Abstract

We describe a birefringence measurement bench that allows fast two-dimensional measurements of low-birefringence fields in large transparent samples. We present calculations that show that, even when a birefringence bench exhibits defects (nonideal components, misalignments, etc.), measurements can be performed under realistic conditions without any a priori knowledge of the origin of the bench defects. This allows the measurement of birefringence fields on large-scale samples by use of an array of detectors instead of a single detector element, with a sensitivity of 3 × 10-4 rad for 2-s data integration.

© 2000 Optical Society of America

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References

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  1. J. Badoz, M. Billardon, J. C. Canit, M. F. Russel, “Sensitive devices to determine the state and degree of polarization of a light beam using a birefringence modulator,” J. Opt. 8, 373–384 (1977).
    [CrossRef]
  2. B. Wang, T. C. Oakberg, “A new instrument for measuring both the magnitude and angle of low level linear birefringence,” Rev. Sci. Instrum. 70, 3847–3854 (1999).
    [CrossRef]
  3. Y. Zhu, T. Koyama, T. Takada, Y. Murooka, T. Otsuka, “Two-dimensional measurement technique for birefringence vector distributions: data processing and experimental verification,” Appl. Opt. 38, 2216–2224 (1999).
    [CrossRef]
  4. Y. Zhu, T. Koyama, T. Takada, Y. Murooka, “Two-dimensional measurement technique for birefringence vector distributions: measurement principle,” Appl. Opt. 38, 2225–2231 (1999).
    [CrossRef]
  5. E. Collett, Polarized Light (Marcel Dekker, New York, 1993), Chap. 10.
  6. A. C. Boccara, F. Charbonnier, D. Fournier, P. Gleyzes, “Method and device for multichannel analog detection,” French patentFR 90.08255 (5June1990) and international extensions266 C US 3221 (December1993).
  7. P. Gleyzes, F. Guernet, A. C. Boccara, “Profilométrie picométrique. II. L’approche multi-détecteur et la détection synchrone multiplexée,” J. Opt. (Paris) 26, 251–265 (1995).
    [CrossRef]
  8. P. Gleyzes, V. Loriette, H. Saint-Jalmes, A. C. Boccara, “Roughness measurements in the picometric range using a polarization interferometer and a multichannel lock-in detection technique,” Int. J. Mach. Tools Manufact. 38, 715–717 (1998).
    [CrossRef]
  9. A. Dubois, A. C. Boccara, M. Lebec, “Real-time reflectivity and topography of depth-resolved microscopic surfaces,” Opt. Lett. 24, 309–311 (1999).
    [CrossRef]
  10. W. J. Smith, “Image formation: geometrical and physical optics,” in Handbook of Optics, 1st ed., W. G. Driscoll, W. Vaughan, eds. (McGraw-Hill, New York, 1978), Chap. 2, pp. 40–41. This reference is not included in the second edition of the handbook.

1999

1998

P. Gleyzes, V. Loriette, H. Saint-Jalmes, A. C. Boccara, “Roughness measurements in the picometric range using a polarization interferometer and a multichannel lock-in detection technique,” Int. J. Mach. Tools Manufact. 38, 715–717 (1998).
[CrossRef]

1995

P. Gleyzes, F. Guernet, A. C. Boccara, “Profilométrie picométrique. II. L’approche multi-détecteur et la détection synchrone multiplexée,” J. Opt. (Paris) 26, 251–265 (1995).
[CrossRef]

1977

J. Badoz, M. Billardon, J. C. Canit, M. F. Russel, “Sensitive devices to determine the state and degree of polarization of a light beam using a birefringence modulator,” J. Opt. 8, 373–384 (1977).
[CrossRef]

Badoz, J.

J. Badoz, M. Billardon, J. C. Canit, M. F. Russel, “Sensitive devices to determine the state and degree of polarization of a light beam using a birefringence modulator,” J. Opt. 8, 373–384 (1977).
[CrossRef]

Billardon, M.

J. Badoz, M. Billardon, J. C. Canit, M. F. Russel, “Sensitive devices to determine the state and degree of polarization of a light beam using a birefringence modulator,” J. Opt. 8, 373–384 (1977).
[CrossRef]

Boccara, A. C.

A. Dubois, A. C. Boccara, M. Lebec, “Real-time reflectivity and topography of depth-resolved microscopic surfaces,” Opt. Lett. 24, 309–311 (1999).
[CrossRef]

P. Gleyzes, V. Loriette, H. Saint-Jalmes, A. C. Boccara, “Roughness measurements in the picometric range using a polarization interferometer and a multichannel lock-in detection technique,” Int. J. Mach. Tools Manufact. 38, 715–717 (1998).
[CrossRef]

P. Gleyzes, F. Guernet, A. C. Boccara, “Profilométrie picométrique. II. L’approche multi-détecteur et la détection synchrone multiplexée,” J. Opt. (Paris) 26, 251–265 (1995).
[CrossRef]

A. C. Boccara, F. Charbonnier, D. Fournier, P. Gleyzes, “Method and device for multichannel analog detection,” French patentFR 90.08255 (5June1990) and international extensions266 C US 3221 (December1993).

Canit, J. C.

J. Badoz, M. Billardon, J. C. Canit, M. F. Russel, “Sensitive devices to determine the state and degree of polarization of a light beam using a birefringence modulator,” J. Opt. 8, 373–384 (1977).
[CrossRef]

Charbonnier, F.

A. C. Boccara, F. Charbonnier, D. Fournier, P. Gleyzes, “Method and device for multichannel analog detection,” French patentFR 90.08255 (5June1990) and international extensions266 C US 3221 (December1993).

Collett, E.

E. Collett, Polarized Light (Marcel Dekker, New York, 1993), Chap. 10.

Dubois, A.

Fournier, D.

A. C. Boccara, F. Charbonnier, D. Fournier, P. Gleyzes, “Method and device for multichannel analog detection,” French patentFR 90.08255 (5June1990) and international extensions266 C US 3221 (December1993).

Gleyzes, P.

P. Gleyzes, V. Loriette, H. Saint-Jalmes, A. C. Boccara, “Roughness measurements in the picometric range using a polarization interferometer and a multichannel lock-in detection technique,” Int. J. Mach. Tools Manufact. 38, 715–717 (1998).
[CrossRef]

P. Gleyzes, F. Guernet, A. C. Boccara, “Profilométrie picométrique. II. L’approche multi-détecteur et la détection synchrone multiplexée,” J. Opt. (Paris) 26, 251–265 (1995).
[CrossRef]

A. C. Boccara, F. Charbonnier, D. Fournier, P. Gleyzes, “Method and device for multichannel analog detection,” French patentFR 90.08255 (5June1990) and international extensions266 C US 3221 (December1993).

Guernet, F.

P. Gleyzes, F. Guernet, A. C. Boccara, “Profilométrie picométrique. II. L’approche multi-détecteur et la détection synchrone multiplexée,” J. Opt. (Paris) 26, 251–265 (1995).
[CrossRef]

Koyama, T.

Lebec, M.

Loriette, V.

P. Gleyzes, V. Loriette, H. Saint-Jalmes, A. C. Boccara, “Roughness measurements in the picometric range using a polarization interferometer and a multichannel lock-in detection technique,” Int. J. Mach. Tools Manufact. 38, 715–717 (1998).
[CrossRef]

Murooka, Y.

Oakberg, T. C.

B. Wang, T. C. Oakberg, “A new instrument for measuring both the magnitude and angle of low level linear birefringence,” Rev. Sci. Instrum. 70, 3847–3854 (1999).
[CrossRef]

Otsuka, T.

Russel, M. F.

J. Badoz, M. Billardon, J. C. Canit, M. F. Russel, “Sensitive devices to determine the state and degree of polarization of a light beam using a birefringence modulator,” J. Opt. 8, 373–384 (1977).
[CrossRef]

Saint-Jalmes, H.

P. Gleyzes, V. Loriette, H. Saint-Jalmes, A. C. Boccara, “Roughness measurements in the picometric range using a polarization interferometer and a multichannel lock-in detection technique,” Int. J. Mach. Tools Manufact. 38, 715–717 (1998).
[CrossRef]

Smith, W. J.

W. J. Smith, “Image formation: geometrical and physical optics,” in Handbook of Optics, 1st ed., W. G. Driscoll, W. Vaughan, eds. (McGraw-Hill, New York, 1978), Chap. 2, pp. 40–41. This reference is not included in the second edition of the handbook.

Takada, T.

Wang, B.

B. Wang, T. C. Oakberg, “A new instrument for measuring both the magnitude and angle of low level linear birefringence,” Rev. Sci. Instrum. 70, 3847–3854 (1999).
[CrossRef]

Zhu, Y.

Appl. Opt.

Int. J. Mach. Tools Manufact.

P. Gleyzes, V. Loriette, H. Saint-Jalmes, A. C. Boccara, “Roughness measurements in the picometric range using a polarization interferometer and a multichannel lock-in detection technique,” Int. J. Mach. Tools Manufact. 38, 715–717 (1998).
[CrossRef]

J. Opt.

J. Badoz, M. Billardon, J. C. Canit, M. F. Russel, “Sensitive devices to determine the state and degree of polarization of a light beam using a birefringence modulator,” J. Opt. 8, 373–384 (1977).
[CrossRef]

J. Opt. (Paris)

P. Gleyzes, F. Guernet, A. C. Boccara, “Profilométrie picométrique. II. L’approche multi-détecteur et la détection synchrone multiplexée,” J. Opt. (Paris) 26, 251–265 (1995).
[CrossRef]

Opt. Lett.

Rev. Sci. Instrum.

B. Wang, T. C. Oakberg, “A new instrument for measuring both the magnitude and angle of low level linear birefringence,” Rev. Sci. Instrum. 70, 3847–3854 (1999).
[CrossRef]

Other

E. Collett, Polarized Light (Marcel Dekker, New York, 1993), Chap. 10.

A. C. Boccara, F. Charbonnier, D. Fournier, P. Gleyzes, “Method and device for multichannel analog detection,” French patentFR 90.08255 (5June1990) and international extensions266 C US 3221 (December1993).

W. J. Smith, “Image formation: geometrical and physical optics,” in Handbook of Optics, 1st ed., W. G. Driscoll, W. Vaughan, eds. (McGraw-Hill, New York, 1978), Chap. 2, pp. 40–41. This reference is not included in the second edition of the handbook.

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Figures (9)

Fig. 1
Fig. 1

Polarization elements in the instrument that measures birefringence: (P), a polarizer with its axis at 0°; (Q1), a quarter-wave plate with its fast axis at +34°; (S), the sample; (Q2), a quarter-wave plate with its fast axis at -45°; (M), a birefringence modulator with its axis at 0°; (A), an analyzer with its axis at +45°.

Fig. 2
Fig. 2

Four signals are acquired, and the light source C p is phase shifted by a quarter of a modulation period at each acquisition.

Fig. 3
Fig. 3

Absolute values of the functions 4Q 1 and 8K 0 at ψ = 2.1 rad and ψ = 4.3 rad, respectively.

Fig. 4
Fig. 4

Each pixel (P) of the CCD array images a 470 µm × 470 µm region (R) of the sample (S), while the image (O2) of the source (O1) illuminates the CCD array uniformly. Q, quarter-wave plate; M, modulator; A, analyzer; Obj., objective.

Fig. 5
Fig. 5

Schematic diagram of the birefringence bench: The upper part of the figure shows in detail the section of the lower part of the figure that is enclosed in the dashed box. The sample (S) is placed in front of a concave mirror (CM). The light source is a LED. The basic birefringence measurement setup is made up of a polarizer (P), a quarter-wave plate (Q) that is crossed twice or two quarter-wave plates, a birefringence modulator (Mod), and an analyzer (A). The imaging system is made up of a microscope objective (MO), which creates an image of the source at a distance of ρ = 2 m from the concave mirror, and a 256 × 256 pixel CCD camera with an f = 80 mm objective (O).

Fig. 6
Fig. 6

Sensitivity (in radians) plotted as a function of the number of sequences averaged; the acquisition time of a single sequence of {S 0, S 1, S 2, S 3} is 20 ms.

Fig. 7
Fig. 7

Birefringence-amplitude maps of a CaF2 sample. The gray scales give the birefringence values in radians. The scales of the coordinates are given in pixels (1 pixel corresponds to 470 µm on the sample). Shown are two consecutive measurements performed (a) before and (b) after a 90° rotation of the sample in its holder.

Fig. 8
Fig. 8

Principal-axes directions in a CaF2 sample. The graduations of the coordinates are in pixels (470 µm × 470 µm). The length of each arrow is proportional to the birefringence amplitude.

Fig. 9
Fig. 9

Effect of the stress-induced birefringence between the two ceramic transducers that are glued onto a photoelastic modulator for different values of applied voltage. The coordinates are in units of pixels. The gray scales give the birefringence amplitude in radians.

Equations (76)

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MA=MAP+ΔAMAΔ,  MB=MBP+ΔBMBΔ,
ΔA,B  1.
ΔAΔB  Δ.
Mpol=1000ideal polarizer+θp0110correction matrix+oθp2.
MS=exp-i Δ200expi Δ2.
MS=-γexp-i Δ200expi Δ2γ,
γcosγsinγ-sinγcosγ,
MS=cos2γexp-i Δ2+sin2γexpi Δ2-i sin2γsinΔ2-i sin2γsinΔ2sin2γexp-i Δ2+cos2γexpi Δ2.
MS=1001-i Δ2cos2γsin2γsin2γ-cos2γ+oΔ2,
MS=1-i Δ2 N2γ+oΔ2.
M=MBMA=MBP+ΔBMBΔMAP+ΔAMAΔ=MBPMAP+ΔBMBΔMAP+ΔAMBPMAΔ+oΔAΔB.
MAP=121ii11000=1210i0.
MBP=1221111exp-i ψ200expi ψ21-i-i1=122exp-i ψ21-i1-i+expi ψ2-i1-i1,
MBP=122exp-i ψ2MB,-P+expi ψ2MB,+P.
MBP=122MB,0P+exp-i ψ2MB,-P+expi ψ2MB,+P.
MBPMAP=exp-i ψ221010.
MBΔ=MB,0Δ+exp-i ψ2MB,-Δ+expi ψ2MB,+Δ,
M=exp-i ψ221010+ΔBMB,0ΔMAP+exp-i ψ2ΔBMB,-ΔMAP+ΔA22 MB,-PMAΔ+expi ψ2ΔBMB,+ΔMAP+ΔA22 MB,+PMAΔ.
M=exp-i ψ221010+ΔBΠ0+exp-i ψ2Δ-Π-+expi ψ2Δ+Π+,
Et=exp-i ψ22Ei,x+ΔBE0Π+exp-i ψ2Δ-E-Π+expi ψ2Δ+E+Π,
E0,-,+ΠΠ0,-,+Ei,  Ei,xEi,x11.
It,dc0=12 Ei,xEi,x*+oΔIi,
It,ω0=exp-i ψ22 ΔBEi,xE0Π*++,
Q=MB1-i Δ2 N2γMA
=M-i Δ2 MBPN2γMAP.
-i Δ2 MBPNγMAP=-i Δ42exp-i ψ21-i1-i+expi ψ2-i1-i1×cos2γsin2γsin2γ-cos2γ1210i0
=-Δ4 exp2iγexpi ψ21010,
Q=exp-i ψ22-Δ4 exp2iγexpi ψ21010+ΔBΠ0+exp-i ψ2Δ-Π-+expi ψ2Δ+Π+.
It,dc=It,dc0=12Ei,xEi,x*+oΔIi,
It,Δ=-Δ2 cosψ+2γEi,xEi,x*+exp-i ψ22 ΔBEi,xE0Π*++
=-Δ cosψ+2γIt,dc+It,Δ0.
cos ψtJ0ψ0+2J2ψ0cos2ωt,  sin ψt2J1ψ0sinωt.
It,Δ=-ΔIt,dcJ0ψ0cos2γ-2J1ψ0sin2γsinωt+2J2ψ0cos2γcos2ωt+It,Δ0.
S0=12Ei,xEi,x*=It,dc,  Sω=2ΔJ1ψ0sin2γIt,dc+Sω,0,  S2ω=-2ΔJ2ψ0cos2γIt,dc+S2ω,0,
Δ sin2γ=Sω-Sω,02J1ψ0S0,
Δ cos2γ=-S2ω+S2ω,02J2ψ0S0.
Δx, ysin2γx, y=Sωx, y-Sω,0x, y2J1ψ0S0x, y * Rx, y,
Δx, ycos2γx, y=-S2ωx, y+S2ω,0x, y2J2ψ0S0x, y * Rx, y,
Q=12cosΔ2exp-i ψ2-sinΔ2exp2iγexp-i ψ21010.
S0=1-sinΔJ0ψ0cos2γIt,dc,  Sω=2 sinΔJ1ψ0sin2γIt,dc,  S2ω=-2 sinΔJ2ψ0cos2γIt,dc.
sinΔsin2γ=Sω2J1ψ0S01-J0ψ0sinΔcos2γ,
sinΔcos2γ=-S2ω2J2ψ0S01-J0ψ0S2ω2J2ψ0S0-1,
It,Δ=-It,dcJ0ψ0cos2γ-2 sin2γn=0+ J2n+1ψ0sin2n+1ωt+2 cos2γn=1+ J2nψ0cos2nωt+It,Δ0.
It,Δt=-It,dcJ0ψ0Δ cos2γ+Δ0 cos2γ0-2Δ sin2γ+Δ0 sin2γ0n=1+ J2n+1ψ0×sin2n+1ωt+2Δ cos2γ+Δ0 cos2γ0n=1+ J2nψ0cos2nωt.
Cpt=Ii14+2 n=1+sinnπ/4nπ cosnωt+p4f+ϕω,
Sp=CptIt,dc+It,Δt=Ii14+2Δ cos2γ+Δ0 cos2γ0Kpψ0, ϕ+2Δ sin2γ+Δ0 sin2γ0Qpψ0, ϕ,
Kpψ0, ϕ=n=1+ J2nψ0sinnπ/22nπ cosnpπ+2ϕ,
QPψ0, ϕ=n=0+ J2n+1ψ0sin2n+1π42n+1π×sin2n+1p π2+ϕ,
φS0-S2S1-S3Q1ψ0Q2ψ0,
Δ sin2γ+Δ0 sin2γ0S0+S1-S2-S34IiQ1ψ0+φQ0ψ0,
Δ cos2γ+Δ0 cos2γ0S0-S1+S2-S38IiK0ψ0,
K0ψ0=1πn=0+-1nJ22n+1ψ022n+1,
Q1ψ0=n=0+ J2n+1ψ0sin2n+1π/42n+1π-1n,
Q0ψ0=22πn=0N-1nJ2n+1ψ0.
LSA=t n2-12n3rρ2,
Sp=CptIt,dc+It,Δt=Ii14+2Δ cos2γ+Δ0 cos2γ0Kpψ0, ϕ+2Δ sin2γ+Δ0 sin2γ0Qpψ0, ϕ,
Kpψ0, φ=n=1+ J2nψ0sinnπ/22nπ×cosnpπ+2φ,
Qpψ0, φ=n=0+ J2n+1ψ0sin2n+1π/42n+1π×sin2n+1p π2+φ
K0=K2=n=1+ J2nψ0sinnπ/22nπ cos2nϕ,
K1=K3=n=1+ J2nψ0sinnπ/22nπ-1n cos2nϕ,
Q0=-Q2=n=0+ J2n+1ψ0sin2n+1π/42n+1π×sin2n+1ϕ,
Q1=-Q3=n=0+ J2n+1ψ0sin2n+1π/42n+1π-1n×cos2n+1ϕ.
S0=Ii14+2Δ cos2γ+Δ0 cos2γ0K0ψ0, ϕ+2Δ sin2γ+Δ0 sin2γ0Q0ψ0, ϕ,
S1=Ii14-2Δ cos2γ+Δ0 cos2γ0K0ψ0, ϕ+2Δ sin2γ+Δ0 sin2γ0Q1ψ0, ϕ,
S2=Ii14+2Δ cos2γ+Δ0 cos2γ0K0ψ0, ϕ-2Δ sin2γ+Δ0 sin2γ0Q0ψ0, ϕ,
S3=Ii14-2Δ cos2γ+Δ0 cos2γ0K0ψ0, ϕ-2Δ sin2γ+Δ0 sin2γ0Q1ψ0, ϕ.
Ii=S0+S1+S2+S3,
Δ sin2γ+Δ0 sin2γ0=S0+S1-S2-S34IiQ0ψ0, ϕ+Q1ψ0, ϕ,
Δ cos2γ+Δ0 cos2γ0=S0-S1+S2-S38IiK0ψ0, ϕ.
K0ψ0=n=1+ J2nψ0sinnπ/22nπ=1πn=0+-1nJ22n+1ψ022n+1,
Q1ψ0=n=0+ J2n+1ψ0sin2n+1π/42n+1π-1n.
Q0ψ0, ϕ  1ϕ 22πn=0N-1nJ2n+1ψ0.
Q0ψ0, ϕ  1ϕQ0ψ0.
ϕS0-S2S1-S3Q1ψ0Q0ψ0,
Δ sin2γ+Δ0 sin2γ0S0+S1-S2-S34IiQ1ψ0+ϕQ0ψ0,
Δ cos2γ+Δ0 cos2γ0S0-S1+S2-S38IiK0ψ0.

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