Abstract

An experimental apparatus has been designed to measure group refractive index (ng) by observing the shift of the fringe visibility envelope upon insertion of a sample into one arm of a Twyman-Green interferometer. A criterion is developed for the limiting bandwidth and thickness for which good visibility may be expected and for predicting the bandwidth for the narrowest visibility curve. It is demonstrated that the measured group index data can be converted to phase index data with a previously described technique [ J. R. Rogers and M. D. Hopler, J. Opt. Soc. Am. A 5, 1595– 1600 ( 1988)] to an accuracy of ~0.0006 across the visible spectrum.

© 1991 Optical Society of America

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References

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  1. J. B. Caldwell, Institute of Optics, U. Rochester; personal communication (1988).
  2. C. P. Saylor, “Accuracy of Microscopical Methods For Determining Refractive Index By Immersion,” J. Res. Natl. Bur. Stand. (U.S.) 15, 277–295 (1935).
    [CrossRef]
  3. A. Johannsen, Manual of Petrographic Methods (McGraw-Hill, New York, 1918), p. 277.
  4. W. H. Steel, Interferometry (Cambridge, U.K., 1967), p. 105.
  5. J. E. Chamberlain, J. E. Gibbs, H. A. Gebbie, “Refractometry in the Far Infra-Red Using a Two-Beam Interferometer,” Nature London 198, 874–875 (1963).
    [CrossRef]
  6. W. H. Knox, N. M. Pearson, K. D. Li, C. A. Hirlimann, “Interferometric Measurements of Femtosecond Group Delay in Optical Components,” Opt. Lett. 13, 574–576 (1988).
    [CrossRef] [PubMed]
  7. J. R. Rogers, M. D. Hopler, “Conversion of Group Refractive Index To Phase Refractive Index,” J. Opt. Soc. Am. A 5, 1595–1600 (1988).
    [CrossRef]
  8. See, for example, M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 321.
  9. Rayleigh, “Investigations in Optics, with Special Reference to the Spectroscope,” Philos. Mag. 8, 403–411 (1879).
    [CrossRef]
  10. G. W. Forbes, “Chromatic Coordinates in Aberration Theory,” J. Opt. Soc. Am. A 1, 344–349 (1984).
    [CrossRef]
  11. M. D. Hopler, “Interferometric Measurement of Refractive Index,” M.S. Thesis, U. Rochester, New York (1987).

1988 (2)

1984 (1)

1963 (1)

J. E. Chamberlain, J. E. Gibbs, H. A. Gebbie, “Refractometry in the Far Infra-Red Using a Two-Beam Interferometer,” Nature London 198, 874–875 (1963).
[CrossRef]

1935 (1)

C. P. Saylor, “Accuracy of Microscopical Methods For Determining Refractive Index By Immersion,” J. Res. Natl. Bur. Stand. (U.S.) 15, 277–295 (1935).
[CrossRef]

1879 (1)

Rayleigh, “Investigations in Optics, with Special Reference to the Spectroscope,” Philos. Mag. 8, 403–411 (1879).
[CrossRef]

Born, M.

See, for example, M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 321.

Caldwell, J. B.

J. B. Caldwell, Institute of Optics, U. Rochester; personal communication (1988).

Chamberlain, J. E.

J. E. Chamberlain, J. E. Gibbs, H. A. Gebbie, “Refractometry in the Far Infra-Red Using a Two-Beam Interferometer,” Nature London 198, 874–875 (1963).
[CrossRef]

Forbes, G. W.

Gebbie, H. A.

J. E. Chamberlain, J. E. Gibbs, H. A. Gebbie, “Refractometry in the Far Infra-Red Using a Two-Beam Interferometer,” Nature London 198, 874–875 (1963).
[CrossRef]

Gibbs, J. E.

J. E. Chamberlain, J. E. Gibbs, H. A. Gebbie, “Refractometry in the Far Infra-Red Using a Two-Beam Interferometer,” Nature London 198, 874–875 (1963).
[CrossRef]

Hirlimann, C. A.

Hopler, M. D.

J. R. Rogers, M. D. Hopler, “Conversion of Group Refractive Index To Phase Refractive Index,” J. Opt. Soc. Am. A 5, 1595–1600 (1988).
[CrossRef]

M. D. Hopler, “Interferometric Measurement of Refractive Index,” M.S. Thesis, U. Rochester, New York (1987).

Johannsen, A.

A. Johannsen, Manual of Petrographic Methods (McGraw-Hill, New York, 1918), p. 277.

Knox, W. H.

Li, K. D.

Pearson, N. M.

Rayleigh,

Rayleigh, “Investigations in Optics, with Special Reference to the Spectroscope,” Philos. Mag. 8, 403–411 (1879).
[CrossRef]

Rogers, J. R.

Saylor, C. P.

C. P. Saylor, “Accuracy of Microscopical Methods For Determining Refractive Index By Immersion,” J. Res. Natl. Bur. Stand. (U.S.) 15, 277–295 (1935).
[CrossRef]

Steel, W. H.

W. H. Steel, Interferometry (Cambridge, U.K., 1967), p. 105.

Wolf, E.

See, for example, M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 321.

J. Opt. Soc. Am. A (2)

J. Res. Natl. Bur. Stand. (U.S.) (1)

C. P. Saylor, “Accuracy of Microscopical Methods For Determining Refractive Index By Immersion,” J. Res. Natl. Bur. Stand. (U.S.) 15, 277–295 (1935).
[CrossRef]

Nature London (1)

J. E. Chamberlain, J. E. Gibbs, H. A. Gebbie, “Refractometry in the Far Infra-Red Using a Two-Beam Interferometer,” Nature London 198, 874–875 (1963).
[CrossRef]

Opt. Lett. (1)

Philos. Mag. (1)

Rayleigh, “Investigations in Optics, with Special Reference to the Spectroscope,” Philos. Mag. 8, 403–411 (1879).
[CrossRef]

Other (5)

M. D. Hopler, “Interferometric Measurement of Refractive Index,” M.S. Thesis, U. Rochester, New York (1987).

See, for example, M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 321.

A. Johannsen, Manual of Petrographic Methods (McGraw-Hill, New York, 1918), p. 277.

W. H. Steel, Interferometry (Cambridge, U.K., 1967), p. 105.

J. B. Caldwell, Institute of Optics, U. Rochester; personal communication (1988).

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Figures (9)

Fig. 1
Fig. 1

Visibility for a sample of SF11 produced with a 10-nm flat-topped source.

Fig. 2
Fig. 2

Visibility for a sample of SF11 produced with a 10-nm Gaussian source.

Fig. 3
Fig. 3

Dependence of visibility width on bandwidth for several thicknesses.

Fig. 4
Fig. 4

Illustration of the deviation from the inverse-bandwidth law for several sample thicknesses.

Fig. 5
Fig. 5

Unshifted intensity record generated with a source spectral distribution (bandpass filter) centered at ~551.6 nm.

Fig. 6
Fig. 6

Variance of the intensity record in Fig. 5. Note that this curve closely resembles a Gaussian distribution.

Fig. 7
Fig. 7

Plot showing variance of the shifted intensity record.

Fig. 8
Fig. 8

Group refractive index measurements for sample 1, BK7.

Fig. 9
Fig. 9

Error between the measured values of phase refractive index and phase refractive index calculated from Schott dispersion coefficients.

Tables (1)

Tables Icon

Table I Standard Deviations of the Group Refractive Index Measurements at Each Wavelength (For Each Sample of Known Index)

Equations (45)

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δ ( ν , x ) = 2 [ n ( ν ) 1 ] t 2 x .
n ( ξ ) = n 0 + ξ n 1 + ξ 2 n 2 2 ! + ξ 3 n 3 3 ! + ,
δ ( ξ , x ) = 2 [ ( n 0 + n 1 ξ + n 2 ξ 2 2 + ) 1 ] t 2 x .
I ( ν ; δ ) = I 1 ( ν ) + I 2 ( ν ) + 2 I 1 I 2 cos ϕ ( ν , δ ) ,
ϕ ( ν ; δ ) = 2 π ν c δ ( ξ , x ) ,
I ( ν ; δ ) = 2 I 0 ( ν ) [ 1 + cos ϕ ( ν ; δ ) ] .
I ( x ) = 2 I 0 ( ν ) ( 1 + cos ϕ ) d ν .
I ( x ) = 2 f ( ξ ) { 1 + cos [ 2 π ( ξ + ν 0 ) c δ ( ξ ; x ) ] } d ξ .
I ( x ) = P + 2 f ( ξ ) cos { 4 π ( ξ + ν 0 ) c × [ ( ( n 0 + n 1 ξ + n 2 ξ 2 2 + ) 1 ) t x ] } d ξ,
P = f ( ξ ) d ξ .
A = ( 4 π c ) { [ ( n 0 + ν 0 n 1 1 ) t x ] ξ + [ ( n 0 1 ) t x ] ν 0 } + H ,
H = 4 π c [ n 1 ξ 2 t + ( ξ + ν 0 ) ( n 2 ξ 2 2 ! + n 3 ξ 3 3 ! + ) t ] .
δ 0 ( x ) = 2 ( n 0 1 ) t 2 x .
n g ( λ ) = n ( λ ) λ d n d λ ,
δ g 0 ( x ) = 2 ( n g 0 1 ) t 2 x .
A = 2 π [ τ g 0 ( x ) ξ + τ 0 ( x ) ν 0 ] ,
τ g 0 ( x ) = δ g 0 ( x ) / c ,
τ 0 ( x ) = δ 0 ( x ) / c .
I ( ξ ) = P + cos ( 2 π τ 0 ν 0 ) f ( ξ ) cos ( 2 π τ g 0 ξ ) d ξ sin ( 2 π τ 0 ν 0 ) f ( ξ ) sin ( 2 π τ g 0 ξ ) d ξ .
C ( τ g 0 ) = f ( ξ ) cos ( 2 π τ g 0 ( x ) ξ ) d ξ ,
S ( τ g 0 ) = f ( ξ ) sin ( 2 π τ g 0 ( x ) ξ ) d ξ ,
I ( ξ ) = P + C ( τ g 0 ) cos ( 2 π τ 0 ν 0 ) + S ( τ g 0 ) sin ( 2 π τ 0 ν 0 ) .
V = I MAX I MIN I MAX + I MIN .
tan ( 2 π τ 0 ν 0 ) = S ( τ g 0 ) C ( τ g 0 ) .
I MAX ( τ g 0 ) = P + C ( τ g 0 ) 2 + S ( τ g 0 ) 2 ,
I MIN ( τ g 0 ) = P C ( τ g 0 ) 2 + S ( τ g 0 ) 2 ,
V ( τ g 0 ) = C ( τ g 0 ) 2 + S ( τ g 0 ) 2 P .
ξ 2 t < c 200 π ( 2 n 1 + n 2 ν 0 ) .
ξ max 2 t < c 8 n 1 + 4 ν 0 n 2 .
n g ( ξ ) = n 0 + ν 0 n 1 + ( 2 n 1 + ν 0 n 2 ) ξ + ( n 2 2 ! ) ξ 2 + .
Δ τ g ( ξ ) = 2 c ( 2 n 1 + ν 0 n 2 ) ξ t .
Δ τ g max < τ c .
H rms = [ f ( ξ ) H 2 ( ξ ) δξ f ( ξ ) δξ ] 1 / 2 .
[ f ( ξ ) ξ 4 δξ f ( ξ ) δξ ] 1 / 2 t < c 5 ( 8 n 1 + 4 ν 0 n 2 ) .
ξ e 2 t < c 15 ( 4 n 1 + 2 ν 0 n 2 ) ,
n = p 0 + p 1 ω + p 2 ω 2 + a λ,
ω = λ λ 0 λ λ* .
x = ( λ* ω 2 2 λ 0 ω + λ 0 λ* λ 0 ) ,
y = [ 2 λ* ω 3 ( λ 3 + 3 λ 0 ) ω 2 + 2 λ 0 ω λ* λ 0 ] .
[ p 0 p 1 p 2 ] = [ Σ 1 Σ x Σ y Σ x Σ x 2 Σ x y Σ y Σ x y Σ y 2 ] 1 [ Σ n g Σ n g x Σ n g y ] ,
a = 2 . 348 × 10 5 + 2 . 172 × 10 5 p 0 + 4 . 691 × 10 4 p 1 + 1 . 063 × 10 3 p 2 ,
n g = Δ x t + 1 .
η = 4 λ .
V = a 0 + a 1 x + a 2 x 2
x MAX = a 1 2 a 2 .

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