Accurate wide-range displacement measurement using tunable diode laser and optical frequency comb generator

Youichi Bitou; Thomas R. Schibli; Kaoru Minoshima

doi:10.1364/OPEX.14.000644

Optics Express
Vol. 14,
Issue 2,
pp. 644-654
(2006)
•https://doi.org/10.1364/OPEX.14.000644

Accurate wide-range displacement measurement using tunable diode laser and optical frequency comb generator

Youichi Bitou, Thomas R. Schibli, and Kaoru Minoshima

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Youichi Bitou, Thomas R. Schibli, and Kaoru Minoshima, "Accurate wide-range displacement measurement using tunable diode laser and optical frequency comb generator," Opt. Express 14, 644-654 (2006)

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Table of Contents Category
- Instrumentation, Measurement, and Metrology
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About this Article
History
- Original Manuscript: December 1, 2005
- Revised Manuscript: January 11, 2006
- Manuscript Accepted: January 14, 2006
- Published: January 23, 2006

Abstract

A laser-frequency-based displacement measurement system with sub-nanometer uncertainty using an optical frequency comb generator is developed. In this method, the optical frequency of a tunable laser is locked to the resonance of a Fabry-Perot cavity. One of the two mirrors of this Fabry-Perot cavity is connected to the element whose displacement is to be measured. Wide range optical frequency and displacement measurements were realized by using an optical frequency comb generator, which consists of an electro-optic modulator placed inside of an optical resonator. We demonstrate a displacement measurement of up to 10 μm with 220 pm uncertainty under the stable condition.

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Fig. 1 Schematic layout of the laser-frequency-based displacement measurement system. ECLD, external cavity laser diode; PBS, polarizing beam splitter; BE, beam expander.

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Fig. 2 Error signal E(f) used for locking optical frequency of ECLD to FP cavity.

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Fig. 3 Experimental setup for the optical frequency measurement using an optical frequency comb generator. AOM, acousto-optic modulator; BS, beam splitter; APD, avalanche photodiode; PZT, piezoelectric transducer.

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Fig. 4 Spectrum of comb envelope observed by an optical spectrum analyzer.

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Fig. 5 Schematic explanation of the relative positions of the optical frequency comb lines and the optical frequency locked to the FP cavity.

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Fig. 6 Example of measured beat signal f _beat after filtering and amplification. The resolution bandwidth was 300 kHz.

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Fig. 7 Variation of the RF power transmitted through the bandpass filter during the tuning of f _FP caused by length changes of the FP cavity.

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Fig. 8 Temporal variation of the measured beat-note signal f _beat under stable environmental condition (temperature and air pressure).

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Fig. 9 Beat frequency change Δf when displacements ΔL of around 1, 5, and 10 μm were sequentially applied to the FP cavity. The mode number changes Δk of comb-lines were 3, 16, and 32 when ΔL = 1, 5, and 10 μm, respectively.

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Equations (12)

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Δ L = - \frac{Δ f}{nf} L,

2 n_{0} L_{0} + λ_{0} \frac{ϕ_{0}}{2 π} = m_{0} \frac{C}{f_{0}},

2 n_{M} L_{0} + λ_{M} \frac{ϕ_{M}}{2 π} = (m_{0} + Δm) \frac{C}{f_{M}},

2 n_{d} (L_{0} + Δ L) + λ_{d} \frac{ϕ_{d}}{2 π} = m_{0} \frac{C}{f_{d}},

Δ L = - \frac{C Δ f Δm}{2 n_{d} f_{d} Δ F} + \frac{1}{2 n_{d}} (\frac{f_{M} Δ f}{f_{d} Δ f} Δ L_{M} - Δ L_{d}) .

Δ L_{M} = 2 Δ n_{M} L_{0} + λ_{M} \frac{ϕ_{M}}{2 π} - λ_{0} \frac{ϕ_{0}}{2 π},

Δ L_{d} = 2 Δ n_{d} L_{0} + λ_{d} \frac{ϕ_{d}}{2 π} - λ_{0} \frac{ϕ_{0}}{2 π},

Δ L = - \frac{C Δ f Δm}{2 n_{d} f_{d} Δ F} .

Δ ≅ - \frac{C Δ f Δm}{2 n_{d} f_{d} Δ F} + \frac{Δ F}{8 n_{d} f_{d}} Δ L_{M}

≅ - \frac{C Δ f Δm}{2 n_{d} f_{d} Δ F} + \frac{Δ F}{8 n_{d} f_{d}} (2 Δ n_{M} L_{0} + λ_{0} \frac{Δ ϕ_{M}}{2 π}),

f_{FP} = f_{Rb} \pm k f_{EO} \pm f_{beat},

{[\frac{u (Δ L)}{Δ L}]}^{2} = {[\frac{u (Δ f)}{Δ f}]}^{2} + {[\frac{u (Δ F)}{Δ F}]}^{2} + {[\frac{u (Δ f_{d})}{Δ f_{d}}]}^{2} + {[\frac{u (Δ n_{d})}{Δ n_{d}}]}^{2} .

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Equations (12)

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