Abstract

A polarization phase shifting interferometer using a cyclic path configuration for measurement of phase nonuniformities in transparent samples is presented. A cube beam splitter masked by two linear polarizers is used to split the source wavefront into two counter propagating linearly polarized beams that pass through the sample. At the output of the interferometer, the two orthogonally polarized beams are rendered circularly polarized in the opposite sense through the use of a quarter wave plate. Finally, phase shifting is achieved by rotating a linear polarizer before the recording plane. In a rectangular path interferometer, although the two counter propagating wavefronts are laterally folded with respect to each other in the interferometer arms, the beams finally emerge mutually unfolded at the output of the interferometer. This phenomenon is utilized to create a reference if the sample is introduced in one lateral half of the beam in any one of the interferometer arms. The polarization phase shifting technique is used to generate four phase-shifted interferograms, which are utilized to evaluate the phase profile of the phase sample. Experimental results presented validate the proposed technique.

© 2012 Optical Society of America

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References

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  1. P. Hariharan, “Sagnac or Michelson-Sagnac interferometer?” Appl. Opt. 14, 2319–2321 (1975).
  2. G. Sagnac, “Sur la preuve de la réalité de l’ether lumineux par l’expérience de l’interférographe tournant,” C. R. Acad. Sci. (Paris) 157, 1410–1413 (1913).
  3. G. A. Pavlath, “Fiber-optic gyroscopes,” in Proceedings of IEEE Lasers and Electro-Optics Society (LEOS) Annual Meeting (IEEE, 1994), pp. 237–238 .
  4. A. Tselikov and J. Blake, “Sagnac-interferometer-based Fresnel flow probe,” Appl. Opt. 37, 6690–6694 (1998).
    [CrossRef]
  5. B. J. Vakoc, M. J. F. Digonnet, and G. S. Kino, “A novel fiber-optic sensor array based on the Sagnac interferometer,” J. Lightwave Technol. 17, 2316–2326 (1999).
    [CrossRef]
  6. K. H. Han, W. J. Lee, and B. Y. Kim, “Fiber-optic sensor array based on Sagnac interferometer with stable phase bias,” IEEE Photon. Technol. Lett. 13, 148–150 (2001).
    [CrossRef]
  7. W. H. Steel, Interferometry (Cambridge University, 1983).
  8. O. D. D. Soares, “Analysis and alignment of cyclic interferometers,” J. Phys. E 11, 773–776 (1978).
    [CrossRef]
  9. A. Cordero-Davila, J. Pedraza-Contreras, O. Cardona-Nuez, and A. Cornejo-Rodriguez, “Cyclic interferometers for optical testing,” Appl. Opt. 22, 2478–2480 (1983).
    [CrossRef]
  10. S. Chatterjee and Y. P. Kumar, “Long trace profiler with cyclic optical configuration,” Appl. Opt. 41, 5857–5859 (2002).
    [CrossRef]
  11. Y. P. Kumar and S. Chatterjee, “Simultaneous measurement of refractive index and wedge angle of optical windows using Fizeau interferometery and a cyclic path optical configuration,” Appl. Opt. 48, 4756–4761 (2009).
    [CrossRef]
  12. P. Hariharan and D. Sen, “Cyclic shearing interferometer,” J. Sci. Instrum. 37, 374 (1960).
    [CrossRef]
  13. M. P. Kothiyal and C. Delisle, “Shearing interferometer for phase shifting interferometry with polarization phase shifter,” Appl. Opt. 24, 4439–4442 (1985).
    [CrossRef]
  14. J. Huanga, T. Hondaa, and N. Ohyamaa, “A phase stepping double-focus interferometer using a cyclic design and polarized light,” Opt. Commun. 76, 297–301 (1990).
    [CrossRef]
  15. Y. P. Kumar and S. Chatterjee, “Thickness measurement of transparent glass plates using a lateral shearing cyclic path optical configuration setup and polarization phase shifting interferometry,” Appl. Opt. 49, 6552–6557 (2010).
    [CrossRef]
  16. N. I. Toto-Arellano, G. Rodriguez-Zurita, C. Meneses-Fabian, and J. F. Vázquez-Castillo, “A single-shot phase-shifting radial-shearing interferometer,” J. Opt. A 11, 045704 (2009).
    [CrossRef]
  17. G. Rodríguez-Zurita, N. I. Toto-Arellano, C. Meneses-Fabian, and J. F. Vázquez-Castillo, “Adjustable lateral-shear single-shot phase-shifting interferometry for moving phase distributions,” Meas. Sci. Technol. 20, 115902 (2009).
    [CrossRef]
  18. K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, Vol. 26, E. Wolf, ed. (North Holland, 1988), Chap. 5, pp. 349–393.
  19. D. Malacara, M. Servin, and Z. Malacara, Phase Detection Algorithms in Interferogram Analysis for Optical Testing (Dekker, 1998).
  20. F. Pi and G. Orriols, “Energy-balance in the superposition of light waves with lossless beam-splitters,” Am. J. Phys. 53, 667–670 (1985).
    [CrossRef]
  21. D. Malacara, Optical Shop Testing (Wiley, 2007).

2010

2009

Y. P. Kumar and S. Chatterjee, “Simultaneous measurement of refractive index and wedge angle of optical windows using Fizeau interferometery and a cyclic path optical configuration,” Appl. Opt. 48, 4756–4761 (2009).
[CrossRef]

N. I. Toto-Arellano, G. Rodriguez-Zurita, C. Meneses-Fabian, and J. F. Vázquez-Castillo, “A single-shot phase-shifting radial-shearing interferometer,” J. Opt. A 11, 045704 (2009).
[CrossRef]

G. Rodríguez-Zurita, N. I. Toto-Arellano, C. Meneses-Fabian, and J. F. Vázquez-Castillo, “Adjustable lateral-shear single-shot phase-shifting interferometry for moving phase distributions,” Meas. Sci. Technol. 20, 115902 (2009).
[CrossRef]

2002

2001

K. H. Han, W. J. Lee, and B. Y. Kim, “Fiber-optic sensor array based on Sagnac interferometer with stable phase bias,” IEEE Photon. Technol. Lett. 13, 148–150 (2001).
[CrossRef]

1999

1998

1990

J. Huanga, T. Hondaa, and N. Ohyamaa, “A phase stepping double-focus interferometer using a cyclic design and polarized light,” Opt. Commun. 76, 297–301 (1990).
[CrossRef]

1985

F. Pi and G. Orriols, “Energy-balance in the superposition of light waves with lossless beam-splitters,” Am. J. Phys. 53, 667–670 (1985).
[CrossRef]

M. P. Kothiyal and C. Delisle, “Shearing interferometer for phase shifting interferometry with polarization phase shifter,” Appl. Opt. 24, 4439–4442 (1985).
[CrossRef]

1983

1978

O. D. D. Soares, “Analysis and alignment of cyclic interferometers,” J. Phys. E 11, 773–776 (1978).
[CrossRef]

1975

1960

P. Hariharan and D. Sen, “Cyclic shearing interferometer,” J. Sci. Instrum. 37, 374 (1960).
[CrossRef]

1913

G. Sagnac, “Sur la preuve de la réalité de l’ether lumineux par l’expérience de l’interférographe tournant,” C. R. Acad. Sci. (Paris) 157, 1410–1413 (1913).

Blake, J.

Cardona-Nuez, O.

Chatterjee, S.

Cordero-Davila, A.

Cornejo-Rodriguez, A.

Creath, K.

K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, Vol. 26, E. Wolf, ed. (North Holland, 1988), Chap. 5, pp. 349–393.

Delisle, C.

Digonnet, M. J. F.

Han, K. H.

K. H. Han, W. J. Lee, and B. Y. Kim, “Fiber-optic sensor array based on Sagnac interferometer with stable phase bias,” IEEE Photon. Technol. Lett. 13, 148–150 (2001).
[CrossRef]

Hariharan, P.

P. Hariharan, “Sagnac or Michelson-Sagnac interferometer?” Appl. Opt. 14, 2319–2321 (1975).

P. Hariharan and D. Sen, “Cyclic shearing interferometer,” J. Sci. Instrum. 37, 374 (1960).
[CrossRef]

Hondaa, T.

J. Huanga, T. Hondaa, and N. Ohyamaa, “A phase stepping double-focus interferometer using a cyclic design and polarized light,” Opt. Commun. 76, 297–301 (1990).
[CrossRef]

Huanga, J.

J. Huanga, T. Hondaa, and N. Ohyamaa, “A phase stepping double-focus interferometer using a cyclic design and polarized light,” Opt. Commun. 76, 297–301 (1990).
[CrossRef]

Kim, B. Y.

K. H. Han, W. J. Lee, and B. Y. Kim, “Fiber-optic sensor array based on Sagnac interferometer with stable phase bias,” IEEE Photon. Technol. Lett. 13, 148–150 (2001).
[CrossRef]

Kino, G. S.

Kothiyal, M. P.

Kumar, Y. P.

Lee, W. J.

K. H. Han, W. J. Lee, and B. Y. Kim, “Fiber-optic sensor array based on Sagnac interferometer with stable phase bias,” IEEE Photon. Technol. Lett. 13, 148–150 (2001).
[CrossRef]

Malacara, D.

D. Malacara, M. Servin, and Z. Malacara, Phase Detection Algorithms in Interferogram Analysis for Optical Testing (Dekker, 1998).

D. Malacara, Optical Shop Testing (Wiley, 2007).

Malacara, Z.

D. Malacara, M. Servin, and Z. Malacara, Phase Detection Algorithms in Interferogram Analysis for Optical Testing (Dekker, 1998).

Meneses-Fabian, C.

G. Rodríguez-Zurita, N. I. Toto-Arellano, C. Meneses-Fabian, and J. F. Vázquez-Castillo, “Adjustable lateral-shear single-shot phase-shifting interferometry for moving phase distributions,” Meas. Sci. Technol. 20, 115902 (2009).
[CrossRef]

N. I. Toto-Arellano, G. Rodriguez-Zurita, C. Meneses-Fabian, and J. F. Vázquez-Castillo, “A single-shot phase-shifting radial-shearing interferometer,” J. Opt. A 11, 045704 (2009).
[CrossRef]

Ohyamaa, N.

J. Huanga, T. Hondaa, and N. Ohyamaa, “A phase stepping double-focus interferometer using a cyclic design and polarized light,” Opt. Commun. 76, 297–301 (1990).
[CrossRef]

Orriols, G.

F. Pi and G. Orriols, “Energy-balance in the superposition of light waves with lossless beam-splitters,” Am. J. Phys. 53, 667–670 (1985).
[CrossRef]

Pavlath, G. A.

G. A. Pavlath, “Fiber-optic gyroscopes,” in Proceedings of IEEE Lasers and Electro-Optics Society (LEOS) Annual Meeting (IEEE, 1994), pp. 237–238 .

Pedraza-Contreras, J.

Pi, F.

F. Pi and G. Orriols, “Energy-balance in the superposition of light waves with lossless beam-splitters,” Am. J. Phys. 53, 667–670 (1985).
[CrossRef]

Rodriguez-Zurita, G.

N. I. Toto-Arellano, G. Rodriguez-Zurita, C. Meneses-Fabian, and J. F. Vázquez-Castillo, “A single-shot phase-shifting radial-shearing interferometer,” J. Opt. A 11, 045704 (2009).
[CrossRef]

Rodríguez-Zurita, G.

G. Rodríguez-Zurita, N. I. Toto-Arellano, C. Meneses-Fabian, and J. F. Vázquez-Castillo, “Adjustable lateral-shear single-shot phase-shifting interferometry for moving phase distributions,” Meas. Sci. Technol. 20, 115902 (2009).
[CrossRef]

Sagnac, G.

G. Sagnac, “Sur la preuve de la réalité de l’ether lumineux par l’expérience de l’interférographe tournant,” C. R. Acad. Sci. (Paris) 157, 1410–1413 (1913).

Sen, D.

P. Hariharan and D. Sen, “Cyclic shearing interferometer,” J. Sci. Instrum. 37, 374 (1960).
[CrossRef]

Servin, M.

D. Malacara, M. Servin, and Z. Malacara, Phase Detection Algorithms in Interferogram Analysis for Optical Testing (Dekker, 1998).

Soares, O. D. D.

O. D. D. Soares, “Analysis and alignment of cyclic interferometers,” J. Phys. E 11, 773–776 (1978).
[CrossRef]

Steel, W. H.

W. H. Steel, Interferometry (Cambridge University, 1983).

Toto-Arellano, N. I.

N. I. Toto-Arellano, G. Rodriguez-Zurita, C. Meneses-Fabian, and J. F. Vázquez-Castillo, “A single-shot phase-shifting radial-shearing interferometer,” J. Opt. A 11, 045704 (2009).
[CrossRef]

G. Rodríguez-Zurita, N. I. Toto-Arellano, C. Meneses-Fabian, and J. F. Vázquez-Castillo, “Adjustable lateral-shear single-shot phase-shifting interferometry for moving phase distributions,” Meas. Sci. Technol. 20, 115902 (2009).
[CrossRef]

Tselikov, A.

Vakoc, B. J.

Vázquez-Castillo, J. F.

N. I. Toto-Arellano, G. Rodriguez-Zurita, C. Meneses-Fabian, and J. F. Vázquez-Castillo, “A single-shot phase-shifting radial-shearing interferometer,” J. Opt. A 11, 045704 (2009).
[CrossRef]

G. Rodríguez-Zurita, N. I. Toto-Arellano, C. Meneses-Fabian, and J. F. Vázquez-Castillo, “Adjustable lateral-shear single-shot phase-shifting interferometry for moving phase distributions,” Meas. Sci. Technol. 20, 115902 (2009).
[CrossRef]

Am. J. Phys.

F. Pi and G. Orriols, “Energy-balance in the superposition of light waves with lossless beam-splitters,” Am. J. Phys. 53, 667–670 (1985).
[CrossRef]

Appl. Opt.

C. R. Acad. Sci. (Paris)

G. Sagnac, “Sur la preuve de la réalité de l’ether lumineux par l’expérience de l’interférographe tournant,” C. R. Acad. Sci. (Paris) 157, 1410–1413 (1913).

IEEE Photon. Technol. Lett.

K. H. Han, W. J. Lee, and B. Y. Kim, “Fiber-optic sensor array based on Sagnac interferometer with stable phase bias,” IEEE Photon. Technol. Lett. 13, 148–150 (2001).
[CrossRef]

J. Lightwave Technol.

J. Opt. A

N. I. Toto-Arellano, G. Rodriguez-Zurita, C. Meneses-Fabian, and J. F. Vázquez-Castillo, “A single-shot phase-shifting radial-shearing interferometer,” J. Opt. A 11, 045704 (2009).
[CrossRef]

J. Phys. E

O. D. D. Soares, “Analysis and alignment of cyclic interferometers,” J. Phys. E 11, 773–776 (1978).
[CrossRef]

J. Sci. Instrum.

P. Hariharan and D. Sen, “Cyclic shearing interferometer,” J. Sci. Instrum. 37, 374 (1960).
[CrossRef]

Meas. Sci. Technol.

G. Rodríguez-Zurita, N. I. Toto-Arellano, C. Meneses-Fabian, and J. F. Vázquez-Castillo, “Adjustable lateral-shear single-shot phase-shifting interferometry for moving phase distributions,” Meas. Sci. Technol. 20, 115902 (2009).
[CrossRef]

Opt. Commun.

J. Huanga, T. Hondaa, and N. Ohyamaa, “A phase stepping double-focus interferometer using a cyclic design and polarized light,” Opt. Commun. 76, 297–301 (1990).
[CrossRef]

Other

D. Malacara, Optical Shop Testing (Wiley, 2007).

K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, Vol. 26, E. Wolf, ed. (North Holland, 1988), Chap. 5, pp. 349–393.

D. Malacara, M. Servin, and Z. Malacara, Phase Detection Algorithms in Interferogram Analysis for Optical Testing (Dekker, 1998).

W. H. Steel, Interferometry (Cambridge University, 1983).

G. A. Pavlath, “Fiber-optic gyroscopes,” in Proceedings of IEEE Lasers and Electro-Optics Society (LEOS) Annual Meeting (IEEE, 1994), pp. 237–238 .

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Figures (5)

Fig. 1.
Fig. 1.

Schematic of the proposed polarization shifting rectangular path cyclic interferometer. The clockwise and anticlockwise propagating wavefronts are denoted as (–►) and (←), respectively, and the symbols (—♦) and (—♦) indicate the lateral orientation of the propagating wavefronts, which are denoted by shaded planes. Double headed arrows (↔) show the direction of linear polarization of the wavefronts. The directions of the reference coordinate axes are denoted by X, Y, and Z.

Fig. 2.
Fig. 2.

Polarization phase-shifted interferograms (a) to (d) for a sample partially inserted in the interferometer arm as shown in Fig. 3(a).

Fig. 3.
Fig. 3.

(a) A phase sample in the form of a wedge-shaped rectangular glass plate with vertical wedge direction partially inserted in one lateral half of the beam. (b) The unwrapped phase. The phase profile shown is along Y=150.

Fig. 4.
Fig. 4.

(a) Aphase sample in the form of a wedge-shaped rectangular glass plate with horizontal wedge direction partially inserted in one lateral half of the beam. (b) The unwrapped phase. The phase profile shown is along y=150.

Fig. 5.
Fig. 5.

The transparent sample consisting of a flat glass plate, half of which is coated with a dielectric material, inserted such that one lateral half of the counter propagating beams intercept the bare substrate, while the other lateral half intercepts the coated substrate. (b) The unwrapped phase. The phase profile shown is along Y=75. Coating thickness (t) at the edge is 184nm.

Equations (24)

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εi=(abeiα),
εa=12.W(π/2,45°)·P(90°)·W(π,45°)·P(0°)·(11),
P(θ)=(cos2θcosθcosθsinθsin2θ).
W(γ,φ)=(cosγ/2+i.sinγ/2.cos2φisinγ/2.sin2φisinγ/2.sin2φcosγ/2i.sinγ/2.cos2φ).
εa=12(1i).
εc=12W(π/2,45°)·P(0°)·W(π,135°)·P(90°)·(11)=i2(1i).
εc=i2(1i)eiβ.
εaoS=12P(θ)(1i)eiδ(x,y)=12(cosθisinθ)eiδ(x,y)(cosθsinθ)=12ei(πθ)eiδ(x,y)(cosθsinθ),
εaoR=12P(θ)(1i)=12(cosθisinθ)(cosθsinθ)=12ei(πθ).(cosθsinθ)
εcoS=i2P(θ)(1i)eiδ(x,y)eiβ=i2(cosθ+isinθ)eiδ(x,y)eiβ(cosθsinθ)=12ei(π/2+θ)eiδ(x,y)eiβ(cosθsinθ),
εcoR=i2P(θ)(1i)eiβ=i2(cosθ+isinθ).eiβ(cosθsinθ)=12ei(π/2+θ)eiβ(cosθsinθ).
IL=|εcoR+εaοS|2=12|ei(π/2+θ)eiβ+ei(πθ).eiδ(x,y)|2=12{1+cos(2θ+βπ/2δ(x,y))},
IR=|εcoS+εaοR|2=12|ei(π/2+θ)eiδ(x,y)eiβ+ei(πθ)|2=12{1+cos(2θ+βπ/2+δ(x,y))}.
I=12{1+cos(2θ+βπ/2δ(x,y))}+12{1+cos(2θ+βπ/2+δ(x,y))}.
I1L=12{1+cos(βδ(x,y))},
I2L=12{1+cos(π/2+βδ(x,y))},
I3L=12{1+cos(π+βδ(x,y))},
I4L=12{1+cos(π/2+βδ(x,y))},
I1R=12{1+cos(β+δ(x,y))},
I2R=12{1+cos(π/2+β+δ(x,y))},
I3R=12{1+cos(π+β+δ(x,y))},
I4R=12{1+cos(π/2+β+δ(x,y))}.
[βδ(x,y)]=arctan[I4LI2LI1LI3L],
[β+δ(x,y)]=arctanI4RI2RI1RI3R.

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