Abstract

A colorimetry-based retardation measurement (CBRM) method is presented. The specimen, between crossed polarizers, is illuminated with a white-light source. The retardation that is due to the birefringence of the specimen produces a white-light interference color. The x, y chromaticity coordinates of the color produced are measured with a spectrophotometer. The resulting x, y values are compared with a retardation x, y database that we obtained by measuring the retardation with an accurate Senarmont compensator and the x, y chromaticity values along the length of a 0–4-order quartz wedge. The technique was validated by the measurement of a variety of retardation plates. The retardation accuracy (mean error) of the CBRM method is shown to be 3.6 nm. The resolution is ±0.2 nm, and the measurement range is 5–2150 nm. The method substitutes for a polariscope and eliminates errors associated with quarter-wave plates. The CBRM method does not utilize any moving parts and thus is fast and can be automated.

© 2002 Optical Society of America

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References

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2000 (1)

1998 (2)

A. Ajovalasit, S. Baronne, G. Petrucci, “A method for reducing the influence of quarter-wave plate errors in phase-stepping photoelasticity,” J. Strain Anal. 33, 207–216 (1998).
[CrossRef]

A. Ajovalasit, S. Baronne, G. Petrucci, “A review of automated methods for the collection and analysis of photoelastic data,” J. Strain Anal. 33, 75–91 (1998).
[CrossRef]

1997 (2)

J. M. Desse, “Three-color differential interferometry,” Appl. Opt. 36, 7150–7156 (1997).
[CrossRef]

G. Petrucci, “Full-field automated evaluation of an isoclinic parameter in white light,” Exp. Mech. 37, 420–426 (1997).
[CrossRef]

1995 (2)

A. Ajovalasit, S. Baronne, G. Petrucci, “Automated photoelasticity in white light: influences of quarter-wave plates,” J. Strain Anal. 30, 29–34 (1995).
[CrossRef]

A. Ajovalasit, S. Baronne, G. Petrucci, “Towards RGB photoelasticity: full-field automated photoelasticity in white light,” Exp. Mech. 35, 193–200 (1995).
[CrossRef]

1991 (1)

1985 (1)

A. Redner, “Photoelastic measurements by means of computer-assisted spectral-contents analysis,” Exp. Mech. 25, 148–153 (1985).
[CrossRef]

1982 (1)

Ahn, T. J.

Ajovalasit, A.

A. Ajovalasit, S. Baronne, G. Petrucci, “A method for reducing the influence of quarter-wave plate errors in phase-stepping photoelasticity,” J. Strain Anal. 33, 207–216 (1998).
[CrossRef]

A. Ajovalasit, S. Baronne, G. Petrucci, “A review of automated methods for the collection and analysis of photoelastic data,” J. Strain Anal. 33, 75–91 (1998).
[CrossRef]

A. Ajovalasit, S. Baronne, G. Petrucci, “Automated photoelasticity in white light: influences of quarter-wave plates,” J. Strain Anal. 30, 29–34 (1995).
[CrossRef]

A. Ajovalasit, S. Baronne, G. Petrucci, “Towards RGB photoelasticity: full-field automated photoelasticity in white light,” Exp. Mech. 35, 193–200 (1995).
[CrossRef]

Arapoyianni, A. T.

Baronne, S.

A. Ajovalasit, S. Baronne, G. Petrucci, “A review of automated methods for the collection and analysis of photoelastic data,” J. Strain Anal. 33, 75–91 (1998).
[CrossRef]

A. Ajovalasit, S. Baronne, G. Petrucci, “A method for reducing the influence of quarter-wave plate errors in phase-stepping photoelasticity,” J. Strain Anal. 33, 207–216 (1998).
[CrossRef]

A. Ajovalasit, S. Baronne, G. Petrucci, “Automated photoelasticity in white light: influences of quarter-wave plates,” J. Strain Anal. 30, 29–34 (1995).
[CrossRef]

A. Ajovalasit, S. Baronne, G. Petrucci, “Towards RGB photoelasticity: full-field automated photoelasticity in white light,” Exp. Mech. 35, 193–200 (1995).
[CrossRef]

Berger-Schunn, A.

A. Berger-Schunn, Practical Color Measurement (Wiley, New York, 1994).

Bloss, F. D.

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

Chu, P. L.

Desse, J. M.

Han, W. T.

Hartshorne, N. H.

N. H. Hartshorne, A. Stuart, Crystals and the Polarizing Microscope (Edward Arnold, London, 1960).

Kim, D. Y.

Kim, Y. H.

Paek, U. C.

Park, Y.

Petrucci, G.

A. Ajovalasit, S. Baronne, G. Petrucci, “A method for reducing the influence of quarter-wave plate errors in phase-stepping photoelasticity,” J. Strain Anal. 33, 207–216 (1998).
[CrossRef]

A. Ajovalasit, S. Baronne, G. Petrucci, “A review of automated methods for the collection and analysis of photoelastic data,” J. Strain Anal. 33, 75–91 (1998).
[CrossRef]

G. Petrucci, “Full-field automated evaluation of an isoclinic parameter in white light,” Exp. Mech. 37, 420–426 (1997).
[CrossRef]

A. Ajovalasit, S. Baronne, G. Petrucci, “Towards RGB photoelasticity: full-field automated photoelasticity in white light,” Exp. Mech. 35, 193–200 (1995).
[CrossRef]

A. Ajovalasit, S. Baronne, G. Petrucci, “Automated photoelasticity in white light: influences of quarter-wave plates,” J. Strain Anal. 30, 29–34 (1995).
[CrossRef]

Redner, A.

A. Redner, “Photoelastic measurements by means of computer-assisted spectral-contents analysis,” Exp. Mech. 25, 148–153 (1985).
[CrossRef]

A. Redner, “Photoelastic measurements of residual stresses for NDE,” in International Conference on Photomechanics and Speckle Metrology, F.-P. Chiang, ed., Proc. SPIE814, 16–19 (1987).
[CrossRef]

Stuart, A.

N. H. Hartshorne, A. Stuart, Crystals and the Polarizing Microscope (Edward Arnold, London, 1960).

Theofanous, N. G.

Whitbread, T.

Appl. Opt. (3)

Exp. Mech. (3)

A. Ajovalasit, S. Baronne, G. Petrucci, “Towards RGB photoelasticity: full-field automated photoelasticity in white light,” Exp. Mech. 35, 193–200 (1995).
[CrossRef]

G. Petrucci, “Full-field automated evaluation of an isoclinic parameter in white light,” Exp. Mech. 37, 420–426 (1997).
[CrossRef]

A. Redner, “Photoelastic measurements by means of computer-assisted spectral-contents analysis,” Exp. Mech. 25, 148–153 (1985).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Strain Anal. (3)

A. Ajovalasit, S. Baronne, G. Petrucci, “A method for reducing the influence of quarter-wave plate errors in phase-stepping photoelasticity,” J. Strain Anal. 33, 207–216 (1998).
[CrossRef]

A. Ajovalasit, S. Baronne, G. Petrucci, “A review of automated methods for the collection and analysis of photoelastic data,” J. Strain Anal. 33, 75–91 (1998).
[CrossRef]

A. Ajovalasit, S. Baronne, G. Petrucci, “Automated photoelasticity in white light: influences of quarter-wave plates,” J. Strain Anal. 30, 29–34 (1995).
[CrossRef]

Other (4)

A. Berger-Schunn, Practical Color Measurement (Wiley, New York, 1994).

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

N. H. Hartshorne, A. Stuart, Crystals and the Polarizing Microscope (Edward Arnold, London, 1960).

A. Redner, “Photoelastic measurements of residual stresses for NDE,” in International Conference on Photomechanics and Speckle Metrology, F.-P. Chiang, ed., Proc. SPIE814, 16–19 (1987).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Quartz wedge made of two pieces of quartz with their optic axes perpendicular to one another. One of the pieces has a linearly increasing thickness. Placed between crossed polarizers and illuminated with white-light, interference colors are produced. o.a., optic axis.

Fig. 2
Fig. 2

Normalized transmittance of the light passing through a quartz wedge for retardations R = 400 nm and R = 700 nm. The two curves represent the transmitted spectral distribution of the light when the quartz wedge is observed between crossed polarizers.

Fig. 3
Fig. 3

Quartz wedge observed between crossed polarizers with a transmitted polarized microscope at magnification 5×. White-light interference colors are produced corresponding to linearly increasing retardation along the wedge. The two transmitted colors corresponding to the spectra plotted in Fig. 2 are indicated.

Fig. 4
Fig. 4

Chromaticity curves of the white-light interference colors as the retardation of the quartz wedge linearly increases. The long-dashed curve represents the chromaticity coordinates calculated for an ideal white-light source and ideal optics of the microscope. The short-dashed curve represents the chromaticity coordinates calculated for the spectral distribution of the halogen source used and ideal microscope optics. The solid curve represents the measurements of the white-light interference colors generated when a quartz wedge is observed between crossed polarizers with a transmission polarization microscope.

Fig. 5
Fig. 5

Quartz-wedge retardation as a function of the micrometer position. The open circles correspond to the Senarmont compensator measurements. The solid curve is the polynomial fit used to calculate the retardation as a function of the position along the quartz wedge.

Fig. 6
Fig. 6

Mean deviation between calculated and measured retardations with polynomials of various degrees.

Fig. 7
Fig. 7

(a) Chromaticity coordinate x of the white-light interference colors as a function of the quartz-wedge retardation. (b) Chromaticity coordinate y of the white-light interference colors as a function of the quartz-wedge retardation. The filled circles represent the measurements made with a spectrophotometer when the quartz wedge is observed between crossed polarizers at a magnification of 50×. The solid curve represents the polynomial fit used to calculate x and y as a function of the retardation.

Fig. 8
Fig. 8

White-light interference colors curve plotted on a chromaticity diagram according to the conventions defined by the CIE in 1931. The filled circles represent the color measurements along a quartz wedge observed between crossed polarizers. The solid curve represents the calculated chromaticity coordinates with the polynomial fits for chromaticity coordinates x and y as a function of the retardation.

Fig. 9
Fig. 9

Retardation measurement with the CBRM method. The color is measured and plotted (×) in a chromaticity diagram. The measurement is projected onto the closest point M on the white-light interference colors curve defined with the polynomial fits that were determined in preliminary experiments. This curve is a fine grid in which the chromaticity coordinates x and y of each point, i.e., the color, are known as well as the corresponding retardation. We then calculated the unknown retardation at point M by assuming linear variation of the retardation in the grid between points A and B.

Fig. 10
Fig. 10

Quartz-wedge retardation measurements by the CBRM method and the Senarmont compensator method. Note how both methods resolve the slight nonlinearity in the quartz-wedge surface.

Fig. 11
Fig. 11

Quartz-wedge retardation deviation from linear retardation for the CBRM method and the Senarmont compensator method. We determined this deviation from linearity by subtracting the two linear functions fit to the retardation measurements made, respectively, with the CBRM method and the Senarmont compensator method and shown in Fig. 10.

Fig. 12
Fig. 12

White-light interference colors curve calculated with the spectral distribution of the halogen source of the transmission polarization microscope and ideal optics. The colors are plotted for retardations from 0 to 4000 nm.

Tables (3)

Tables Icon

Table 1 Comparison of Retardation Measurements of Quartz Wedge by the Senarmont Compensator Method and the CBRM Method

Tables Icon

Table 2 Retardation Measurements of Antireflection-Coated Retardation Plates: Comparison of Senarmont Compensator and CBRM Methods

Tables Icon

Table 3 Retardation Measurements of Non-Antireflection-Coated Retardation Plates: Comparison of Senarmont Compensator and CBRM Methods

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

TR, λ=I0 sin2πRλ,
x=XX+Y+Z,  y=YX+Y+Z,  z=ZX+Y+Z.
X=λ SλTλTopticsλx¯λ, Y=λ SλTλTopticsλy¯λ, Z=λ SλTλTopticsλz¯λ,
R=m-θ180°546 nm,
RM=RB+DAMDABRA-RB,
R=dRdd d+R0,

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