Abstract

The polarization-transforming properties of rotational prisms are analyzed with polarized light by using the Jones calculus and the exact ray-trace. A general expression of the Jones matrix for a rotational prism is derived, incorporating an explicit dependence on the image-rotation angle or the wave-front-rotation angle. The Jones matrix for the Pechan, Dove, Reversion, and Delta prisms is derived where the explicit dependence on the angle of rotation of the image is given. An experiment with a rotating Dove prism is also conducted to determine the output states of polarization for incident linearly polarized light. Experimental results agree with theoretical expectations.

© 2004 Optical Society of America

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References

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  1. W. L. Wolfe, “Nondispersive prisms,” in Handbook of Optics, 2nd ed., M. Bass, E. W. Van Stryland, D. R. Williams, W. L. Wolfe, eds. (McGraw-Hill, New York, 1995), pp. 4.1–4.29.
  2. D. L. Sullivan, “Alignment of rotational prisms,” Appl. Opt. 11, 2028–2032 (1972).
    [CrossRef] [PubMed]
  3. P. R. Yoder, Design and Mounting of Prisms and Small Mirrors in Optical Instruments (SPIE Press, Bellingham, Wash., 1998), Vol. TT32, pp. 19–57.
  4. H. Fujii, Y. Ohtsuka, “Rotational filtering for randomly oriented pattern recognition,” Opt. Commun. 36, 255–257 (1981).
    [CrossRef]
  5. I. Moreno, G. Paez, M. Strojnik, “Dove prism with increased throughput for implementation in a rotational-shearing interferometer,” Appl. Opt. 42, 4514–4521 (2003).
    [CrossRef] [PubMed]
  6. M. Strojnik Scholl, G. Paez, “Simulated interferometric patterns generated by a nearby star-planet system and detected by a rotational shearing interferometer,” J. Opt. Soc. Am. A 16, 2019–2024 (1999).
    [CrossRef]
  7. C. Perez-Lopez, F. Mendoza Santoyo, G. Pedrini, S. Schedin, H. J. Tiziani, “Pulsed digital holographic interferometry for dynamic measurement of rotating objects with an optical derotator,” Appl. Opt. 40, 5106–5110 (2001).
    [CrossRef]
  8. P. Z. Takacs, E. L. Church, C. J. Bresloff, L. Assoufid, “Improvements in the accuracy and the repeatability of long trace profiler measurements,” Appl. Opt. 38, 5468–5479 (1999).
    [CrossRef]
  9. A. V. Smith, M. S. Bowers, “Image-rotating cavity designs for improved beam quality in nanosecond optical parametric oscillators,” J. Opt. Soc. Am. B 18, 706–713 (2001).
    [CrossRef]
  10. D. J. Armstrong, A. V. Smith, “Demonstration of improved beam quality in an image-rotating optical parametric oscillator,” Opt. Lett. 27, 40–42 (2002).
    [CrossRef]
  11. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
    [CrossRef] [PubMed]
  12. J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, M. J. Padgett, “Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum,” Phys. Rev. Lett. 80, 3217–3219 (1998).
    [CrossRef]
  13. I. Moreno, P. Velasquez, C. R. Fernandez-Pousa, M. M. Sanchez-Lopez, “Jones matrix method for predicting and optimizing the optical modulation properties of a liquid-crystal display,” J. Appl. Phys. 94, 3697–3702 (2003).
    [CrossRef]
  14. M. J. Padgett, J. P. Lesso, “Dove prisms and polarized light,” J. Mod. Opt. 46, 175–179 (1999).
  15. E. J. Galvez, C. D. Holmes, “Geometric phase of optical rotators,” J. Opt. Soc. Am. A 16, 1981–1985 (1999).
    [CrossRef]
  16. R. A. Chipman, “Mechanics of polarization ray tracing,” Opt. Eng. 34, 1636–1645 (1995).
    [CrossRef]
  17. E. Waluschka, “Polarization ray trace,” Opt. Eng. 28, 86–89 (1989).
    [CrossRef]
  18. M. Strojnik Scholl, “Ray trace through a corner-cube retroreflector with complex reflection coefficients,” J. Opt. Soc. Am. A 12, 1589–1592 (1995).
    [CrossRef]
  19. M. Strojnik, G. Paez, “Radiometry,” in Handbook of Optical Engineering, D. Malacara, B. Thompson, eds. (Marcel Dekker, New York, 2001), Chap. 18, pp. 649–699.
  20. W. A. Shurcliff, Polarized Light (Harvard University, Cambridge, Mass., 1957).
  21. J. M. Bennett, “Polarization,” in Handbook of Optics, 2nd ed., M. Bass, E. W. Van Stryland, D. R. Williams, W. L. Wolfe, eds. (McGraw-Hill, New York, 1995), p. 5.1.

2003

I. Moreno, G. Paez, M. Strojnik, “Dove prism with increased throughput for implementation in a rotational-shearing interferometer,” Appl. Opt. 42, 4514–4521 (2003).
[CrossRef] [PubMed]

I. Moreno, P. Velasquez, C. R. Fernandez-Pousa, M. M. Sanchez-Lopez, “Jones matrix method for predicting and optimizing the optical modulation properties of a liquid-crystal display,” J. Appl. Phys. 94, 3697–3702 (2003).
[CrossRef]

2002

D. J. Armstrong, A. V. Smith, “Demonstration of improved beam quality in an image-rotating optical parametric oscillator,” Opt. Lett. 27, 40–42 (2002).
[CrossRef]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

2001

1999

1998

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, M. J. Padgett, “Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum,” Phys. Rev. Lett. 80, 3217–3219 (1998).
[CrossRef]

1995

1989

E. Waluschka, “Polarization ray trace,” Opt. Eng. 28, 86–89 (1989).
[CrossRef]

1981

H. Fujii, Y. Ohtsuka, “Rotational filtering for randomly oriented pattern recognition,” Opt. Commun. 36, 255–257 (1981).
[CrossRef]

1972

Allen, L.

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, M. J. Padgett, “Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum,” Phys. Rev. Lett. 80, 3217–3219 (1998).
[CrossRef]

Armstrong, D. J.

Assoufid, L.

Barnett, S. M.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

Bennett, J. M.

J. M. Bennett, “Polarization,” in Handbook of Optics, 2nd ed., M. Bass, E. W. Van Stryland, D. R. Williams, W. L. Wolfe, eds. (McGraw-Hill, New York, 1995), p. 5.1.

Bowers, M. S.

Bresloff, C. J.

Chipman, R. A.

R. A. Chipman, “Mechanics of polarization ray tracing,” Opt. Eng. 34, 1636–1645 (1995).
[CrossRef]

Church, E. L.

Courtial, J.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, M. J. Padgett, “Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum,” Phys. Rev. Lett. 80, 3217–3219 (1998).
[CrossRef]

Dholakia, K.

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, M. J. Padgett, “Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum,” Phys. Rev. Lett. 80, 3217–3219 (1998).
[CrossRef]

Fernandez-Pousa, C. R.

I. Moreno, P. Velasquez, C. R. Fernandez-Pousa, M. M. Sanchez-Lopez, “Jones matrix method for predicting and optimizing the optical modulation properties of a liquid-crystal display,” J. Appl. Phys. 94, 3697–3702 (2003).
[CrossRef]

Franke-Arnold, S.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

Fujii, H.

H. Fujii, Y. Ohtsuka, “Rotational filtering for randomly oriented pattern recognition,” Opt. Commun. 36, 255–257 (1981).
[CrossRef]

Galvez, E. J.

Holmes, C. D.

Leach, J.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

Lesso, J. P.

M. J. Padgett, J. P. Lesso, “Dove prisms and polarized light,” J. Mod. Opt. 46, 175–179 (1999).

Mendoza Santoyo, F.

Moreno, I.

I. Moreno, P. Velasquez, C. R. Fernandez-Pousa, M. M. Sanchez-Lopez, “Jones matrix method for predicting and optimizing the optical modulation properties of a liquid-crystal display,” J. Appl. Phys. 94, 3697–3702 (2003).
[CrossRef]

I. Moreno, G. Paez, M. Strojnik, “Dove prism with increased throughput for implementation in a rotational-shearing interferometer,” Appl. Opt. 42, 4514–4521 (2003).
[CrossRef] [PubMed]

Ohtsuka, Y.

H. Fujii, Y. Ohtsuka, “Rotational filtering for randomly oriented pattern recognition,” Opt. Commun. 36, 255–257 (1981).
[CrossRef]

Padgett, M. J.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

M. J. Padgett, J. P. Lesso, “Dove prisms and polarized light,” J. Mod. Opt. 46, 175–179 (1999).

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, M. J. Padgett, “Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum,” Phys. Rev. Lett. 80, 3217–3219 (1998).
[CrossRef]

Paez, G.

Pedrini, G.

Perez-Lopez, C.

Robertson, D. A.

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, M. J. Padgett, “Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum,” Phys. Rev. Lett. 80, 3217–3219 (1998).
[CrossRef]

Sanchez-Lopez, M. M.

I. Moreno, P. Velasquez, C. R. Fernandez-Pousa, M. M. Sanchez-Lopez, “Jones matrix method for predicting and optimizing the optical modulation properties of a liquid-crystal display,” J. Appl. Phys. 94, 3697–3702 (2003).
[CrossRef]

Schedin, S.

Shurcliff, W. A.

W. A. Shurcliff, Polarized Light (Harvard University, Cambridge, Mass., 1957).

Smith, A. V.

Strojnik, M.

I. Moreno, G. Paez, M. Strojnik, “Dove prism with increased throughput for implementation in a rotational-shearing interferometer,” Appl. Opt. 42, 4514–4521 (2003).
[CrossRef] [PubMed]

M. Strojnik, G. Paez, “Radiometry,” in Handbook of Optical Engineering, D. Malacara, B. Thompson, eds. (Marcel Dekker, New York, 2001), Chap. 18, pp. 649–699.

Strojnik Scholl, M.

Sullivan, D. L.

Takacs, P. Z.

Tiziani, H. J.

Velasquez, P.

I. Moreno, P. Velasquez, C. R. Fernandez-Pousa, M. M. Sanchez-Lopez, “Jones matrix method for predicting and optimizing the optical modulation properties of a liquid-crystal display,” J. Appl. Phys. 94, 3697–3702 (2003).
[CrossRef]

Waluschka, E.

E. Waluschka, “Polarization ray trace,” Opt. Eng. 28, 86–89 (1989).
[CrossRef]

Wolfe, W. L.

W. L. Wolfe, “Nondispersive prisms,” in Handbook of Optics, 2nd ed., M. Bass, E. W. Van Stryland, D. R. Williams, W. L. Wolfe, eds. (McGraw-Hill, New York, 1995), pp. 4.1–4.29.

Yoder, P. R.

P. R. Yoder, Design and Mounting of Prisms and Small Mirrors in Optical Instruments (SPIE Press, Bellingham, Wash., 1998), Vol. TT32, pp. 19–57.

Appl. Opt.

J. Appl. Phys.

I. Moreno, P. Velasquez, C. R. Fernandez-Pousa, M. M. Sanchez-Lopez, “Jones matrix method for predicting and optimizing the optical modulation properties of a liquid-crystal display,” J. Appl. Phys. 94, 3697–3702 (2003).
[CrossRef]

J. Mod. Opt.

M. J. Padgett, J. P. Lesso, “Dove prisms and polarized light,” J. Mod. Opt. 46, 175–179 (1999).

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Commun.

H. Fujii, Y. Ohtsuka, “Rotational filtering for randomly oriented pattern recognition,” Opt. Commun. 36, 255–257 (1981).
[CrossRef]

Opt. Eng.

R. A. Chipman, “Mechanics of polarization ray tracing,” Opt. Eng. 34, 1636–1645 (1995).
[CrossRef]

E. Waluschka, “Polarization ray trace,” Opt. Eng. 28, 86–89 (1989).
[CrossRef]

Opt. Lett.

Phys. Rev. Lett.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, M. J. Padgett, “Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum,” Phys. Rev. Lett. 80, 3217–3219 (1998).
[CrossRef]

Other

W. L. Wolfe, “Nondispersive prisms,” in Handbook of Optics, 2nd ed., M. Bass, E. W. Van Stryland, D. R. Williams, W. L. Wolfe, eds. (McGraw-Hill, New York, 1995), pp. 4.1–4.29.

P. R. Yoder, Design and Mounting of Prisms and Small Mirrors in Optical Instruments (SPIE Press, Bellingham, Wash., 1998), Vol. TT32, pp. 19–57.

M. Strojnik, G. Paez, “Radiometry,” in Handbook of Optical Engineering, D. Malacara, B. Thompson, eds. (Marcel Dekker, New York, 2001), Chap. 18, pp. 649–699.

W. A. Shurcliff, Polarized Light (Harvard University, Cambridge, Mass., 1957).

J. M. Bennett, “Polarization,” in Handbook of Optics, 2nd ed., M. Bass, E. W. Van Stryland, D. R. Williams, W. L. Wolfe, eds. (McGraw-Hill, New York, 1995), p. 5.1.

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Figures (6)

Fig. 1
Fig. 1

Arbitrary propagation of polarized light through a simple rotational prism (Dove prism). The ray path is given by unit vectors k in, k 1, k 2, and k out.

Fig. 2
Fig. 2

Calculated polarization states at the output of a rotated Dove prism. The simulated prism is a Dove prism designed with a base angle of α = 45° and optical glass BK7 (n = 1.515 for λ = 632.8 nm). (a) Dove prism rotated by angle φ with linearly polarized incident light. (b) State of polarization at the output of a Dove prism for different image-rotation angles 2φ. (The Dove prism is rotated by φ.) The output state of polarization is mildly elliptically polarized.

Fig. 3
Fig. 3

Calculated polarization states at the output of a rotated Delta prism. The simulated prism is a Delta prism designed with an apex angle of 2θ = 60° and optical glass SF11 (n = 1.785 for λ = 546 nm) and n M = 0.82–i5.99 (aluminum film at λ = 546 nm). (a) Delta prism rotated by angle φ with linearly polarized incident light. (b) State of polarization at the output of a Delta prism for different image-rotation angles 2φ. (The Delta prism is rotated by φ.) The output state of polarization is elliptically polarized.

Fig. 4
Fig. 4

Calculated polarization states at the output of a rotated Reversion prism. The simulated prism is a Reversion prism designed with optical glass BK7 (n = 1.519 for λ = 546 nm) with n M = 0.82 - i5.99 (aluminum film at λ = 546 nm) and a typical optical cement of n c = 1.53. (a) Reversion prism rotated by an angle φ with linearly polarized light incident. (b) State of polarization at the output of the Reversion prism for different image-rotation angles 2φ. (The Reversion prism is rotated by φ.) The output state of polarization is strongly elliptically polarized.

Fig. 5
Fig. 5

Calculated polarization states at the output of a rotated Pechan prism. The simulated prism is a Pechan prism designed with optical glass BK7 (n = 1.515 for λ = 632.8 nm) and n M = 1.44 + i5.23 (aluminum film at λ = 632.8 nm). (a) Pechan prism rotated by angle φ with linearly polarized incident light. (b) State of polarization at the output of a Pechan prism for different image-rotation angles 2φ. (The Pechan prism is rotated by φ.) The output state of polarization is linearly polarized, and the polarization plane is rotated by φ.

Fig. 6
Fig. 6

Comparison between the theoretical and the experimental results for incidances I 0, I 1, I 2, and I 3 at the output of the Dove prism as a function of the prism angle of rotation φ. These are the incidance measurements of the beam transmitted through the filters corresponding to the Stokes parameters. (a) The first filter is isotropic, passing all states equally and transmitting light with incidance I 0. (b) The second filter is a linear polarizer with the horizontal transmission axis (i.e., along the y direction) transmitting I 1. (c) The third is a linear polarizer with the transmission axis at +45°, transmitting I 2. (d) The fourth filter is a circular polarizer opaque to left-circularly polarized light, transmitting I 3.

Equations (33)

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Eoutφ=τ2rn,, r2r1τ1Ein,
τ1=1|kin×nin|×t1kin×nin · s-t1kin×nin · pt1kin×nin · pt1kin×nin · s,
rm=1|km×nmkm×nm-1|rmkm×nm · km×nm-1-rmkm×nm · nm-1rmkm×nm · nm-1rmkm×nm · km×nm-1,
τ2=1|kn+1×noutkn+1×nn|t2kn+1×nout · kn+1×nn-t2kn+1×nout · nnt2kn+1×nout · nnt2kn+1×nout · kn+1×nn,
Tφ=Bi,jTN,, T1,
Bi,j=cos φ-sin φsin φcos φ.
Tφ=-T cos2 φ-T sin2 φT-Tcos φ sin φT-Tcos φ sin φ-T sin2 φ-T cos2 φ,
T=t1t2, , t2Nr1r2, , rn,
T=t1t2, , t2Nr1r2, , rn,
Eoutφ=TφEin.
Tφ=Bi,jτ2r1τ1.
Tφ=-t1t2r1 cos2 φ-t1t2r1 sin2 φt1t2r1-t1t2r1cos φ sin φt1t2r1-t1t2r1cos φ sin φ-t1t2r1 sin2 φ-t1t2r1 cos2 φ,
T=t1t2r1=4n2 sin αn2-cos2 α1/2n2 sin α+n2-cos2 α1/22 ×cosα+α-inn2 sin2α+α-11/2cosα+α+inn2 sin2α+α-11/2,
T=t1t2r1=4 sin αn2-cos2 α1/2sin α+n2-cos2 α1/22 ×n cosα+α-in2 sin2α+α-11/2n cosα+α+in2 sin2α+α-11/2,
α=arcsincos αn.
Tφ=Bi,jτ2r3r2r1τ1.
Tφ=-t1t2r1r2r3 cos2 φ-t1t2r1r2r3 sin2 φt1t2r1r2r3-t1t2r1r2r3cos φ sin φt1t2r1r2r3-t1t2r1r2r3cos φ sin φ-t1t2r1r2r3 sin2 φ-t1t2r1r2r3 cos2 φ,
T=t1t2r1r2r3=4n2 cos θn2-sin2 θ1/2n2 cos θ+n2-sin2 θ1/22 ×cos2θ-θ-inn2 sin22θ-θ-11/2cos2θ-θ+inn2 sin22θ-θ-11/22×nM2 sin3θ-θ-nnM2-n2 cos23θ-θ1/2nM2sin3θ-θ+nnM2-n2 cos23θ-θ1/2,
T=t1t2r1r2r3=4 cos θn2-sin2 θ1/2cos θ+n2-sin2 θ1/22 ×n cos2θ-θ-in2 sin22θ-θ-11/2n cos2θ-θ+in2 sin22θ-θ-11/22×n sin3θ-θ-nM2-n2 cos23θ-θ1/2n sin3θ-θ+nM2-n2 cos23θ-θ1/2,
θ=arcsinsin θn,
Tφ=Bi,jτ4r3τ3τ2r2r1τ1.
Tφ=-t1t2t3t4r1r2r3 cos2 φ+r1r2r3 sin2 φt1t2t3t4r1r2r3-r1r2r3cos φ sin φt1t2t3t4r1r2r3-r1r2r3cos φ sin φ-t1t2t3t4r1r2r3 sin2 φ+r1r2r3 cos2 φ,
T=t1t2t3t4r1r2r3=2nn+122nnc1/2n+nc21-in3n2-41/21+in3n2-41/22×3nM2-n4nM2-n21/23nM2+n4nM2-n21/2,
T=t1t2t3t4r1r2r3=2nn+122nnc1/2n+nc2n-i3n2-41/2n+i3n2-41/22×3n-4nM2-n21/23n+4nM2-n21/2,
Tφ=Bi,jτ4r5r4r3τ3τ2r2r1τ1.
Tφ=t1t2t3t4r1r2r3r4r5 cos2 φ+r1r2r3r4r5 sin2 φ-t1t2t3t4r1r2r3r4r5-r1r2r3r4r5cos φ sin φ-t1t2t3t4r1r2r3r4r5-r1r2r3r4r5cos φ sin φt1t2t3t4r1r2r3r4r5 sin2 φ+r1r2r3r4r5 cos2 φ,
T=t1t2t3t4r1r2r3r4r5=2nn+141-inn2-21/21+inn2-23×nM2 cos π8-nnM2-n2 sin2 π81/2nM2 cos π8+nnM2-n2 sin2 π81/22,
T=t1t2t3t4r1r2r3r4r5=2nn+14n-in2-21/2n+in2-21/23×nM cosπ8-nM2-n2 sin2π81/2nM cosπ8+nM2-n2 sin2π81/22,
Tφ=0.9244-0.3568icos2 φ+0.6248-0.6539isin2 φ 0.2996+0.2971icos φ sin φ0.2996+0.2971icos φ sin φ 0.9244-0.3568isin2 φ+0.6248-0.6539icos2 φ.
I0φ=|Eoutφ|2|Ein|2,
I1φ=|JyEoutφ|2|Ein|2,
I2φ=|J+45Eoutφ|2|Ein|2,
I3φ=|J+45Jλ/4Eoutφ|2|Ein|2,

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