Abstract

On the basis of white-light interferometry with spectrally integrated detection and Fourier transform (FT) analysis, we demonstrate a novel technique for measuring the spectrally-resolved absolute phase difference between orthogonal optical modes with milliradian precision. The phase difference is evaluated from a nonlinear beat signal, occurring in the phase spectrum when independent interferograms, formed by individual modes, are recorded simultaneously. Although scanning white-light FT interferometry is a linear technique in general, the nonlinear beat signal is due to spectral amplitude variations in each mode. These proof-of-principle absolute phase difference measurements were carried out with polarization and spatial fiber modes.

© 2005 Optical Society of America

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References

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Appl. Opt. (1)

Appl. Phys. Lett. (2)

H. K. Heinrich, D. M. Bloom, and B. R. Hemenway, "Noninvasive sheet charge density probe for integrated silicon devices," Appl. Phys. Lett. 48, 1066-1068 (1986).
[CrossRef]

U. Keller, S. K. Diamond, B. A. Auld, and D. M. Bloom, "A noninvasive optical probe of free charge and applied voltage in GaAs devices," Appl. Phys. Lett. 53, 388-390 (1988).
[CrossRef]

J. Opt. Soc. Am B (1)

A. Gosteva, M. Haiml, R. Paschotta, and U. Keller, "Noise-related resolution limit of dispersion measurements with white-light interferometers," J. Opt. Soc. Am B 22, 1868-1874 (2005).
[CrossRef]

J. Opt. Soc. Am. B (1)

Mitsuo Takeda, Hideki Ina, and Seiji Kobayashi, "Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry," J. Opt. Soc. Am. B 72, 156-160 (1982).
[CrossRef]

Jap. J. Appl. Phys. (1)

G. Nomarski, "A double-shear differential interferometer using birefringent beamsplitter," Jap. J. Appl. Phys. 14, 363-368 (1975).

Opt. Lett. (5)

Rev. Sci. Instrum. (1)

I. A. Walmsley, L. Waxer, and C. Dorrer, "The role of dispersion in optics," Rev. Sci. Instrum. 72, 1-29 (2001).
[CrossRef]

Other (2)

M. Born and E. Wolf, Principles of Optics. (U. K.: Pergamon, London, 1984).

R.M.A. Azzam and N.M. Bashara, Ellipsometry and Polarized Light. (North-Holland, Amsterdam, 1987).

Supplementary Material (2)

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» Media 2: GIF (74 KB)     

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Figures (7)

Fig. 1.
Fig. 1.

Collinear fiber-coupled scanning white light interferometer for measurements of the spectrally resolved phase difference between polarization modes (a), and spatial fiber modes (b). SOA - centered at 1550 nm unseeded semiconductor optical amplifier (white light source), DFB LD - distributed feedback laser diode, He-Ne laser - Helium Neon laser (632 nm), M1, M2, M3 - silver mirrors, L - lenses, P - polarizer, BS - non-polarizing beam splitter, DM - dichroic mirror, G - thick glass plate, C- compensator. Collinear to the white light, the He-Ne (or DFB LD) beam is used to monitor the displacement of the reference mirror.

Fig. 2.
Fig. 2.

Shows two interferograms (dotted red curve and solid green curve) of two individual modes, whose phases experience different phase retardations inside the interferometer. The phase and the complex amplitude are extracted from the Fourier transform of a linear superposition (black curve) of these two interferograms.

Fig. 3.
Fig. 3.

(110 kB) Vector diagram for two individual modes (green and red vectors), and their sum (black vector). (a) The phase spectrum is the average when the amplitude ratio between the modes is constant. (b) The amplitude ratio between the modes changes, but the phase retardation for each mode is the same. (c) The phase for each mode is constant, but the amplitudes change with frequency. The sum exhibits strong phase oscillations.

Fig. 4.
Fig. 4.

(301 kB) (a) Measured GDD (D2) for s- , p- and unpolarized light. (b) Spectral amplitude of the s- and p- polarized light. Solid black curves are the simulated GDD (a), and amplitude spectrum (b) when the phase difference between the modes increases.

Fig. 5.
Fig. 5.

(a) Measured GDD values (D2) for the unpolarized light and the fit to the measurement. (b) The retrieved spectrally-resolved absolute phase difference between s- and p- polarizations.

Fig. 6.
Fig. 6.

(a) Measured GDD (D2) and the spectrum for the P1-3224-FC-2 (dashed blue curves), for the unbent P1-5624-FC-2 (solid red curves), and the bended P1-5624-FC-2 (dotted green curves). (b) Simulated spectrum for the unbent P1-5624-FC-2 fiber (solid red curve), and the corresponding spectrums of the fundamental (dashed black curve) and higher excited mode (dotted violet curve).

Fig. 7.
Fig. 7.

(a) Measured (red curve, open circles) and simulated (solid black curve) GDD (D2) for the unbent P1-5624-FC-2 fiber when the phase difference between the fundamental and higher excited mode is equal to (-0.15 rad). (b) Measured (green curve, filled circles) and simulated (solid black curve) GDD (D2) for the slightly bended P1-5624-FC-2 fiber when the phase difference between the modes is equal to (0.21 rad). (c) Spectrally-resolved absolute phase difference between the fundamental and higher excited modes.

Equations (3)

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φ s ( ω ) = D 2 , s ( ω ) d ω 2 + C 1 , s ω + C 0 , s
φ p ( ω ) = D 2 , p ( ω ) d ω 2 + C 1 , p ω + C 0 , p
Δ φ = C 1 ω + C 0

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