Synthetic phase-shifting for optical testing: Point-diffraction interferometry without null optics or phase shifters

Ryeojin Park; Dae Wook Kim; Harrison H. Barrett

doi:10.1364/OE.21.026398

Optics Express
Vol. 21,
Issue 22,
pp. 26398-26417
(2013)
•https://doi.org/10.1364/OE.21.026398

Synthetic phase-shifting for optical testing: Point-diffraction interferometry without null optics or phase shifters

Ryeojin Park, Dae Wook Kim, and Harrison H. Barrett

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Ryeojin Park, Dae Wook Kim, and Harrison H. Barrett, "Synthetic phase-shifting for optical testing: Point-diffraction interferometry without null optics or phase shifters," Opt. Express 21, 26398-26417 (2013)

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- Instrumentation, Measurement, and Metrology
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About this Article
History
- Original Manuscript: July 1, 2013
- Revised Manuscript: October 13, 2013
- Manuscript Accepted: October 21, 2013
- Published: October 28, 2013

Abstract

An innovative iterative search method called the synthetic phase-shifting (SPS) algorithm is proposed. This search algorithm is used for maximum-likelihood (ML) estimation of a wavefront that is described by a finite set of Zernike Fringe polynomials. In this paper, we estimate the coefficient, or parameter, values of the wavefront using a single interferogram obtained from a point-diffraction interferometer (PDI). In order to find the estimates, we first calculate the squared-difference between the measured and simulated interferograms. Under certain assumptions, this squared-difference image can be treated as an interferogram showing the phase difference between the true wavefront deviation and simulated wavefront deviation. The wavefront deviation is the difference between the reference and the test wavefronts. We calculate the phase difference using a traditional phase-shifting technique without physical phase-shifters. We present a detailed forward model for the PDI interferogram, including the effect of the finite size of a detector pixel. The algorithm was validated with computational studies and its performance and constraints are discussed. A prototype PDI was built and the algorithm was also experimentally validated. A large wavefront deviation was successfully estimated without using null optics or physical phase-shifters. The experimental result shows that the proposed algorithm has great potential to provide an accurate tool for non-null testing.

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Fig. 1 The schematic of the PDI for non-null testing. Unlike a conventional PDI, physical phase-shifters are not required. The dashed line represents the test beam and its wavefront, and the solid line represents the reference wavefront generated by pinhole diffraction at the PDI plate.

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Fig. 2 Graphical demonstration of the synthetic four-step phase-shifting algorithm. (a) The experimental result was generated with the true parameter values based on the forward model, including noise. (b)-(e) The simulated interferograms, without noise, with nominal parameter values at phase-steps

0, π / 2, π, 3 π / 2 .

(f)-(i) The squared-difference images between (a) and (b)-(e) respectively. The squared-difference images show the slowly varying

ϕ_{d i f f} (r) .

(Note: some spurious fringes may also be visible due to the aliasing by the monitor display sampling.)

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Fig. 3 The unwrapped-phase results after (a) 1, (b) 3, (c) 5, (d) 7, (e) 16, and (f) 25 iterations ofthe synthetic phase-shifting algorithm. The scale bars are in units of waves.

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Fig. 4 The flow chart of the iterative synthetic phase-shifting algorithm. The box-averaging filtering process is optional for noise suppression, but it may improve the convergence rate.

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Fig. 5 (a) The first simulated noisy interferogram using the true values in Table 2 and (b) is the estimation result after 10 iterations of the SPS algorithm. Even though the spatial frequencies of the interferograms satisfy the Nyquist condition, some spurious fringes appear due to the aliasing by the monitor display sampling.

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Fig. 6 (a) The simulated noisy interferogram was generated using the true values in Table 4 (Subsection 3.2) and (b) the estimated result after 50 iterations of the SPS algorithm. (Note: some spurious fringes may also be visible due to the aliasing by the monitor display sampling.)

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Fig. 7 The absolute differences (in waves) between the true ZF coefficients of the test wavefront used in Subsection 3.2 and their estimates after 50 iterations of the SPS algorithm.

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Fig. 8 (a) A single line profile of the true test wavefront used in Subsection 3.2 and its estimate after 50 iterations of the SPS algorithm. The true values and the estimates are almost overlapping each other. (b) The zoomed image of (a) at pixel 1022.

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Fig. 9 The simulated noisy interferograms using the true values in Table 2 with a pixel size of (a)

48 μ m \times 48 μ m,

and (c)

96 μ m \times 96 μ m .

(b) and (d) are the estimation results after the SPS algorithm was applied. (Note: some spurious fringes may also be visible due to the aliasing by the monitor display sampling.)

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Fig. 10 The effect of different ranges of deviation between the initial values and the true values on the convergence to a threshold value.

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Fig. 11 The experimental setup of the prototype point-diffraction interferometer.

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Fig. 12 The focused ion-beam etched pinhole on a chromium layer deposited on a fused silica plate. Images were taken with a scanning electron microscope.

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Fig. 13 (a) The measured interferogram obtained by using a f/#_W = 2.537 aspheric lens. (b) The estimated interferogram after 13 iterations of the SPS algorithm. Pearson’s correlation coefficient between the two images is ~0.95. (Note: some spurious fringes may also be visible due to the aliasing by the monitor display sampling.)

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Fig. 14 A single line profile, through the x and y axes, comparison between the measured data [(a) and (c)] and the estimated results [(b) and (d)].

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Tables (8)

Table 1 The system construction parameter values for the reference beam at the detector plane used in Section 3.1. The values were chosen such that the departure of the test wavefront from the reference wavefront changes by less than half a wave over two adjacent pixels.

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Table 2 The true ZF coefficients, the initial values and the final estimates of the test wavefront at the detector plane used in Subsection 3.1. The values of terms that are not listed are zero. All ZF coefficient values are in waves.

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Table 3 The system construction parameters for the reference beam used in Subsection 3.2.

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Table 4 The true ZF coefficients and the initial values of the test wavefront used in Subsection 3.2. The estimation results are after 50 iterations of the SPS algorithm. All ZF coefficient values are in waves. The corresponding aberration types for some indices are shown in Table 2.

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Table 5 The piston-corrected RMSDs of the test wavefront at the detector plane with different pixel sizes. The true ZF coefficients of the test wavefront are shown in Table 2.

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Table 6 ZF coefficients, peak-to-valley and RMS of the exit pupil aberration of the aspheric lens (Edmund 49104) obtained from Zemax. The center of the reference sphere is located at the paraxial image plane. The exit pupil diameter is 20.01 mm. The values of non-symmetric terms are zero.

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Table 7 The design ZF coefficient values at the detector plane obtained from Zemax. The final estimates were obtained after subtracting the alignment errors (tip/tilt and distances between the elements). The corresponding aberration types for some indices are shown in Table 2.

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Table 8 All system construction parameters were experimentally measured using a micrometer, except C_ref which was calculated by a local Michelson contrast measurement. The measurement error of d_PDI was large due to the limited access to the CCD plane.

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Equations (27)

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I (x, y) = I_{1} + I_{2} + 2 \sqrt{I_{1} I_{2}} c o s [ϕ_{t e s t} - ϕ_{r e f}],

pr (g | θ) = \prod_{m = 1}^{M} \frac{1}{\sqrt{2 π σ_{m}^{2}}} \exp {- \frac{{[g_{m} - {\bar{g}}_{m} (θ)]}^{2}}{2 σ_{m}^{2}}},

{\hat{θ}}_{M L} \equiv \underset{θ}{\arg \max} pr (g | θ),

{\hat{θ}}_{M L} \equiv \underset{θ}{\arg \min} \sum_{m = 1}^{M} {[g_{m} - {\bar{g}}_{m} (θ)]}^{2},

Δ g_{m}^{2} \equiv {[g_{m} - {\bar{g}}_{m} (θ)]}^{2} .

Δ g_{m}^{2} \propto \cos [2 π ϕ_{d i f f} (r_{m})],

ϕ_{d i f f} (r) = [ϕ_{t e s t} (r) - ϕ_{r e f} (r)] - [{\hat{ϕ}}_{t e s t} (r | θ) - {\hat{ϕ}}_{r e f} (r)] = ϕ_{e x p} (r) - ϕ_{s i m} (r) .

ϕ_{t e s t} (r_{m}) \approx {\hat{ϕ}}_{t e s t} (r_{m} | θ) + ϕ_{d i f f} (r_{m}) .

{\hat{θ}}^{j + 1} = {\hat{θ}}^{j} + θ_{d i f f}^{j} .

σ_{R M S D} = \sqrt{\frac{\sum_{m = 1}^{M} {[ϕ_{t e s t} (r_{m}) - {\hat{ϕ}}_{t e s t} (r_{m})]}^{2}}{M}},

σ_{R M S} = \sqrt{\frac{\sum_{m = 1}^{M} {[ϕ_{d i f f} (r_{m}) - c_{d i f f}]}^{2}}{M}},

σ_{R S S} = (n - 1) {(σ_{t e s t}^{2} + σ_{t e s t}^{2})}^{1 / 2},

[\sum_{p = 1}^{P} θ_{p} Z_{p} (r) - ϕ_{r e f} (r)] = - [\sum_{p = 1}^{P} {θ^{'}}_{p} Z_{p} (r) - ϕ_{r e f} (r)],

E (r | θ) = {| U_{t e s t} (r) + U_{r e f} (r) |}^{2},

U_{t e s t} (r) = α (r) \exp [2 π i {\hat{ϕ}}_{t e s t} (r)],

{\hat{ϕ}}_{t e s t} (r) = \sum_{p = 1}^{P} θ_{p} Z_{p} (r),

U_{r e f} (r) = β (r) \exp (i k \sqrt{{| r - r_{c} |}^{2} + d_{P ​ D ​ I}^{2}}) = β (r) \exp [2 π i {\hat{ϕ}}_{r e f} (r)],

\begin{matrix} {\bar{g}}_{m} (θ) \equiv R {\bar{N}}_{m} = R η T_{f r} \int_{A_{m}} \bar{E} (r | θ) d r \\ = R η T_{f r} (α {(r)}^{2} + β {(r)}^{2} + 2 α (r) β (r) \cos {2 π [{\hat{ϕ}}_{t e s t} (r | θ) - {\hat{ϕ}}_{r e f} (r)]}) d r, \end{matrix}

{\bar{g}}_{m} (θ) = \int_{A_{m}} {\hat{B} (r) + \hat{γ} (r) \cos [2 π ϕ_{s i m} (r | θ)]} d r,

g_{m} = \int_{A_{m}} {B (r) + γ (r) \cos [2 π ϕ_{e x p} (r)]} d r + n_{m},

Δ g_{m}^{2} \equiv {[g_{m} - {\bar{g}}_{m} (θ)]}^{2} .

Δ g_{m}^{2} \approx {[\int_{A_{m}} (γ (r) {\cos [2 π ϕ_{e x p} (r)] - \cos [2 π ϕ_{s i m} (r)]} + B (r) - \hat{B} (r)) d r + n_{m}]}^{2} .

Δ g_{m}^{2} \approx {(\begin{array}{l} \int_{A_{m}} 2 γ (r) \sin {π [ϕ_{e x p} (r) - ϕ_{s i m} (r)]} \sin {π [ϕ_{e x p} (r) + ϕ_{s i m} (r)]} d r \\ + \int_{A_{m}} [B (r) - \hat{B} (r)] d r + n_{m} \end{array})}^{2} .

Δ g_{m}^{2} \approx {({B^{'}}_{m} + {γ^{'}}_{m} \sin [π ϕ_{d i f f} (r_{m})])}^{2},

{B^{'}}_{m} \equiv \int_{A_{m}} {B (r) - \hat{B} (r)} d r + n_{m} .

{γ^{'}}_{m} \equiv \int_{A_{m}} {2 γ (r) \sin [π ϕ_{s u m} (r)]} d r .

Δ g_{m}^{2} \approx {B^{″}}_{m} + {γ^{″}}_{m} \cos [2 π ϕ_{d i f f} (r_{m})],

Index	Aberration Type	True	Initial	Estimate
2	Tilt x	0	1.2	0
3	Tilt y	0	-0.4	0
4	Defocus	592.7707	591.9707	592.7681
7	Coma x	0	0.3	0
8	Coma y	0	-0.5	0
9	Spherical Aberration, Primary	-32.9418	-32.5418	-32.932
16	Spherical Aberration, Secondary	5.4881	5.5681	5.4954
25	Spherical Aberration, Tertiary	-2.1501	-2.1501	-2.1605
36	Spherical Aberration, Quaternary	0.1490	0.1490	0.1338
37	Spherical Aberration, 12^th- order term	-0.3450	-0.3450	-0.3373

Index	True	Initial	Estimate	Index	True	Initial	Estimate
2	-2.1123	0.0	-2.1120	20	2.1394	0.0	2.1311
3	5.662	0.0	5.660	21	-9.6740	0.0	-9.6731
4	598.7395	592.7707	598.7425	22	-5.1423	0.0	-5.1387
5	8.2324	0.0	8.2346	23	-7.2554	0.0	-7.2573
6	-6.0490	0.0	-6.0488	24	6.0835	0.0	6.0741
7	-3.2955	0.0	-3.2955	25	-9.0165	-2.1501	-9.0088
8	5.3646	0.0	5.3600	26	-1.9811	0.0	-1.9839
9	-37.3863	-32.9418	-37.3810	27	-7.4042	0.0	-7.4072
10	1.0794	0.0	1.0823	28	-7.8238	0.0	-7.8302
11	-0.4521	0.0	-0.4583	29	9.9785	0.0	9.9719
12	2.5774	0.0	2.5793	30	-5.6349	0.0	-5.6279
13	-2.7043	0.0	-2.7025	31	0.2586	0.0	0.2498
14	0.2680	0.0	0.2665	32	6.7822	0.0	6.7837
15	9.0466	0.0	9.0368	33	2.2528	0.0	2.2583
16	13.8120	5.4881	13.8185	34	-4.0794	0.0	-4.0811
17	2.7142	0.0	2.7093	35	2.7510	0.0	2.7420
18	4.3459	0.0	4.3407	36	0.6347	0.1490		0.6404
19	-7.1679	0.0	-7.1628	37	-0.4730	-0.3450		-0.4700

Index	Aberration Type	Design Value
4	Defocus	43.7432894
9	Spherical Aberration, Primary	14.1472115
16	Spherical Aberration, Secondary	-0.2436503
25	Spherical Aberration, Tertiary	-0.0167761
36	Spherical Aberration, Quaternary	-0.0005934
37	Spherical Aberration, 12^th- order term	-0.0000688
Peak-to-valley	N/A	86.9980866
RMS [waves]	N/A	25.7345771
Variance [waves²]	N/A	662.2684598

Index	Design^a	Estimate	Index	Design^a	Estimate
2	0	-0.763	20	0	-0.046
3	0	-0.278	21	0	0.104
4	565.495	565.802	22	0	0.053
5	0	0.321	23	0	-0.088
6	0	-0.290	24	0	-0.022
7	0	-0.167	25	-1.181	-1.186
8	0	-0.007	26	0	-0.111
9	-27.591	-26.098	27	0	-0.061
10	0	-0.044	28	0	0.107
11	0	-0.076	29	0	0.078
12	0	-0.047	30	0	-0.090
13	0	0.076	31	0	-0.041
14	0	-0.114	32	0	0.085
15	0	0.250	33	0	0.059
16	4.756	3.755	34	0	-0.071
17	0	0.049	35	0	0.096
18	0	0.070	36	0.3	0.351
19	0	-0.065	37	-0.121	-0.392

Index	Aberration Type	True	Initial	Estimate
2	Tilt x	0	1.2	0
3	Tilt y	0	-0.4	0
4	Defocus	592.7707	591.9707	592.7681
7	Coma x	0	0.3	0
8	Coma y	0	-0.5	0
9	Spherical Aberration, Primary	-32.9418	-32.5418	-32.932
16	Spherical Aberration, Secondary	5.4881	5.5681	5.4954
25	Spherical Aberration, Tertiary	-2.1501	-2.1501	-2.1605
36	Spherical Aberration, Quaternary	0.1490	0.1490	0.1338
37	Spherical Aberration, 12^th- order term	-0.3450	-0.3450	-0.3373

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Equations (27)

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