponding to the 0.5 μm coordinate reconstruction error for 0.60 NA fibers and 250 μm lateral displacement between two fibers. To further remove the residual systematic error, a second-step calibration, which is based on symmetric lateral displacement compensation, needs to be carried out.
5.2.2. Second-step calibration based on symmetric lateral displacement compensation
The expression for OPD in Eq. (12) can be simplified as follows:
where refers to
Similarly, the reconstructed optical path difference (though with certain coordinate reconstruction error) follows the relationship given by Eq. (16). Thus, the residual systematic error after first-step calibration, according to Eqs. (15) and (16), can be written as follows:
According to the analysis above, the geometric error can be further reduced by superposing two measurements with opposite shear directions, corresponding to lateral displacement and , respectively. Denoting the preliminarily calibrated wavefront data corresponding to lateral displacement and as and , and the true shearing wavefront under test , we have
According to Eqs. (19) and (20), the shearing wavefront can be obtained with
To get the measurement data with the lateral displacement , one may rotate the SMA fiber projector by 180°. Figure 17(a) and (b) shows the residual systematic errors corresponding to various lateral displacement amounts with different NAs with traditional method and the double-step calibration method. The traditional method is not valid when the lateral displacement is over 50 μm because the residual error RMS is larger than with the 0.10 NA fiber. The residual error also increases significantly with the lateral displacement, especially for the case with high NA fiber. In Figure 17(b), the residual error is less than RMS within a 300 μm lateral displacement even for the 0.60 NA fiber, it confirms the feasibility of the double-step calibration in Section 5.2.
To obtain point-diffraction spherical wavefront in the experiment, the projector should be placed at the far-field zone that is the distance from the fibers , where
With the true shearing wavefronts in
In this chapter, the application of PDI system in the wavefront and surface testing is introduced. Various PDI setups, including the fiber PDI and pinhole PDI, have developed for high-accuracy optical testing, and the measurement accuracy up to subnanometer was reported. Different from the pinhole PDI, the adjustable fringe contrast is easily realized with fiber PDI, and it can be applied to test the surface with various reflectivities. To realize the adjustable fringe contrast with pinhole method, the polarizing elements with special structure is needed to transform the polarization states and adjust the relative intensities of the interfering testing and reference beams. High measurable NA can be achieved with the pinhole; however, the light transmission is quite low; high light transmittance can be obtained with single-mode fiber; however, the measurable NA is limited by the NA of the fiber. To obtain both the high diffraction light power and high-NA spherical wavefront, a SMA fiber with cone-shaped exit end has been proposed, and it is considered as a feasible way to extend the measurement range of the system. Based on the SMA fiber, a PDI system for the absolute 3D coordinate measurement was proposed. The system utilizes the SMA fiber to get spherical wave with both high NA and high light intensity, and large measurable range is achieved.
The numerical study of point-diffraction wavefront both with pinhole and SMA fiber is performed, which is based on the FDTD method. The aperture angle, light transmittance, and wavefront error are analyzed. According to simulation results, the ideal spherical wavefront with the accuracy better than subnanometer can be obtained both with pinhole and SMA fiber, and the increase in NA range and diffraction aperture size could lead to the growth of diffracted wavefront error. It can be seen from the comparison of pinhole diffraction and SMA fiber diffraction, the light transmittance in both the methods grows with the increase in aperture size, and the similar aperture angles can be obtained with the same diffraction aperture size. However, the light transmittance in SMA fiber diffraction is far larger than that in pinhole diffraction. Thus, the SMA fiber provides a feasible way to obtain the high light intensity required in the optical testing, enabling the extension of measurement range with PDI system.
The experimental measurement of SMA fiber point-diffraction wavefront is carried out to evaluate its accuracy in practical application, which is based on shearing interferometry. To realize the high-precision calibration of the systematic error introduced by the lateral displacement between SMA fibers, a double-step calibration method based on three-dimensional coordinate reconstruction and symmetric lateral displacement compensation is used to remove the geometric aberration from the shearing wavefront. Besides, the differential Zernike polynomials fitting method is used to reconstruct the point-diffraction wavefront. The spherical wavefront with the accuracy reaching the order of is obtained in the experiment. The calibration method can be performed without any preknowledge about the measurement system configuration. It realizes the high-accuracy measurement of point-diffraction wavefront and also provides a feasible method for geometric aberration calibration in interferometric system.
This work was partially supported by the National Natural Science Foundation of China (11404312), Zhejiang Provincial Natural Science Foundation of China (LY17E050014), and Zhejiang Key Discipline of Instrument Science and Technology (JL150508). Authors are grateful to editors for giving them the chance to contribution to this book.
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