Abstract

A closed-loop phase-shifting Fizeau-type interferometer was constructed that uses direct frequency modulation of a laser diode. The interferometer is servo controlled entirely in the phase domain, where optical phases are detected by two-frequency optical heterodyning. A detailed study of stabilization of the interferometer under feedback control was conducted both experimentally and theoretically. The interferometer showed good stability against vibration up to 200 Hz. The stabilization factors obtained experimentally are in good agreement with the theoretical calculations. The phase-shifting experiment was accomplished with high precision as well as with high stability against external disturbances. The profile measurement of a mirror surface was made with a phase-shifting analysis algorithm, and good measurement reproducibility of λ/60 in the root-mean-square value was obtained for ten measurements within a period of 20 min.

© 2001 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]

1998

1992

1991

1989

1987

Chen, J.

Frankena, H. J.

Hariharan, P.

Ishii, Y.

Kubota, T.

Larkin, K. G.

Lee, B. S.

Mnatzakanian, S.

Murata, K.

Nara, M.

Oreb, B. F.

Smorenburg, C.

Strand, T.

Wingerden, J. V.

Yamaguchi, H.

Yoshino, T.

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Figures (8)

Fig. 1
Fig. 1

Schematic of the system for stabilized phase-shifting interferometry: PC, personal computer; OSC, oscilloscope; DVM, digital voltmeter; ISO, Faraday optical isolator; BS1–BS4, beam splitters; LD, laser diode; Amp, amplifier; other abbreviations defined in text (Pol1 and Pol2 are described in Ref. 7).

Fig. 2
Fig. 2

Calculated dependence of the stabilization factor on voltage gain of the amplifier.

Fig. 3
Fig. 3

Output of the vector voltmeter in the interferometer with feedback on and off and a voltage gain of 1.38.

Fig. 4
Fig. 4

Output of the vector voltmeter in the interferometer with feedback on and off and a voltage gain of 2.46.

Fig. 5
Fig. 5

Output of the vector voltmeter in the interferometer subject to a 40-Hz vibrational disturbance with feedback on and off.

Fig. 6
Fig. 6

Dependence of measured stabilization factor S on the frequency of applied mechanical vibration for three voltage gains, together with the calculated values denoted by open symbols at the left.

Fig. 7
Fig. 7

Four-step phase-shifting experiment under feedback control.

Fig. 8
Fig. 8

Three-dimensional display of the measured height profile of an aluminum mirror surface.

Equations (11)

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δ=4πfnd/c,
V=Bδ-2mπm an integer,
I=I0+AGV-V0,
f=f0+χI,
V=V0-f0χAG-I0AG+c4πχndAGδ.
δ=V0-f0χAG-I0AG+2mBπB-c4πχndAG.
Δδoff=4πDc Δf0+4πfc ΔD,
Δδon=-4πD4πχAGBD-c Δf0-4πf4πχAGBD-c ΔD,
S=Δδoff/Δδon.
S=-4πχAGBD-cc.
r=θsθc=DsDc,

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