Abstract

The Talbot effect and moiré fringes are combined to measure the refractive indices of a birefringent material. Techniques based on the visibility of moiré fringes in finite fringe mode and based on fringe count are presented. These techniques are sensitive and advantageous. They are described and analyzed.

© 2001 Optical Society of America

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References

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  1. F. D. Bloss, An Introduction to the Methods of Optical Crystallography (Holt, New York, 1961).
  2. D. F. Heller, O. Kafri, J. Krasinski, “Direct birefringence measurements using moiré ray deflection techniques,” Appl. Opt. 33, 3037–3040 (1985).
    [CrossRef]
  3. J. C. Bhattacharya, A. K. Aggarwal, “Finite fringe moiré deflectometry for measurement of radius of curvature: an alternative approach,” Opt. Laser Technol. 25, 167–169 (1993).
    [CrossRef]
  4. F. Talbot, “Facts relating to optical science. IV,” Philos. Mag. 9, 401–407 (1936).
  5. J. C. Bhattacharya, “Measurement of the refractive index using the Talbot effect and a moiré technique,” Appl. Opt. 28, 2600–2604 (1989).
    [CrossRef] [PubMed]
  6. M. Francon, S. Mallick, Polarization Interferometers (Wiley-Interscience, New York, 1971), Appendix A.
  7. R. W. Ditchburn, Light, 3rd ed. (Academic, London, 1976), p. 458.

1993

J. C. Bhattacharya, A. K. Aggarwal, “Finite fringe moiré deflectometry for measurement of radius of curvature: an alternative approach,” Opt. Laser Technol. 25, 167–169 (1993).
[CrossRef]

1989

1985

D. F. Heller, O. Kafri, J. Krasinski, “Direct birefringence measurements using moiré ray deflection techniques,” Appl. Opt. 33, 3037–3040 (1985).
[CrossRef]

1936

F. Talbot, “Facts relating to optical science. IV,” Philos. Mag. 9, 401–407 (1936).

Aggarwal, A. K.

J. C. Bhattacharya, A. K. Aggarwal, “Finite fringe moiré deflectometry for measurement of radius of curvature: an alternative approach,” Opt. Laser Technol. 25, 167–169 (1993).
[CrossRef]

Bhattacharya, J. C.

J. C. Bhattacharya, A. K. Aggarwal, “Finite fringe moiré deflectometry for measurement of radius of curvature: an alternative approach,” Opt. Laser Technol. 25, 167–169 (1993).
[CrossRef]

J. C. Bhattacharya, “Measurement of the refractive index using the Talbot effect and a moiré technique,” Appl. Opt. 28, 2600–2604 (1989).
[CrossRef] [PubMed]

Bloss, F. D.

F. D. Bloss, An Introduction to the Methods of Optical Crystallography (Holt, New York, 1961).

Ditchburn, R. W.

R. W. Ditchburn, Light, 3rd ed. (Academic, London, 1976), p. 458.

Francon, M.

M. Francon, S. Mallick, Polarization Interferometers (Wiley-Interscience, New York, 1971), Appendix A.

Heller, D. F.

D. F. Heller, O. Kafri, J. Krasinski, “Direct birefringence measurements using moiré ray deflection techniques,” Appl. Opt. 33, 3037–3040 (1985).
[CrossRef]

Kafri, O.

D. F. Heller, O. Kafri, J. Krasinski, “Direct birefringence measurements using moiré ray deflection techniques,” Appl. Opt. 33, 3037–3040 (1985).
[CrossRef]

Krasinski, J.

D. F. Heller, O. Kafri, J. Krasinski, “Direct birefringence measurements using moiré ray deflection techniques,” Appl. Opt. 33, 3037–3040 (1985).
[CrossRef]

Mallick, S.

M. Francon, S. Mallick, Polarization Interferometers (Wiley-Interscience, New York, 1971), Appendix A.

Talbot, F.

F. Talbot, “Facts relating to optical science. IV,” Philos. Mag. 9, 401–407 (1936).

Appl. Opt.

D. F. Heller, O. Kafri, J. Krasinski, “Direct birefringence measurements using moiré ray deflection techniques,” Appl. Opt. 33, 3037–3040 (1985).
[CrossRef]

J. C. Bhattacharya, “Measurement of the refractive index using the Talbot effect and a moiré technique,” Appl. Opt. 28, 2600–2604 (1989).
[CrossRef] [PubMed]

Opt. Laser Technol.

J. C. Bhattacharya, A. K. Aggarwal, “Finite fringe moiré deflectometry for measurement of radius of curvature: an alternative approach,” Opt. Laser Technol. 25, 167–169 (1993).
[CrossRef]

Philos. Mag.

F. Talbot, “Facts relating to optical science. IV,” Philos. Mag. 9, 401–407 (1936).

Other

M. Francon, S. Mallick, Polarization Interferometers (Wiley-Interscience, New York, 1971), Appendix A.

R. W. Ditchburn, Light, 3rd ed. (Academic, London, 1976), p. 458.

F. D. Bloss, An Introduction to the Methods of Optical Crystallography (Holt, New York, 1961).

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Figures (4)

Fig. 1
Fig. 1

Experimental arrangement for forming moiré fringes by doubly refractive beams.

Fig. 2
Fig. 2

Sensitivity versus angle of incidence graph for a calcite crystal plate.

Fig. 3
Fig. 3

Sensitivity versus extraordinary refractive-index graphs for p = 0.05 and 0.1 mm.

Fig. 4
Fig. 4

Tolerance in period versus extraordinary refractive-index graphs for p = 0.05 and 0.1 mm.

Tables (2)

Tables Icon

Table 1 Error Summary for a Plane-Parallel Calcite Plate, μ2 = 1.34

Tables Icon

Table 2 Error Summary for μe = 1.49

Equations (17)

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m1-m2p=kp,
m1-m2p=k+0.5p.
μo=sin isin ro,  μ2=sin isin re,
m1p=t sin i1-1-sin2 iμo2-sin2 i1/2,
m2p=t sin i1-1-sin2 iμ22-sin2 i1/2.
m1-m2p=kp=t sin 2i21μ22-sin2 i1/2-1μo2-sin2 i1/2,
tan re=μo sin iμeμe2-sin2 i1/2.
μe2μe2-sin2 i=μo2μ22-sin2 i.
μoμe2-1μo=2kpt sin 2i
μoμo21-Δμ/μo2-1μo=2kpt sin 2i.
Δμ/μo  1,
Δμ/μo2kp/t sin 2i.
S=dk×dμe×1×10-4.
S=t sin 2iμo2p1μe2-sin2 i1/21μe2-sin2 i+1μe2×1×10-4.
kp=t sin 2i2μeμeμe2-sin2 i1/2-1μo2-sin2 i1/2.
p/μe=t sin 2i2kμoμe2-sin2 i1/2×1μe2+1μe2-sin2 i1/2.
Tp=p/μe×10-4=t sin 2i2kμoμe2-sin2 i1/21μe2+1μe2-sin2 i1/2×10-4.

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