Linear diffraction grating interferometer with high alignment tolerance and high accuracy

Fang Cheng; Kuang-Chao Fan

doi:10.1364/AO.50.004550

Applied Optics
Vol. 50,
Issue 22,
pp. 4550-4556
(2011)
•https://doi.org/10.1364/AO.50.004550

Linear diffraction grating interferometer with high alignment tolerance and high accuracy

Fang Cheng and Kuang-Chao Fan

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Fang Cheng and Kuang-Chao Fan, "Linear diffraction grating interferometer with high alignment tolerance and high accuracy," Appl. Opt. 50, 4550-4556 (2011)

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Abstract

We present an innovative structure of a linear diffraction grating interferometer as a long stroke and nanometer resolution displacement sensor for any linear stage. The principle of this diffractive interferometer is based on the phase information encoded by the $\pm 1$ st order beams diffracted by a holographic grating. Properly interfering these two beams leads to modulation similar to a Doppler frequency shift that can be translated to displacement measurements via phase decoding. A self-compensation structure is developed to improve the alignment tolerance. LightTool analysis shows that this new structure is completely immune to alignment errors of offset, standoff, yaw, and roll. The tolerance of the pitch is also acceptable for most installation conditions. In order to compact the structure and improve the signal quality, a new optical bonding technology by mechanical fixture is presented so that the miniature optics can be permanently bonded together without an air gap in between. For the output waveform signals, a software module is developed for fast real-time pulse counting and phase subdivision. A laser interferometer HP5529A is employed to test the repeatability of the whole system. Experimental data show that within $15 mm$ travel length, the repeatability is within $15 nm$ .

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Equations (23)

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Polarization	Jones Vector	Optics	Transfer Matrix
Horizontal Polarization	$[\begin{matrix} 1 \\ 0 \end{matrix}]$	PBS, horizontally installed	Through: $[\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}]$ Reflecting: $[\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}]$
Vertical Polarization	$[\begin{matrix} 0 \\ 1 \end{matrix}]$	PBS, with $45 °$ rotation of fast axis	Through: $\frac{1}{2} [\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}]$ Reflecting: $\frac{1}{2} [\begin{matrix} 1 & - 1 \\ - 1 & 1 \end{matrix}]$
Right Circular Polarization	$\frac{\sqrt{2}}{2} [\begin{matrix} 1 \\ - i \end{matrix}]$	$1 / 4$ wave plate, fast axis with $45 °$ decline	$\frac{\sqrt{2}}{2} [\begin{matrix} 1 & i \\ i & 1 \end{matrix}]$
Left Circular Polarization	$\frac{\sqrt{2}}{2} [\begin{matrix} 1 \\ i \end{matrix}]$

Geometric Errors	Spots Movement on the Photodetector	Tolerance (10% intensity loss)
Yaw		—
Roll		—
Pitch		$0.15 °$
Standoff		—
Offset		—

Position (mm)	0.1	1	5	10	15
Error 1 (nm)	1	96	432	882	1315
Error 2 (nm)	2	105	421	881	1338
Error 3 (nm)	$- 1$	88	439	861	1332
Error 4 (nm)	0	87	418	875	1321
Error 5 (nm)	3	99	425	876	1329
Average Error (nm)	1	95	427	877	1327
Standard deviation (nm)	1.6	7.6	8.5	8.0	9.1

Position (mm)	0.1	1	5	10	15
Error 1 (nm)	$- 7$	11	0	7	$- 12$
Error 2 (nm)	$- 6$	20	$- 11$	6	11
Error 3 (nm)	$- 9$	3	7	$- 14$	5
Error 4 (nm)	$- 8$	2	$- 14$	0	$- 6$
Error 5 (nm)	$- 5$	14	$- 7$	1	2
Average Error (nm)	$- 7$	10	$- 5$	2	0

Polarization	Jones Vector	Optics	Transfer Matrix
Horizontal Polarization	$[\begin{matrix} 1 \\ 0 \end{matrix}]$	PBS, horizontally installed	Through: $[\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}]$ Reflecting: $[\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}]$
Vertical Polarization	$[\begin{matrix} 0 \\ 1 \end{matrix}]$	PBS, with $45 °$ rotation of fast axis	Through: $\frac{1}{2} [\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}]$ Reflecting: $\frac{1}{2} [\begin{matrix} 1 & - 1 \\ - 1 & 1 \end{matrix}]$
Right Circular Polarization	$\frac{\sqrt{2}}{2} [\begin{matrix} 1 \\ - i \end{matrix}]$	$1 / 4$ wave plate, fast axis with $45 °$ decline	$\frac{\sqrt{2}}{2} [\begin{matrix} 1 & i \\ i & 1 \end{matrix}]$
Left Circular Polarization	$\frac{\sqrt{2}}{2} [\begin{matrix} 1 \\ i \end{matrix}]$

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Equations (23)

Applied Optics