Abstract

Gradient-index lenses are samples whose special characteristics must be taken into account to design the optical polariscopes that can be applied in the evaluation of their birefringence. We discuss the main infidelity sources that modify the conoscopic patterns when a traditional polariscopic setup is used.

© 2002 Optical Society of America

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References

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  1. F. D. Bloss, Introducción a Los Métodos de la Cristalografia óptica, 5th ed. (Ediciones Omega, Barcelona, 1994), pp. 124–128 [Translated from An Introduction to the Methods of Optical Crystallography (Holt, Rinehart & Winston, New York, 1961)].
  2. S. Tolansky, An Introduction to Interferometry, 2nd ed. (Longman, London, 1973), p. 203.
  3. SELFOC Product Catalog 1999 (NSG America, Inc., N.J., 1999), p. 7.
  4. W. A. Wožniak, “Residual birefringence in gradient index lenses,” Opt. Appl. 19, 429–437 (1989).
  5. W. Su, J. A. Gilbert, “Birefringent properties of diametrically loaded gradient-index lenses,” Appl. Opt. 35, 4772–4781 (1986).
    [CrossRef]
  6. D. Tentori, J. Camacho, “Birefringence characterization of one-quarter-pitch GRIN lenses,” Opt. Eng. 41, 2468–2475 (2002).
    [CrossRef]
  7. J. Camacho, D. Tentori, “Polarization optics of GRIN lenses,” J. Opt. A: Pure Appl. Opt. 3, 89–95 (2001).
    [CrossRef]
  8. R. C. Jones, “A new calculus for the treatment of optical systems. I. Description and discussion of the calculus,” J. Opt. Soc. Am. 31, 488–493 (1941).
    [CrossRef]
  9. E. W. Marchand, Gradient Index Optics (Academic, New York, 1978), pp. 53–60.
  10. M. Françon, S. Mallick, Polarization Interferometers: Applications in Microscopy and Macroscopy (Wiley Interscience, London, 1971), pp. 140.

2002 (1)

D. Tentori, J. Camacho, “Birefringence characterization of one-quarter-pitch GRIN lenses,” Opt. Eng. 41, 2468–2475 (2002).
[CrossRef]

2001 (1)

J. Camacho, D. Tentori, “Polarization optics of GRIN lenses,” J. Opt. A: Pure Appl. Opt. 3, 89–95 (2001).
[CrossRef]

1989 (1)

W. A. Wožniak, “Residual birefringence in gradient index lenses,” Opt. Appl. 19, 429–437 (1989).

1986 (1)

1941 (1)

Bloss, F. D.

F. D. Bloss, Introducción a Los Métodos de la Cristalografia óptica, 5th ed. (Ediciones Omega, Barcelona, 1994), pp. 124–128 [Translated from An Introduction to the Methods of Optical Crystallography (Holt, Rinehart & Winston, New York, 1961)].

Camacho, J.

D. Tentori, J. Camacho, “Birefringence characterization of one-quarter-pitch GRIN lenses,” Opt. Eng. 41, 2468–2475 (2002).
[CrossRef]

J. Camacho, D. Tentori, “Polarization optics of GRIN lenses,” J. Opt. A: Pure Appl. Opt. 3, 89–95 (2001).
[CrossRef]

Françon, M.

M. Françon, S. Mallick, Polarization Interferometers: Applications in Microscopy and Macroscopy (Wiley Interscience, London, 1971), pp. 140.

Gilbert, J. A.

Jones, R. C.

Mallick, S.

M. Françon, S. Mallick, Polarization Interferometers: Applications in Microscopy and Macroscopy (Wiley Interscience, London, 1971), pp. 140.

Marchand, E. W.

E. W. Marchand, Gradient Index Optics (Academic, New York, 1978), pp. 53–60.

Su, W.

Tentori, D.

D. Tentori, J. Camacho, “Birefringence characterization of one-quarter-pitch GRIN lenses,” Opt. Eng. 41, 2468–2475 (2002).
[CrossRef]

J. Camacho, D. Tentori, “Polarization optics of GRIN lenses,” J. Opt. A: Pure Appl. Opt. 3, 89–95 (2001).
[CrossRef]

Tolansky, S.

S. Tolansky, An Introduction to Interferometry, 2nd ed. (Longman, London, 1973), p. 203.

Wožniak, W. A.

W. A. Wožniak, “Residual birefringence in gradient index lenses,” Opt. Appl. 19, 429–437 (1989).

Appl. Opt. (1)

J. Opt. A: Pure Appl. Opt. (1)

J. Camacho, D. Tentori, “Polarization optics of GRIN lenses,” J. Opt. A: Pure Appl. Opt. 3, 89–95 (2001).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Appl. (1)

W. A. Wožniak, “Residual birefringence in gradient index lenses,” Opt. Appl. 19, 429–437 (1989).

Opt. Eng. (1)

D. Tentori, J. Camacho, “Birefringence characterization of one-quarter-pitch GRIN lenses,” Opt. Eng. 41, 2468–2475 (2002).
[CrossRef]

Other (5)

F. D. Bloss, Introducción a Los Métodos de la Cristalografia óptica, 5th ed. (Ediciones Omega, Barcelona, 1994), pp. 124–128 [Translated from An Introduction to the Methods of Optical Crystallography (Holt, Rinehart & Winston, New York, 1961)].

S. Tolansky, An Introduction to Interferometry, 2nd ed. (Longman, London, 1973), p. 203.

SELFOC Product Catalog 1999 (NSG America, Inc., N.J., 1999), p. 7.

E. W. Marchand, Gradient Index Optics (Academic, New York, 1978), pp. 53–60.

M. Françon, S. Mallick, Polarization Interferometers: Applications in Microscopy and Macroscopy (Wiley Interscience, London, 1971), pp. 140.

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Figures (14)

Fig. 1
Fig. 1

In traditional polariscopes the sample is placed between two orthogonal polarizers. (a) Orthoscopic setup. (b) Conoscopic setup. In both cases the observation is made at the output, on a plane perpendicular to the optical axis located after the analyzer.

Fig. 2
Fig. 2

GRIN sample in a null linear polarizer. Along each thin meridional plane, the GRIN lens behaves like a linear retarder whose retardation value has a radial dependence.

Fig. 3
Fig. 3

Conoscopic pattern obtained in a null linear polariscope with a convergent beam of light (not from a spatial filter) focused close to the input face of a one-quarter-pitch GRIN lens. The observation plane is the output face of the GRIN sample. As the exposure time increases, the dark square produced by the Fresnel contribution to intensity becomes negligible. (a) 20 ms, (b) 50 ms, (c) 80 ms.

Fig. 4
Fig. 4

Ray tracing along a meridional plane. (a) Convergent incident beam focused close to the input face of the GRIN lens. (b) Collimated incident beam. The angles of the director cosines of the ray inside the GRIN medium are α, 0, and γ. The angle of incidence at the input face is ξ. The angle of incidence at the output face is ξ′.

Fig. 5
Fig. 5

Ray-tracing calculations for a collimated beam of light traveling parallel to the optical axis. The sample is a one-quarter-pitch GRIN lens for 633 nm. (a) The intersection of each incident ray with the optical axis varies with its input radial position. (b) Close to the focusing point, some of the output radial positions overlap an ∼0.8-mm region along the optical axis.

Fig. 6
Fig. 6

Conoscopic images obtained near the focusing point; the same exposure time was used. The sample was a one-quarter-pitch GRIN lens illuminated with a collimated beam of light parallel to the optical axis. The observation plane was behind the output face (inside the GRIN lens). From left to right the distance from the observation plane to the output face of the GRIN lens was increased (∼1 to 3 mm). For images (a), (c), and (e), the analyzer and the polarizer were aligned. Images (b), (d), and (f) were obtained at the same positions as the corresponding images (a), (c), (e), with the analyzer orthogonal to the polarizer.

Fig. 7
Fig. 7

GRIN sample in the circular polariscope used in this paper. A linear polarizer with an azimuth angle equal to 45° and a quarter-wave plate (azimuth angle of 0°) formed the input circular polarizer. A quarter-wave plate (azimuth angle of 0°) and a linear polarizer whose azimuth angle can be 45° (null circular polariscope) or -45° formed the circular analyzer.

Fig. 8
Fig. 8

Conoscopic patterns obtained for a one-quarter-pitch GRIN lens illuminated with a convergent light beam focused inside the sample. (a) Observation plane located inside the sample. (b) Observation plane located outside the sample.

Fig. 9
Fig. 9

Optical path calculated for a one-quarter-pitch GRIN lens for 633 nm. (a) The incident beam is a convergent beam focused on the vertex of the GRIN lens’s input face. (b) The incident beam is a collimated light beam traveling parallel to the optical axis.

Fig. 10
Fig. 10

Comparison between two conoscopic patterns obtained with a linear polariscope. The sample (a half-pitch GRIN lens) was illuminated with a collimated beam of light. (top) The analyzer was orthogonal to the input polarizer. (bottom) The analyzer was aligned with the input polarizer.

Fig. 11
Fig. 11

Conoscopic pattern obtained in a linear null polariscope for a one-quarter-pitch GRIN lens illuminated by a convergent light beam focused close to the vertex of the input face. The numerical aperture of the incident beam is larger than the numerical aperture of the GRIN lens. The observation plane is located inside the GRIN lens.

Fig. 12
Fig. 12

Diagram of the experimental arrangement used for the conoscopic evaluation of the birefringence of one-quarter-pitch GRIN samples. In combination with the circular polariscope, we use a plate polarizer P0 at the input to control the light intensity. The incident convergent beam is produced with a spatial filter. The projecting lens L is a dissection microscope objective (inverted orientation). The conoscopic pattern on the output face of the GRIN lens is projected on a CCD arrangement.

Fig. 13
Fig. 13

When the directly transmitted beam of light is combined with beams reflected and redirected along the optical axis in the forward direction, we obtain at the pinhole the interference produced by their superposition. The circle at the right is the output face of the GRIN lens.

Fig. 14
Fig. 14

When the convergent beam of light is focused at the vertex of the GRIN sample, a 90° rotation of the linear polarizer in the circular analyzer produces a contrast inversion, as is predicted by the theory. (a) Conoscopic pattern obtained with a null circular polariscope. (b) Conoscopic pattern obtained when the input circular polarizer and the circular analyzer were aligned. (c) If the input beam is not focused at the vertex of the input face, there is an additional optical phase that varies with the angle of incidence, modifying the typical result obtained for plane-incident wave fronts. This image was obtained with the same sample in the same null circular polariscope.

Equations (25)

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MGr, θ=cosϕr2+i cos 2θ sinϕr2i sin 2θ sinϕr2i sin 2θ sinϕr2cosϕr2-i cos 2θ sinϕr2,
n2r=N02-B2r2,
x=x0 cosz¯+p0Bsinz¯, y=y0 cosz¯+q0Bsinz¯,
z¯=B/l0z.
p0=nr0cos α, q0=nr0cos β, l0=nr0cos γ,
cos α=drdx, cos β=drdy, cos γ=drdz.
x=x0 cosz¯+sin ξBsinz¯,
z¯=BzN02-B2r02-sin2 ξ1/2.
x=sin ξBsinz¯,
z¯=BzN02-sin2 ξ1/2.
Lf=2πBN02-sin2 ξ,
ξ=arcsin-sin ξN02-sin2 ξ1/2cosBzN02-sin2 ξ1/2.
x=x0 cosz¯,
z¯=BzN02-B2x021/2.
Lc=2πBN02-B2x02,
ξ=arcsin-x0BN02-B2x021/2cosBzN02-B2x021/2.
141i±1-iMGr, θ1-i=12sinϕr2sin 2θ+i cos 2θ1112cosϕr21-1.
I  sin2ϕr2cos2ϕr2.
ϕr=2m+1π, ϕr=2mπ,
ϕr=2πλOPDr, z,
OPr, z=1l00z n2rdz,
OPR,Tr, z=N02N02-sin2 ξ1/2-sin2 ξ2N02-sin2 ξz+sin2 ξ4BR,Tsin2BR,TzN02-sin2 ξ1/2,
OPR,Tr, z=zN02-BR,T2x02/2N02-BR,T2x021/2-BR,Tx024×sin2BR,TzN02-BR,T2x021/2.
OPDr, z=OPRr, z-OPTr, z.
OPDr, z=m+12λ, OPDr, z=mλ,

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