Abstract

We describe procedures for constructing inexpensive wave plates of desired retardation out of ordinary commercially available transparencies. Various relevant properties of the transparencies are investigated: the dependence of retardation on rotation of the film, tilt, wavelength, position, and temperature. A transparency is typically a multiple-order wave plate with the difference of in-plane refractive indices of 0.07 and a temperature dependence of retardation 0.02rad/K. Constructing wave plates out of combinations of transparency sheets is also explored.

© 2007 Optical Society of America

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References

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  1. E. Hecht and A. Zajac, Optics, 4th ed. (Addison-Wesley, 1979).
  2. G. D. Walter and V. William, Handbook of Optics Sponsored by OSA (McGraw-Hill, 1978).
  3. Melles Griot catalogue (1997-1998).
  4. R. P. Feynman, The Feynman Lectures on Physics, 6th ed. (Addison-Wesley, 1977), Vol. I.
  5. M. Ortiz-Gutiérrez, A. Olivares-Peréz, J. L. Juárez-Pérez, and V. Sánchez-Villicaña, "3M PP2500TM film as quarter wave retarder for light at λ = 632.8 nm," Opt. Mater. 14, 41-48 (2000).
    [CrossRef]
  6. M. Ortiz-Gutiérrez, M. A. Salgado V., A. M. Martínez-Basurto, A. Olivares-Pérez, J. L. Juárez-Pérez, M. Pérez-Cortés, and J. C. Ibarra-Torres, "Refraction indexes of the polyester film 3M PP2500TM," Proc. SPIE 5363, 137-140 (2004).
    [CrossRef]
  7. J. C. Allred, R. N. Lyman, T. W. Kornack, and M. V. Romalis, "High-sensitivity atomic magnetometer unaffected by spin-exchange relaxation," Phys. Rev. Lett. 89, 130801 (2002).
    [CrossRef] [PubMed]
  8. J. F. Elman, J. Greener, C. M. Herzinger, and B. Johs, "Characterization of biaxially-stretched plastic films by generalized ellipsometry," Thin Solid Films 313-314, 814-818 (1998).
    [CrossRef]
  9. V. Ratta, G. L. Wilkes, and T. K. Su, "Structure-property-processing investigations of the tenter-frame process for making biaxially oriented HDPE film. I. Base sheet and draw along the MD," Polymer 42, 9059-9071 (2001).
    [CrossRef]
  10. R. S. Stein, H. H. Winter, J. Müller, and M. Srinivasarao, "Optical properties of polymers and liquid crystals in electrical, magnetic, and hydrodynamic fields," Pure Appl. Chem. 67, 1971-1982 (1995).
    [CrossRef]
  11. M. Born and E. Wolf, Principles of Optics (Pergamon, 1993).
  12. P. Velasquez, M. del Mar Sánchez-López, I. Moreno, D. Puerto, and F. Mateos, "Interference birefringent filters fabricated with low cost commercial polymers," Am. J. Phys. 73, 357-361 (2005).
    [CrossRef]
  13. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
    [CrossRef]
  14. J. C. Martínez-Antón and E. Bernabeu, "Spectrogoniometry and the wanted method for refractive index determination," Thin Solid Films 313-314, 85-89 (1998).
    [CrossRef]
  15. G. W. C. Kaye and T. H. Laby, Tables of Physical and Chemical Constants, 16th ed. (Longman, 1995).
  16. H. Xia, A. Ben-Amar Baranga, D. Hoffman, and M. V. Romalis, "Magnetoencephalography with an atomic magnetometer," Appl. Phys. Lett. 89, 211104 (2006).
    [CrossRef]
  17. D. Budker, D. F. Kimbal, and D. P. DeMille, Atomic Physics (Oxford U. Press, 2004).
  18. B. Charm, E. Babcock, L. W. Anderson, and T. G. Walker, "Skew light propagation in optically thick optical pumping cells," Phys. Rev. A 66, 033406 (2002).

2006 (1)

H. Xia, A. Ben-Amar Baranga, D. Hoffman, and M. V. Romalis, "Magnetoencephalography with an atomic magnetometer," Appl. Phys. Lett. 89, 211104 (2006).
[CrossRef]

2005 (1)

P. Velasquez, M. del Mar Sánchez-López, I. Moreno, D. Puerto, and F. Mateos, "Interference birefringent filters fabricated with low cost commercial polymers," Am. J. Phys. 73, 357-361 (2005).
[CrossRef]

2004 (2)

M. Ortiz-Gutiérrez, M. A. Salgado V., A. M. Martínez-Basurto, A. Olivares-Pérez, J. L. Juárez-Pérez, M. Pérez-Cortés, and J. C. Ibarra-Torres, "Refraction indexes of the polyester film 3M PP2500TM," Proc. SPIE 5363, 137-140 (2004).
[CrossRef]

D. Budker, D. F. Kimbal, and D. P. DeMille, Atomic Physics (Oxford U. Press, 2004).

2002 (2)

B. Charm, E. Babcock, L. W. Anderson, and T. G. Walker, "Skew light propagation in optically thick optical pumping cells," Phys. Rev. A 66, 033406 (2002).

J. C. Allred, R. N. Lyman, T. W. Kornack, and M. V. Romalis, "High-sensitivity atomic magnetometer unaffected by spin-exchange relaxation," Phys. Rev. Lett. 89, 130801 (2002).
[CrossRef] [PubMed]

2001 (1)

V. Ratta, G. L. Wilkes, and T. K. Su, "Structure-property-processing investigations of the tenter-frame process for making biaxially oriented HDPE film. I. Base sheet and draw along the MD," Polymer 42, 9059-9071 (2001).
[CrossRef]

2000 (1)

M. Ortiz-Gutiérrez, A. Olivares-Peréz, J. L. Juárez-Pérez, and V. Sánchez-Villicaña, "3M PP2500TM film as quarter wave retarder for light at λ = 632.8 nm," Opt. Mater. 14, 41-48 (2000).
[CrossRef]

1998 (2)

J. C. Martínez-Antón and E. Bernabeu, "Spectrogoniometry and the wanted method for refractive index determination," Thin Solid Films 313-314, 85-89 (1998).
[CrossRef]

J. F. Elman, J. Greener, C. M. Herzinger, and B. Johs, "Characterization of biaxially-stretched plastic films by generalized ellipsometry," Thin Solid Films 313-314, 814-818 (1998).
[CrossRef]

1995 (2)

R. S. Stein, H. H. Winter, J. Müller, and M. Srinivasarao, "Optical properties of polymers and liquid crystals in electrical, magnetic, and hydrodynamic fields," Pure Appl. Chem. 67, 1971-1982 (1995).
[CrossRef]

G. W. C. Kaye and T. H. Laby, Tables of Physical and Chemical Constants, 16th ed. (Longman, 1995).

1993 (1)

M. Born and E. Wolf, Principles of Optics (Pergamon, 1993).

1991 (1)

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
[CrossRef]

1979 (1)

E. Hecht and A. Zajac, Optics, 4th ed. (Addison-Wesley, 1979).

1978 (1)

G. D. Walter and V. William, Handbook of Optics Sponsored by OSA (McGraw-Hill, 1978).

1977 (1)

R. P. Feynman, The Feynman Lectures on Physics, 6th ed. (Addison-Wesley, 1977), Vol. I.

Allred, J. C.

J. C. Allred, R. N. Lyman, T. W. Kornack, and M. V. Romalis, "High-sensitivity atomic magnetometer unaffected by spin-exchange relaxation," Phys. Rev. Lett. 89, 130801 (2002).
[CrossRef] [PubMed]

Anderson, L. W.

B. Charm, E. Babcock, L. W. Anderson, and T. G. Walker, "Skew light propagation in optically thick optical pumping cells," Phys. Rev. A 66, 033406 (2002).

Babcock, E.

B. Charm, E. Babcock, L. W. Anderson, and T. G. Walker, "Skew light propagation in optically thick optical pumping cells," Phys. Rev. A 66, 033406 (2002).

Baranga, A. Ben-Amar

H. Xia, A. Ben-Amar Baranga, D. Hoffman, and M. V. Romalis, "Magnetoencephalography with an atomic magnetometer," Appl. Phys. Lett. 89, 211104 (2006).
[CrossRef]

Bernabeu, E.

J. C. Martínez-Antón and E. Bernabeu, "Spectrogoniometry and the wanted method for refractive index determination," Thin Solid Films 313-314, 85-89 (1998).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1993).

Budker, D.

D. Budker, D. F. Kimbal, and D. P. DeMille, Atomic Physics (Oxford U. Press, 2004).

Charm, B.

B. Charm, E. Babcock, L. W. Anderson, and T. G. Walker, "Skew light propagation in optically thick optical pumping cells," Phys. Rev. A 66, 033406 (2002).

del Mar Sánchez-López, M.

P. Velasquez, M. del Mar Sánchez-López, I. Moreno, D. Puerto, and F. Mateos, "Interference birefringent filters fabricated with low cost commercial polymers," Am. J. Phys. 73, 357-361 (2005).
[CrossRef]

DeMille, D. P.

D. Budker, D. F. Kimbal, and D. P. DeMille, Atomic Physics (Oxford U. Press, 2004).

Elman, J. F.

J. F. Elman, J. Greener, C. M. Herzinger, and B. Johs, "Characterization of biaxially-stretched plastic films by generalized ellipsometry," Thin Solid Films 313-314, 814-818 (1998).
[CrossRef]

Feynman, R. P.

R. P. Feynman, The Feynman Lectures on Physics, 6th ed. (Addison-Wesley, 1977), Vol. I.

Greener, J.

J. F. Elman, J. Greener, C. M. Herzinger, and B. Johs, "Characterization of biaxially-stretched plastic films by generalized ellipsometry," Thin Solid Films 313-314, 814-818 (1998).
[CrossRef]

Hecht, E.

E. Hecht and A. Zajac, Optics, 4th ed. (Addison-Wesley, 1979).

Herzinger, C. M.

J. F. Elman, J. Greener, C. M. Herzinger, and B. Johs, "Characterization of biaxially-stretched plastic films by generalized ellipsometry," Thin Solid Films 313-314, 814-818 (1998).
[CrossRef]

Hoffman, D.

H. Xia, A. Ben-Amar Baranga, D. Hoffman, and M. V. Romalis, "Magnetoencephalography with an atomic magnetometer," Appl. Phys. Lett. 89, 211104 (2006).
[CrossRef]

Ibarra-Torres, J. C.

M. Ortiz-Gutiérrez, M. A. Salgado V., A. M. Martínez-Basurto, A. Olivares-Pérez, J. L. Juárez-Pérez, M. Pérez-Cortés, and J. C. Ibarra-Torres, "Refraction indexes of the polyester film 3M PP2500TM," Proc. SPIE 5363, 137-140 (2004).
[CrossRef]

Johs, B.

J. F. Elman, J. Greener, C. M. Herzinger, and B. Johs, "Characterization of biaxially-stretched plastic films by generalized ellipsometry," Thin Solid Films 313-314, 814-818 (1998).
[CrossRef]

Juárez-Pérez, J. L.

M. Ortiz-Gutiérrez, M. A. Salgado V., A. M. Martínez-Basurto, A. Olivares-Pérez, J. L. Juárez-Pérez, M. Pérez-Cortés, and J. C. Ibarra-Torres, "Refraction indexes of the polyester film 3M PP2500TM," Proc. SPIE 5363, 137-140 (2004).
[CrossRef]

M. Ortiz-Gutiérrez, A. Olivares-Peréz, J. L. Juárez-Pérez, and V. Sánchez-Villicaña, "3M PP2500TM film as quarter wave retarder for light at λ = 632.8 nm," Opt. Mater. 14, 41-48 (2000).
[CrossRef]

Kaye, G. W. C.

G. W. C. Kaye and T. H. Laby, Tables of Physical and Chemical Constants, 16th ed. (Longman, 1995).

Kimbal, D. F.

D. Budker, D. F. Kimbal, and D. P. DeMille, Atomic Physics (Oxford U. Press, 2004).

Kornack, T. W.

J. C. Allred, R. N. Lyman, T. W. Kornack, and M. V. Romalis, "High-sensitivity atomic magnetometer unaffected by spin-exchange relaxation," Phys. Rev. Lett. 89, 130801 (2002).
[CrossRef] [PubMed]

Laby, T. H.

G. W. C. Kaye and T. H. Laby, Tables of Physical and Chemical Constants, 16th ed. (Longman, 1995).

Lyman, R. N.

J. C. Allred, R. N. Lyman, T. W. Kornack, and M. V. Romalis, "High-sensitivity atomic magnetometer unaffected by spin-exchange relaxation," Phys. Rev. Lett. 89, 130801 (2002).
[CrossRef] [PubMed]

Martínez-Antón, J. C.

J. C. Martínez-Antón and E. Bernabeu, "Spectrogoniometry and the wanted method for refractive index determination," Thin Solid Films 313-314, 85-89 (1998).
[CrossRef]

Martínez-Basurto, A. M.

M. Ortiz-Gutiérrez, M. A. Salgado V., A. M. Martínez-Basurto, A. Olivares-Pérez, J. L. Juárez-Pérez, M. Pérez-Cortés, and J. C. Ibarra-Torres, "Refraction indexes of the polyester film 3M PP2500TM," Proc. SPIE 5363, 137-140 (2004).
[CrossRef]

Mateos, F.

P. Velasquez, M. del Mar Sánchez-López, I. Moreno, D. Puerto, and F. Mateos, "Interference birefringent filters fabricated with low cost commercial polymers," Am. J. Phys. 73, 357-361 (2005).
[CrossRef]

Moreno, I.

P. Velasquez, M. del Mar Sánchez-López, I. Moreno, D. Puerto, and F. Mateos, "Interference birefringent filters fabricated with low cost commercial polymers," Am. J. Phys. 73, 357-361 (2005).
[CrossRef]

Müller, J.

R. S. Stein, H. H. Winter, J. Müller, and M. Srinivasarao, "Optical properties of polymers and liquid crystals in electrical, magnetic, and hydrodynamic fields," Pure Appl. Chem. 67, 1971-1982 (1995).
[CrossRef]

Olivares-Peréz, A.

M. Ortiz-Gutiérrez, A. Olivares-Peréz, J. L. Juárez-Pérez, and V. Sánchez-Villicaña, "3M PP2500TM film as quarter wave retarder for light at λ = 632.8 nm," Opt. Mater. 14, 41-48 (2000).
[CrossRef]

Olivares-Pérez, A.

M. Ortiz-Gutiérrez, M. A. Salgado V., A. M. Martínez-Basurto, A. Olivares-Pérez, J. L. Juárez-Pérez, M. Pérez-Cortés, and J. C. Ibarra-Torres, "Refraction indexes of the polyester film 3M PP2500TM," Proc. SPIE 5363, 137-140 (2004).
[CrossRef]

Ortiz-Gutiérrez, M.

M. Ortiz-Gutiérrez, M. A. Salgado V., A. M. Martínez-Basurto, A. Olivares-Pérez, J. L. Juárez-Pérez, M. Pérez-Cortés, and J. C. Ibarra-Torres, "Refraction indexes of the polyester film 3M PP2500TM," Proc. SPIE 5363, 137-140 (2004).
[CrossRef]

M. Ortiz-Gutiérrez, A. Olivares-Peréz, J. L. Juárez-Pérez, and V. Sánchez-Villicaña, "3M PP2500TM film as quarter wave retarder for light at λ = 632.8 nm," Opt. Mater. 14, 41-48 (2000).
[CrossRef]

Pérez-Cortés, M.

M. Ortiz-Gutiérrez, M. A. Salgado V., A. M. Martínez-Basurto, A. Olivares-Pérez, J. L. Juárez-Pérez, M. Pérez-Cortés, and J. C. Ibarra-Torres, "Refraction indexes of the polyester film 3M PP2500TM," Proc. SPIE 5363, 137-140 (2004).
[CrossRef]

Puerto, D.

P. Velasquez, M. del Mar Sánchez-López, I. Moreno, D. Puerto, and F. Mateos, "Interference birefringent filters fabricated with low cost commercial polymers," Am. J. Phys. 73, 357-361 (2005).
[CrossRef]

Ratta, V.

V. Ratta, G. L. Wilkes, and T. K. Su, "Structure-property-processing investigations of the tenter-frame process for making biaxially oriented HDPE film. I. Base sheet and draw along the MD," Polymer 42, 9059-9071 (2001).
[CrossRef]

Romalis, M. V.

H. Xia, A. Ben-Amar Baranga, D. Hoffman, and M. V. Romalis, "Magnetoencephalography with an atomic magnetometer," Appl. Phys. Lett. 89, 211104 (2006).
[CrossRef]

J. C. Allred, R. N. Lyman, T. W. Kornack, and M. V. Romalis, "High-sensitivity atomic magnetometer unaffected by spin-exchange relaxation," Phys. Rev. Lett. 89, 130801 (2002).
[CrossRef] [PubMed]

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
[CrossRef]

Sánchez-Villicaña, V.

M. Ortiz-Gutiérrez, A. Olivares-Peréz, J. L. Juárez-Pérez, and V. Sánchez-Villicaña, "3M PP2500TM film as quarter wave retarder for light at λ = 632.8 nm," Opt. Mater. 14, 41-48 (2000).
[CrossRef]

Srinivasarao, M.

R. S. Stein, H. H. Winter, J. Müller, and M. Srinivasarao, "Optical properties of polymers and liquid crystals in electrical, magnetic, and hydrodynamic fields," Pure Appl. Chem. 67, 1971-1982 (1995).
[CrossRef]

Stein, R. S.

R. S. Stein, H. H. Winter, J. Müller, and M. Srinivasarao, "Optical properties of polymers and liquid crystals in electrical, magnetic, and hydrodynamic fields," Pure Appl. Chem. 67, 1971-1982 (1995).
[CrossRef]

Su, T. K.

V. Ratta, G. L. Wilkes, and T. K. Su, "Structure-property-processing investigations of the tenter-frame process for making biaxially oriented HDPE film. I. Base sheet and draw along the MD," Polymer 42, 9059-9071 (2001).
[CrossRef]

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
[CrossRef]

V., M. A. Salgado

M. Ortiz-Gutiérrez, M. A. Salgado V., A. M. Martínez-Basurto, A. Olivares-Pérez, J. L. Juárez-Pérez, M. Pérez-Cortés, and J. C. Ibarra-Torres, "Refraction indexes of the polyester film 3M PP2500TM," Proc. SPIE 5363, 137-140 (2004).
[CrossRef]

Velasquez, P.

P. Velasquez, M. del Mar Sánchez-López, I. Moreno, D. Puerto, and F. Mateos, "Interference birefringent filters fabricated with low cost commercial polymers," Am. J. Phys. 73, 357-361 (2005).
[CrossRef]

Walker, T. G.

B. Charm, E. Babcock, L. W. Anderson, and T. G. Walker, "Skew light propagation in optically thick optical pumping cells," Phys. Rev. A 66, 033406 (2002).

Walter, G. D.

G. D. Walter and V. William, Handbook of Optics Sponsored by OSA (McGraw-Hill, 1978).

Wilkes, G. L.

V. Ratta, G. L. Wilkes, and T. K. Su, "Structure-property-processing investigations of the tenter-frame process for making biaxially oriented HDPE film. I. Base sheet and draw along the MD," Polymer 42, 9059-9071 (2001).
[CrossRef]

William, V.

G. D. Walter and V. William, Handbook of Optics Sponsored by OSA (McGraw-Hill, 1978).

Winter, H. H.

R. S. Stein, H. H. Winter, J. Müller, and M. Srinivasarao, "Optical properties of polymers and liquid crystals in electrical, magnetic, and hydrodynamic fields," Pure Appl. Chem. 67, 1971-1982 (1995).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1993).

Xia, H.

H. Xia, A. Ben-Amar Baranga, D. Hoffman, and M. V. Romalis, "Magnetoencephalography with an atomic magnetometer," Appl. Phys. Lett. 89, 211104 (2006).
[CrossRef]

Zajac, A.

E. Hecht and A. Zajac, Optics, 4th ed. (Addison-Wesley, 1979).

Am. J. Phys. (1)

P. Velasquez, M. del Mar Sánchez-López, I. Moreno, D. Puerto, and F. Mateos, "Interference birefringent filters fabricated with low cost commercial polymers," Am. J. Phys. 73, 357-361 (2005).
[CrossRef]

Appl. Phys. Lett. (1)

H. Xia, A. Ben-Amar Baranga, D. Hoffman, and M. V. Romalis, "Magnetoencephalography with an atomic magnetometer," Appl. Phys. Lett. 89, 211104 (2006).
[CrossRef]

Opt. Mater. (1)

M. Ortiz-Gutiérrez, A. Olivares-Peréz, J. L. Juárez-Pérez, and V. Sánchez-Villicaña, "3M PP2500TM film as quarter wave retarder for light at λ = 632.8 nm," Opt. Mater. 14, 41-48 (2000).
[CrossRef]

Phys. Rev. A (1)

B. Charm, E. Babcock, L. W. Anderson, and T. G. Walker, "Skew light propagation in optically thick optical pumping cells," Phys. Rev. A 66, 033406 (2002).

Phys. Rev. Lett. (1)

J. C. Allred, R. N. Lyman, T. W. Kornack, and M. V. Romalis, "High-sensitivity atomic magnetometer unaffected by spin-exchange relaxation," Phys. Rev. Lett. 89, 130801 (2002).
[CrossRef] [PubMed]

Polymer (1)

V. Ratta, G. L. Wilkes, and T. K. Su, "Structure-property-processing investigations of the tenter-frame process for making biaxially oriented HDPE film. I. Base sheet and draw along the MD," Polymer 42, 9059-9071 (2001).
[CrossRef]

Proc. SPIE (1)

M. Ortiz-Gutiérrez, M. A. Salgado V., A. M. Martínez-Basurto, A. Olivares-Pérez, J. L. Juárez-Pérez, M. Pérez-Cortés, and J. C. Ibarra-Torres, "Refraction indexes of the polyester film 3M PP2500TM," Proc. SPIE 5363, 137-140 (2004).
[CrossRef]

Pure Appl. Chem. (1)

R. S. Stein, H. H. Winter, J. Müller, and M. Srinivasarao, "Optical properties of polymers and liquid crystals in electrical, magnetic, and hydrodynamic fields," Pure Appl. Chem. 67, 1971-1982 (1995).
[CrossRef]

Thin Solid Films (2)

J. C. Martínez-Antón and E. Bernabeu, "Spectrogoniometry and the wanted method for refractive index determination," Thin Solid Films 313-314, 85-89 (1998).
[CrossRef]

J. F. Elman, J. Greener, C. M. Herzinger, and B. Johs, "Characterization of biaxially-stretched plastic films by generalized ellipsometry," Thin Solid Films 313-314, 814-818 (1998).
[CrossRef]

Other (8)

E. Hecht and A. Zajac, Optics, 4th ed. (Addison-Wesley, 1979).

G. D. Walter and V. William, Handbook of Optics Sponsored by OSA (McGraw-Hill, 1978).

Melles Griot catalogue (1997-1998).

R. P. Feynman, The Feynman Lectures on Physics, 6th ed. (Addison-Wesley, 1977), Vol. I.

G. W. C. Kaye and T. H. Laby, Tables of Physical and Chemical Constants, 16th ed. (Longman, 1995).

D. Budker, D. F. Kimbal, and D. P. DeMille, Atomic Physics (Oxford U. Press, 2004).

M. Born and E. Wolf, Principles of Optics (Pergamon, 1993).

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Price of wave plates versus size. The data are from a CASIX catalog. Price grows linearly with the square of the size of the wave plate.

Fig. 2
Fig. 2

On the left: Definition of the coordinate system of a transparency wave plate. The principal axes x and y of the index ellipsoid, shown on the right, lie in the plane of transparency film, while the z axis is perpendicular to this plane. The “rotation” is defined as the rotation around the z axis through angle θ. The tilt is defined as the rotation around the tilt axis oriented at some angle α with respect to x in the transparency plane. On the right: The index ellipsoid and its intersection with the plane perpendicular to a wave normal k. The intersection is an ellipse whose half-axes n a and n b are responsible for the retardation produced by the wave plate. The tilt around y (the case when α = 90 ° ) is illustrated.

Fig. 3
Fig. 3

Ellipticity as a function of transparency rotation. (HP color LaserJet transparency C2934A.) The data shown with squares agree well with a fit using Eq. (3) (solid curve), in which δ = 0.91 rad and an arbitrary offset in the rotation angle is allowed. Some aperiodicity can be observed due to the fact that under rotation slightly different regions of the transparency are probed. The rotation of the transparency is the most convenient way to adjust ellipticity to a desirable level between maximal and minimal values.

Fig. 4
Fig. 4

Ellipticity as a function of transparency translation. (HP color LaserJet transparency C2934A.) The size of probing laser beam is approximately 2 m m . In the areas where the spatial variation of retardation is significant across the cross-section of the light beam, the variation of the polarization state of the resultant light beam may lead to effective depolarization of the beam. The data exhibit smooth behavior with a typical continuous variation of 5° per 10 m m and ellipticity changing slowly in a large range. From applications point of view, this means that wave plates of given retardation and of the size 10 30 m m (depending on retardation tolerance) can be made by selecting an appropriate region. This behavior is similar for different transparencies.

Fig. 5
Fig. 5

(Color online) Large-scale retardation uniformity of a transparency sheet (HP color LaserJet transparency C2934A) in two dimensions. The transparency is placed between a 17 in. thin-film transistor–liquid crystal display (TFT-LCD) computer monitor, source of polarized light with polarization at 45° with respect to vertical, and a square polaroid film of similar size oriented to extinguish light in the absence of the transparency. The transparency has its edges aligned with the screen. The color segments are generated in the computer by drawing filled rectangles in a graphics program. The display colors have been chosen as the standard red∕green∕blue set (note that the colors may appear altered as an artifact of digital photography and background light). Three “pure” color quadrants provide information on uniformity at different wavelengths. The fourth white quadrant shows that the superposition of nonuniformities for three different colors is spatially averaged. In general it is expected that birefringence variation caused by nonuniformity of stretching should be similar for the three different colors, although somewhat smaller for red.

Fig. 6
Fig. 6

(Color online) Uniformity of almost zero ellipticity across large area of a transparency film (HP color LaserJet transparency C2934A) for a small angle between the fast axis and the polarization axis, meaning that the orientation of the index ellipsoid does not depend on position.

Fig. 7
Fig. 7

Ellipticity as a function of light wavelength for 3M AF4300 Write-On Transparency Film. The data follow Eq. (3) with θ = 45 ° and δ [ r a d ] = 46120 / λ [ n m ] . At 840 n m , the wave-plate retardation is 17.5 π .

Fig. 8
Fig. 8

Refractive index difference n x n y of 3M AF4300 Write-On Transparency Film (circles and dotted curve) and HP color LaserJet transparency C2934A (solid squares and curve) measured with a spectrometer and two crossed polarizers with a transparency in between. The dotted curve is the fit with n x n y = 0.0806 13100 / λ 2 + 3.5 × 10 9 / λ 4 , and the solid curve is the fit with n x n y = 0.05628 5500 / λ 2 + 2.2 × 10 9 / λ 4 .

Fig. 9
Fig. 9

(Color online) Dependence of ellipticity in degrees on the tilt in two directions (first around the vertical axis [Tilt 1] and then around the horizontal axis [Tilt 2]) for 3M AF4300 Write-On Transparency Film. The ellipticity was maximized by rotating the transparency around its normal.

Fig. 10
Fig. 10

Dependence of ellipticity on the tilt angle t of an HP color LaserJet transparency C2934A for the tilt around a principal axis of the refractive index ellipsoid. Solid curve is the fit based on Eqs. (3) and (6).

Fig. 11
Fig. 11

Dependence of retardation of a combination of two wave plates on the angle between their symmetry axes. Points are experimental data for 3M AF4300 Write-On Transparency Film. They allow accurate fit with analytical equation Eq. (8), the solid curve going through the points, with ϕ 2 = 72.3 ° and ϕ 1 = 38.7 ° . Measurements for individual wave plates gave ϕ 2 = 74.6 ° and ϕ 1 = 37 ° , in agreement. The small deviation is due to spatial variations of retardation when transparency is rotated. The analytical equation [Eq. (8)] for retardations ϕ 1 = 23 ° and ϕ 2 = 57 ° , the other solid curve, is also tested by comparison with numerical simulations based on the Jones-matrix formalism, dotted curve.

Fig. 12
Fig. 12

Dependence of retardation of a HP color LaserJet transparency C2934A on temperature at 633 n m . Open circles show a set of data in one run, and the squares show a set of data in the second run after the heating–cooling cycle to check reproducibility of the data. The dependence fits well with a curve, ϕ = 0.50 ( 1 ) + 0.0187 ( 4 ) T , where T is the temperature. Large offset 20 π r a d which can be estimated from the refractive index for the color transparency (Fig. 8) is not included.

Equations (8)

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χ = 1 2 arcsin s 3 = 1 2 arcsin 2 a x a y sin δ a x 2 + a y 2 ,
ψ = 1 2 arctan s 2 s 1 = 1 2 arctan 2 a x a y cos δ a x 2 a y 2 .
χ = 1 2 arcsin [ sin 2 θ sin δ ] ,
ψ = 1 2 arctan [ tan 2 θ cos δ ] .
δ = 2 π Δ n ( λ ) l λ 2 π Δ n ( λ 0 ) l λ 0 2 π Δ n ( λ 0 ) l ( λ λ 0 ) λ 0 2 ,
n a n b = n x n y ( n z n y ) sin ( t x / n ) 2
n a n b = n x n y + ( n z n x ) sin ( t y / n ) 2 ,
ϕ = ( ϕ 1 + ϕ 2 ) cos 2 θ 12 + ( ϕ 1 ϕ 2 ) sin 2 θ 12 .

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