Abstract

We describe a quantitative method for measuring the phase of a propagating wave field in three dimensions by use of a scanning optical-fiber interferometer. Because phase modulation in the reference arm is exploited, this technique is insensitive to large variations in the intensity of the field being studied and is therefore highly suitable for measurement of phase within spatially confined optical beams. It uses only a single detector and is not reliant on lock-in electronics. The technique is applied to the measurement of the near field of a cleaved optical fiber and is shown to produce results in good agreement with theory.

© 1999 Optical Society of America

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References

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  1. W. H. Steel, Interferometry, 2nd ed. (Cambridge U Press, Cambridge, UK, 1983).
  2. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980).
  3. E. G. Newman, Single-Mode Fibers (Fundamentals) (Springer-Verlag, Berlin, 1988).
    [CrossRef]
  4. M. Bachmann, P. A. Besse, H. Melchior, “General self-imaging properties in N × N multimode interference couplers including phase relations,” Appl. Opt. 33, 3905–3911 (1994).
    [CrossRef] [PubMed]
  5. S. Kareenahalli, M. Dagenais, D. Stone, T. J. Tayag, “Experimental confirmation of phase relationships of multimode interference splitters using a shearing-type Sagnac interferometer,” IEEE Photonics Technol. Lett. 9, 937–939 (1997).
    [CrossRef]
  6. C.-P. Cherng, T. C. Salvi, M. Osiński, J. McInerney, “Near field wavefront measurements of semiconductor laser arrays by shearing interferometry,” Appl. Opt. 29, 2701–2706 (1990).
    [CrossRef] [PubMed]
  7. M. Iiyama, T. Kamiya, H. Yanai, “Optical field mapping using single-mode optical fibers,” Appl. Opt. 17, 1965–1971 (1978).
    [CrossRef]
  8. M. Iiyama, T. Kamiya, H. Yanai, “Automated optical field mapping using a single-mode fiber interferometer,” Appl. Opt. 20, 4296–4301 (1981).
    [CrossRef]
  9. C. D. Butter, G. B. Hocker, “Fiber optics strain gauge,” Appl. Opt. 17, 2867–2869 (1978).
    [CrossRef] [PubMed]
  10. G. B. Hocker, “Fiber-optic sensing of pressure and temperature,” Appl. Opt. 18, 1445–1448 (1979).
    [CrossRef] [PubMed]
  11. M. Takeda, Z. Tung, “Subfringe holographic interferometry by computer-based spatial-carrier fringe-pattern analysis,” J. Opt. 16, 127–131 (1985); T. Kreis, “Digital holographic interference-phase measurement using the Fourier-transform method,” J. Opt. Soc. Am. A 3, 847–855 (1986).
    [CrossRef]
  12. For example, D. A. Jackson, A. D. Kersey, M. Corke, J. D. C. Jones, “Pseudoheterodyne detection scheme for optical interferometers,” Electron. Lett. 18, 1081–1083 (1982).
    [CrossRef]
  13. D. A. Jackson, R. Priest, A. Dandridge, A. B. Tveten, “Elimination of drift in a single-mode optical fiber interferometer using a piezoelectrically stretched coiled optical fiber,” Appl. Opt. 19, 2926–2929 (1980).
    [CrossRef] [PubMed]
  14. V. S. Sudarshanam, K. Srinivasan, “Linear readout of dynamic phase change in a fiber-optic homodyne interferometer,” Opt. Lett. 14, 140–142 (1989).
    [CrossRef] [PubMed]
  15. V. S. Sudarshanam, K. Srinivasan, “Static phase change in a fiber optic coil hydrophone,” Appl. Opt. 29, 855–863 (1990).
    [CrossRef] [PubMed]
  16. V. S. Sudarshanam, “Direct static strain measurement utilizing signal fading and spectrum analysis in a fiber optic interferometric sensor,” J. Mod. Opt. 41, 683–694 (1994).
    [CrossRef]
  17. K. Freischlad, C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7, 542–551 (1990).
    [CrossRef]
  18. J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
    [CrossRef]
  19. J. C. Wyant, “Use of an ac heterodyne lateral shear interferometer with real-time wave-front correction systems,” Appl. Opt. 14, 2622–2626 (1975).
    [CrossRef] [PubMed]
  20. G. Martini, “Analysis of a single-mode optical fiber piezoceramic phase modulator,” J. Quant. Elect. 19, 179–190 (1987).
    [CrossRef]
  21. V. S. Sudarshanam, S. B. Desu, “Fiber-optic polarization and phase modulator utilizing transparent piezofilm with indium tin oxide electrodes,” Appl. Opt. 34, 1177–1189 (1995).
    [CrossRef] [PubMed]
  22. S. A. Kingsley, “Optical-fiber phase modulator,” Electron. Lett. 11, 453–454 (1975).
    [CrossRef]
  23. D. J. Butler, K. A. Nugent, A. Roberts, “Characterization of optical fibers using near-field scanning optical microscopy,” J. Appl. Phys. 75, 2753–2756 (1994).
    [CrossRef]
  24. E. Betzig, J. K. Trautman, “Near-field optics: microscopy, spectroscopy and surface modification beyond the diffraction limit,” Science 257, 189–195 (1992).
    [CrossRef] [PubMed]

1997

S. Kareenahalli, M. Dagenais, D. Stone, T. J. Tayag, “Experimental confirmation of phase relationships of multimode interference splitters using a shearing-type Sagnac interferometer,” IEEE Photonics Technol. Lett. 9, 937–939 (1997).
[CrossRef]

1995

1994

M. Bachmann, P. A. Besse, H. Melchior, “General self-imaging properties in N × N multimode interference couplers including phase relations,” Appl. Opt. 33, 3905–3911 (1994).
[CrossRef] [PubMed]

V. S. Sudarshanam, “Direct static strain measurement utilizing signal fading and spectrum analysis in a fiber optic interferometric sensor,” J. Mod. Opt. 41, 683–694 (1994).
[CrossRef]

D. J. Butler, K. A. Nugent, A. Roberts, “Characterization of optical fibers using near-field scanning optical microscopy,” J. Appl. Phys. 75, 2753–2756 (1994).
[CrossRef]

1992

E. Betzig, J. K. Trautman, “Near-field optics: microscopy, spectroscopy and surface modification beyond the diffraction limit,” Science 257, 189–195 (1992).
[CrossRef] [PubMed]

1990

1989

1987

G. Martini, “Analysis of a single-mode optical fiber piezoceramic phase modulator,” J. Quant. Elect. 19, 179–190 (1987).
[CrossRef]

1985

M. Takeda, Z. Tung, “Subfringe holographic interferometry by computer-based spatial-carrier fringe-pattern analysis,” J. Opt. 16, 127–131 (1985); T. Kreis, “Digital holographic interference-phase measurement using the Fourier-transform method,” J. Opt. Soc. Am. A 3, 847–855 (1986).
[CrossRef]

1984

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
[CrossRef]

1982

For example, D. A. Jackson, A. D. Kersey, M. Corke, J. D. C. Jones, “Pseudoheterodyne detection scheme for optical interferometers,” Electron. Lett. 18, 1081–1083 (1982).
[CrossRef]

1981

1980

1979

1978

1975

Bachmann, M.

Besse, P. A.

Betzig, E.

E. Betzig, J. K. Trautman, “Near-field optics: microscopy, spectroscopy and surface modification beyond the diffraction limit,” Science 257, 189–195 (1992).
[CrossRef] [PubMed]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980).

Butler, D. J.

D. J. Butler, K. A. Nugent, A. Roberts, “Characterization of optical fibers using near-field scanning optical microscopy,” J. Appl. Phys. 75, 2753–2756 (1994).
[CrossRef]

Butter, C. D.

Cherng, C.-P.

Corke, M.

For example, D. A. Jackson, A. D. Kersey, M. Corke, J. D. C. Jones, “Pseudoheterodyne detection scheme for optical interferometers,” Electron. Lett. 18, 1081–1083 (1982).
[CrossRef]

Dagenais, M.

S. Kareenahalli, M. Dagenais, D. Stone, T. J. Tayag, “Experimental confirmation of phase relationships of multimode interference splitters using a shearing-type Sagnac interferometer,” IEEE Photonics Technol. Lett. 9, 937–939 (1997).
[CrossRef]

Dandridge, A.

Desu, S. B.

Freischlad, K.

Greivenkamp, J. E.

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
[CrossRef]

Hocker, G. B.

Iiyama, M.

Jackson, D. A.

For example, D. A. Jackson, A. D. Kersey, M. Corke, J. D. C. Jones, “Pseudoheterodyne detection scheme for optical interferometers,” Electron. Lett. 18, 1081–1083 (1982).
[CrossRef]

D. A. Jackson, R. Priest, A. Dandridge, A. B. Tveten, “Elimination of drift in a single-mode optical fiber interferometer using a piezoelectrically stretched coiled optical fiber,” Appl. Opt. 19, 2926–2929 (1980).
[CrossRef] [PubMed]

Jones, J. D. C.

For example, D. A. Jackson, A. D. Kersey, M. Corke, J. D. C. Jones, “Pseudoheterodyne detection scheme for optical interferometers,” Electron. Lett. 18, 1081–1083 (1982).
[CrossRef]

Kamiya, T.

Kareenahalli, S.

S. Kareenahalli, M. Dagenais, D. Stone, T. J. Tayag, “Experimental confirmation of phase relationships of multimode interference splitters using a shearing-type Sagnac interferometer,” IEEE Photonics Technol. Lett. 9, 937–939 (1997).
[CrossRef]

Kersey, A. D.

For example, D. A. Jackson, A. D. Kersey, M. Corke, J. D. C. Jones, “Pseudoheterodyne detection scheme for optical interferometers,” Electron. Lett. 18, 1081–1083 (1982).
[CrossRef]

Kingsley, S. A.

S. A. Kingsley, “Optical-fiber phase modulator,” Electron. Lett. 11, 453–454 (1975).
[CrossRef]

Koliopoulos, C. L.

Martini, G.

G. Martini, “Analysis of a single-mode optical fiber piezoceramic phase modulator,” J. Quant. Elect. 19, 179–190 (1987).
[CrossRef]

McInerney, J.

Melchior, H.

Newman, E. G.

E. G. Newman, Single-Mode Fibers (Fundamentals) (Springer-Verlag, Berlin, 1988).
[CrossRef]

Nugent, K. A.

D. J. Butler, K. A. Nugent, A. Roberts, “Characterization of optical fibers using near-field scanning optical microscopy,” J. Appl. Phys. 75, 2753–2756 (1994).
[CrossRef]

Osinski, M.

Priest, R.

Roberts, A.

D. J. Butler, K. A. Nugent, A. Roberts, “Characterization of optical fibers using near-field scanning optical microscopy,” J. Appl. Phys. 75, 2753–2756 (1994).
[CrossRef]

Salvi, T. C.

Srinivasan, K.

Steel, W. H.

W. H. Steel, Interferometry, 2nd ed. (Cambridge U Press, Cambridge, UK, 1983).

Stone, D.

S. Kareenahalli, M. Dagenais, D. Stone, T. J. Tayag, “Experimental confirmation of phase relationships of multimode interference splitters using a shearing-type Sagnac interferometer,” IEEE Photonics Technol. Lett. 9, 937–939 (1997).
[CrossRef]

Sudarshanam, V. S.

Takeda, M.

M. Takeda, Z. Tung, “Subfringe holographic interferometry by computer-based spatial-carrier fringe-pattern analysis,” J. Opt. 16, 127–131 (1985); T. Kreis, “Digital holographic interference-phase measurement using the Fourier-transform method,” J. Opt. Soc. Am. A 3, 847–855 (1986).
[CrossRef]

Tayag, T. J.

S. Kareenahalli, M. Dagenais, D. Stone, T. J. Tayag, “Experimental confirmation of phase relationships of multimode interference splitters using a shearing-type Sagnac interferometer,” IEEE Photonics Technol. Lett. 9, 937–939 (1997).
[CrossRef]

Trautman, J. K.

E. Betzig, J. K. Trautman, “Near-field optics: microscopy, spectroscopy and surface modification beyond the diffraction limit,” Science 257, 189–195 (1992).
[CrossRef] [PubMed]

Tung, Z.

M. Takeda, Z. Tung, “Subfringe holographic interferometry by computer-based spatial-carrier fringe-pattern analysis,” J. Opt. 16, 127–131 (1985); T. Kreis, “Digital holographic interference-phase measurement using the Fourier-transform method,” J. Opt. Soc. Am. A 3, 847–855 (1986).
[CrossRef]

Tveten, A. B.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980).

Wyant, J. C.

Yanai, H.

Appl. Opt.

J. C. Wyant, “Use of an ac heterodyne lateral shear interferometer with real-time wave-front correction systems,” Appl. Opt. 14, 2622–2626 (1975).
[CrossRef] [PubMed]

M. Iiyama, T. Kamiya, H. Yanai, “Optical field mapping using single-mode optical fibers,” Appl. Opt. 17, 1965–1971 (1978).
[CrossRef]

G. B. Hocker, “Fiber-optic sensing of pressure and temperature,” Appl. Opt. 18, 1445–1448 (1979).
[CrossRef] [PubMed]

D. A. Jackson, R. Priest, A. Dandridge, A. B. Tveten, “Elimination of drift in a single-mode optical fiber interferometer using a piezoelectrically stretched coiled optical fiber,” Appl. Opt. 19, 2926–2929 (1980).
[CrossRef] [PubMed]

M. Iiyama, T. Kamiya, H. Yanai, “Automated optical field mapping using a single-mode fiber interferometer,” Appl. Opt. 20, 4296–4301 (1981).
[CrossRef]

V. S. Sudarshanam, K. Srinivasan, “Static phase change in a fiber optic coil hydrophone,” Appl. Opt. 29, 855–863 (1990).
[CrossRef] [PubMed]

C.-P. Cherng, T. C. Salvi, M. Osiński, J. McInerney, “Near field wavefront measurements of semiconductor laser arrays by shearing interferometry,” Appl. Opt. 29, 2701–2706 (1990).
[CrossRef] [PubMed]

M. Bachmann, P. A. Besse, H. Melchior, “General self-imaging properties in N × N multimode interference couplers including phase relations,” Appl. Opt. 33, 3905–3911 (1994).
[CrossRef] [PubMed]

V. S. Sudarshanam, S. B. Desu, “Fiber-optic polarization and phase modulator utilizing transparent piezofilm with indium tin oxide electrodes,” Appl. Opt. 34, 1177–1189 (1995).
[CrossRef] [PubMed]

C. D. Butter, G. B. Hocker, “Fiber optics strain gauge,” Appl. Opt. 17, 2867–2869 (1978).
[CrossRef] [PubMed]

Electron. Lett.

For example, D. A. Jackson, A. D. Kersey, M. Corke, J. D. C. Jones, “Pseudoheterodyne detection scheme for optical interferometers,” Electron. Lett. 18, 1081–1083 (1982).
[CrossRef]

S. A. Kingsley, “Optical-fiber phase modulator,” Electron. Lett. 11, 453–454 (1975).
[CrossRef]

IEEE Photonics Technol. Lett.

S. Kareenahalli, M. Dagenais, D. Stone, T. J. Tayag, “Experimental confirmation of phase relationships of multimode interference splitters using a shearing-type Sagnac interferometer,” IEEE Photonics Technol. Lett. 9, 937–939 (1997).
[CrossRef]

J. Appl. Phys.

D. J. Butler, K. A. Nugent, A. Roberts, “Characterization of optical fibers using near-field scanning optical microscopy,” J. Appl. Phys. 75, 2753–2756 (1994).
[CrossRef]

J. Mod. Opt.

V. S. Sudarshanam, “Direct static strain measurement utilizing signal fading and spectrum analysis in a fiber optic interferometric sensor,” J. Mod. Opt. 41, 683–694 (1994).
[CrossRef]

J. Opt.

M. Takeda, Z. Tung, “Subfringe holographic interferometry by computer-based spatial-carrier fringe-pattern analysis,” J. Opt. 16, 127–131 (1985); T. Kreis, “Digital holographic interference-phase measurement using the Fourier-transform method,” J. Opt. Soc. Am. A 3, 847–855 (1986).
[CrossRef]

J. Opt. Soc. Am. A

J. Quant. Elect.

G. Martini, “Analysis of a single-mode optical fiber piezoceramic phase modulator,” J. Quant. Elect. 19, 179–190 (1987).
[CrossRef]

Opt. Eng.

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
[CrossRef]

Opt. Lett.

Science

E. Betzig, J. K. Trautman, “Near-field optics: microscopy, spectroscopy and surface modification beyond the diffraction limit,” Science 257, 189–195 (1992).
[CrossRef] [PubMed]

Other

W. H. Steel, Interferometry, 2nd ed. (Cambridge U Press, Cambridge, UK, 1983).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980).

E. G. Newman, Single-Mode Fibers (Fundamentals) (Springer-Verlag, Berlin, 1988).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Schematic of device, including He–Ne laser, phase modulator Φ, and photomultiplier tube PMT.

Fig. 2
Fig. 2

Design for fiber phase modulator driven by a piezostack, capable of imparting a phase shift of 8π rad with a driving voltage of amplitude 40 V.

Fig. 3
Fig. 3

(a) Interferogram obtained when the phase modulator in the reference arm is operated. The interferometric signal is plotted versus the driving voltage applied to the phase modulator. The dashed curve represents that part of the signal recorded whilst the fiber is being strained (increasing modulator voltage/piezoexpansion), and the solid curve is the interferogram recorded whilst the fiber is being relaxed (decreasing modulator voltage/piezocontraction). (b) Phase (or fiber elongation) imparted by phase modulator, calculated through a polynomial sine fit to the interferogram shown in (a).

Fig. 4
Fig. 4

(a) Interferogram and (b) calibration curve obtained when the data-acquisition time about the increasing part of the driving voltage signal only is windowed.

Fig. 5
Fig. 5

Two phase maps of a Gaussian beam recorded at different distances from the beam waist, by the probe scanning in a plane, including the beam axis. The beam is propagating upward in the images. The images are approximately 8 µm across and 2 µm along the optical axis direction (vertical axis in the image). The gray scale represents the phase of the beam wrapped into the 0-to-2π range. (a) Image recorded at a very large distance from the waist, where the radius of curvature is large. (b) Image recorded at approximately the Rayleigh length from the waist, where the radius of curvature is minimum.

Fig. 6
Fig. 6

Two images showing the superposition of intensity data with phase data. (a) Contour plot of intensity (hollow contours) superimposed over a contour plot of phase (shaded regions). (b) Isometric plot of the field, with shading representing the cosine of the phase and the height of the surface representing beam intensity.

Fig. 7
Fig. 7

Slice through the phase image showing the data to match the expected quadratic form. The solid curve is a least-squares fit.

Fig. 8
Fig. 8

Radius of curvature measured at different distances from the fiber waist (discrete points) and with a fit based on the theoretical form of the radius of curvature. Error bars represent the error in the quadratic coefficient.

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

I=12IS+IR+2ISIR1/2 cos δ,
Ix, y, t=ISx, y+IR+2ISx, yIR1/2 cosδx, y, t,
δx, y, t=ϕprobex, y+ϕmodt.
ϕmodV=fV(t.
Ijx, y=ISx, y+IR+2ISx, yIR1/2×cosϕprobex, y+ϕj,
I=A+B cosϕmodV+C sinϕmodV
AIS+IR, B2ISIR1/2 cosϕprobe, C2ISIR1/2 sinϕprobe.
 sin2ϕj sinϕjcosϕj sinϕj sinϕjcosϕj cos2ϕj cosϕj sinϕj cosϕjN×ABC= yj sinϕj yj cosϕj yj,
I±=A±A2-B2-C21/22,
Ij=a+b cosc+fVj,
fVj=a0+a1Vj+a2Vj2+a3Vj3.
Er, z=E0, 0ω0ωzexp-r2ω2z×exp-ikz-Φz+kr22Rz,
R=-πλa2.
Rz=z1+z0z2.

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