Abstract

A Michelson interferometer setup was used to determine refractive index and thickness of a fused-quartz sample with no knowledge of either parameter. At small angles, <10°, the interferometer equation follows a fourth-order polynomial in the sample refractive index alone, effectively decoupling the sample thickness from the equation. The incident angle of the He–Ne laser beam versus fringe shift was fitted to the polynomial, and its coefficients obtained. These were used to determine refractive index to within 6×104 of the known value with an accuracy of ±1.3%. Sample thickness was determined to an accuracy of ±2.5%. Reproducibility of the rotating table was determined to be ±2×103 degrees.

© 2007 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
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  7. "TIE-29: Refractive Index and Dispersion," in Technical Information--Optics for Devices (Schott North America, 2005), p. 5.
  8. Optical Glass Catalogue (Schott North America, 2006).
  9. N. K. Govil, H. N. Mhaskar, R. N. Mohapatra, Z. Nashed, and J. Szabados, Frontiers in Interpolation and Approximation, 1st ed. (Chapman and Hall/CRC, 2006), pp. 187-188.
  10. IBM Holographic Storage Team, "Optical data storage enters a new dimension," Phys. World 2000, 37-42.
  11. M. H. Yükselici, R. Ince, and A. T. Ince, "Data storage characteristics of iron doped LiNbO3 under a 90° geometry two-beam coupling configuration," Opt. Lasers Eng. 42, 277-287 (2004).
    [CrossRef]

2005

2004

M. H. Yükselici, R. Ince, and A. T. Ince, "Data storage characteristics of iron doped LiNbO3 under a 90° geometry two-beam coupling configuration," Opt. Lasers Eng. 42, 277-287 (2004).
[CrossRef]

2003

G. Coppola, P. Ferraro, M. Iodice, and S. De Nicola, "Method for measuring the refractive index and the thickness of transparent plates with a lateral-shear wavelength-scanning interferometer," Appl. Opt. 41, 3882-3887 (2003).
[CrossRef]

1999

1990

1971

Bashara, N. M.

Coppola, G.

G. Coppola, P. Ferraro, M. Iodice, and S. De Nicola, "Method for measuring the refractive index and the thickness of transparent plates with a lateral-shear wavelength-scanning interferometer," Appl. Opt. 41, 3882-3887 (2003).
[CrossRef]

De Nicola, S.

G. Coppola, P. Ferraro, M. Iodice, and S. De Nicola, "Method for measuring the refractive index and the thickness of transparent plates with a lateral-shear wavelength-scanning interferometer," Appl. Opt. 41, 3882-3887 (2003).
[CrossRef]

Ferraro, P.

G. Coppola, P. Ferraro, M. Iodice, and S. De Nicola, "Method for measuring the refractive index and the thickness of transparent plates with a lateral-shear wavelength-scanning interferometer," Appl. Opt. 41, 3882-3887 (2003).
[CrossRef]

Fukano, T.

Gillen, G. D.

Govil, N. K.

N. K. Govil, H. N. Mhaskar, R. N. Mohapatra, Z. Nashed, and J. Szabados, Frontiers in Interpolation and Approximation, 1st ed. (Chapman and Hall/CRC, 2006), pp. 187-188.

Guha, S.

Ibrahim, M. M.

Ince, A. T.

M. H. Yükselici, R. Ince, and A. T. Ince, "Data storage characteristics of iron doped LiNbO3 under a 90° geometry two-beam coupling configuration," Opt. Lasers Eng. 42, 277-287 (2004).
[CrossRef]

Ince, R.

M. H. Yükselici, R. Ince, and A. T. Ince, "Data storage characteristics of iron doped LiNbO3 under a 90° geometry two-beam coupling configuration," Opt. Lasers Eng. 42, 277-287 (2004).
[CrossRef]

Iodice, M.

G. Coppola, P. Ferraro, M. Iodice, and S. De Nicola, "Method for measuring the refractive index and the thickness of transparent plates with a lateral-shear wavelength-scanning interferometer," Appl. Opt. 41, 3882-3887 (2003).
[CrossRef]

Jenkins, F. A.

F. A. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, 2001).

Kihara, T.

Mhaskar, H. N.

N. K. Govil, H. N. Mhaskar, R. N. Mohapatra, Z. Nashed, and J. Szabados, Frontiers in Interpolation and Approximation, 1st ed. (Chapman and Hall/CRC, 2006), pp. 187-188.

Mohapatra, R. N.

N. K. Govil, H. N. Mhaskar, R. N. Mohapatra, Z. Nashed, and J. Szabados, Frontiers in Interpolation and Approximation, 1st ed. (Chapman and Hall/CRC, 2006), pp. 187-188.

Nashed, Z.

N. K. Govil, H. N. Mhaskar, R. N. Mohapatra, Z. Nashed, and J. Szabados, Frontiers in Interpolation and Approximation, 1st ed. (Chapman and Hall/CRC, 2006), pp. 187-188.

Szabados, J.

N. K. Govil, H. N. Mhaskar, R. N. Mohapatra, Z. Nashed, and J. Szabados, Frontiers in Interpolation and Approximation, 1st ed. (Chapman and Hall/CRC, 2006), pp. 187-188.

Team, IBM Holographic Storage

IBM Holographic Storage Team, "Optical data storage enters a new dimension," Phys. World 2000, 37-42.

White, H. E.

F. A. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, 2001).

Yamaguchi, I.

Yokomori, K.

Yükselici, M. H.

M. H. Yükselici, R. Ince, and A. T. Ince, "Data storage characteristics of iron doped LiNbO3 under a 90° geometry two-beam coupling configuration," Opt. Lasers Eng. 42, 277-287 (2004).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am.

Opt. Lasers Eng.

M. H. Yükselici, R. Ince, and A. T. Ince, "Data storage characteristics of iron doped LiNbO3 under a 90° geometry two-beam coupling configuration," Opt. Lasers Eng. 42, 277-287 (2004).
[CrossRef]

Phys. World

IBM Holographic Storage Team, "Optical data storage enters a new dimension," Phys. World 2000, 37-42.

Other

F. A. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, 2001).

"TIE-29: Refractive Index and Dispersion," in Technical Information--Optics for Devices (Schott North America, 2005), p. 5.

Optical Glass Catalogue (Schott North America, 2006).

N. K. Govil, H. N. Mhaskar, R. N. Mohapatra, Z. Nashed, and J. Szabados, Frontiers in Interpolation and Approximation, 1st ed. (Chapman and Hall/CRC, 2006), pp. 187-188.

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Figures (5)

Fig. 1
Fig. 1

Change in the optical path of the laser beam due to rotation of the sample on a rotation stage. θ t = angle of refraction, θ i = angle of incidence.

Fig. 2
Fig. 2

Geometry involved in rotation of the sample.

Fig. 3
Fig. 3

Michelson interferometer experimental setup.

Fig. 4
Fig. 4

Variation of computed refractive index of sample versus initial incident angle.

Fig. 5
Fig. 5

Graph of fringe shift versus angle in radians.

Tables (2)

Tables Icon

Table 1 Angle of Rotation θ i Versus Number of Fringes Passing, Δm, Passing the Field of View

Tables Icon

Table 2 Computed Refractive Index of Sample for a Decreasing Range of Data Points

Equations (115)

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< 10 °
6 × 10 4
± 1.3 %
± 2.5 %
± 2 × 10 3
31 ° C
d n / d T
11.9 × 10 6 / K
Δ Λ
Δ m λ = Δ Λ .
Λ = ( n s × r s ) + ( n a × r a ) ,
r s
n s
r a
n a
One   way   change   in   optical   path = ( n s × Δ r s ) + ( n a × Δ r a ) ,
Δ r s
Δ r a
Change   in   optical   on   return = Δ Λ = 2 [ ( n s × Δ r s ) + ( n a × Δ r a ) ] .
I n c r e a s e i n g e o m e t r i c p a t h o f s a m p l e = Δ r s = t / cos θ t t ,
Δ r s = t ( cos θ t 1 1 ) ,
Δ r s = t [ ( 1 sin 2 θ t ) 1 / 2 1 ] .
sin θ t
n a sin θ i = n s sin θ t
n a , n s =
θ i
θ t
sin θ t = ( n air / n sample ) sin θ i ,
sin θ t = n sin θ i ,
n = n a / n s
Δ r s
Δ r s = t [ ( 1 n 2 sin 2 θ i ) 1 / 2 1 ] .
Δ r = t / cos θ i t ,
Δ r = t ( 1 / cos θ i 1 ) ,
Δ r = t [ ( 1 sin 2 θ i ) 1 / 2 1 ] .
γ = 90 θ i ,
ϕ = [ 180 ( 90 θ i ) ( θ i θ t ) ] = 90 + θ t .
r / sin ( θ i θ t ) = ( t / cos θ i ) × [ 1 / sin ( 90 + θ t ) ] ,
r = [ t sin ( θ i θ t ) / cos θ i ] × [ 1 / sin ( 90 + θ t ) ] .
w = r sin θ i ,
w = [ t sin ( θ i θ t ) tan θ i ] / sin ( 90 + θ t ) ,
w = t sin ( θ i θ t ) tan θ i / cos θ t ,
w = [ t sin ( θ i θ t ) tan θ i ] / ( 1 sin 2 θ t ) 1 / 2 .
w = [ t sin ( θ i θ t ) tan θ i ] / ( 1 n 2 sin 2 θ i ) 1 / 2 .
Δ r a = w Δ r = [ t sin ( θ i θ t ) tan θ i ] / ( 1 n 2 sin 2 θ i ) 1 / 2 t [ ( 1 sin 2 θ i ) 1 / 2 1 ] .
Δ Λ = 2 n s t [ ( 1 / ( 1 n 2 sin 2 θ i ) 1 / 2 1 ) ] + 2 n a t { [ sin ( θ i θ t ) tan θ i ] / ( 1 n 2 sin 2 θ i ) 1 / 2 [ 1 / ( 1 sin 2 θ i ) 1 / 2 1 ] } .
Δ m = 2 n s t λ [ 1 / ( 1 n 2 sin 2 θ i ) 1 / 2 1 ] + 2 n a t λ { [ sin ( θ i θ t ) tan θ i ] / ( 1 n 2 sin 2 θ i ) 1 / 2 [ 1 / ( 1 sin 2 θ i ) 1 / 2 1 ] } .
θ i
( < 10 ° ) sin θ i tan θ i θ i
θ t = n θ i
Δ m = 2 n s t λ [ 1 / ( 1 n 2 θ i 2 ) 1 / 2 1 ] + 2 n a t λ { [ ( θ i n θ i ) θ i ] / ( 1 n 2 θ i 2 ) 1 / 2 [ 1 / ( 1 θ i 2 ) 1 / 2 1 ] } .
Δ m = 2 n s t λ [ ( 1 n 2 θ i 2 ) ( 1 / 2 ) 1 ] + 2 n a t λ { θ i ( θ i n θ i ) × ( 1 n 2 θ i 2 ) ( 1 / 2 ) [ ( 1 θ i 2 ) ( 1 / 2 ) 1 ] } .
θ i
θ t
Δ m = 2 n s t λ [ ( 1 + 1 2 n 2 θ i 2 ) 1 ] + 2 n a t λ { θ i ( θ i n θ i ) × ( 1 + 1 2 n 2 θ i 2 ) [ ( 1 + 1 2 θ i 2 ) 1 ] } ,
Δ m = 2 n s t λ [ 1 2 n 2 θ i 2 ] + 2 n a t λ [ θ i ( θ i n θ i ) ( 1 + 1 2 n 2 θ i 2 ) ( 1 2 θ i 2 ) ] ,
Δ m = 2 t λ [ 1 2 n s n 2 θ i 2 ] + 2 t λ [ ( n a θ i 2 n a n θ i 2 ) ( 1 + 1 2 n 2 θ i 2 ) ( 1 2 n a θ i 2 ) ] ,
Δ m = 2 t λ [ ( 1 2 n s n 2 θ i 2 + 1 2 n a θ i 2 n a n θ i 2 + 1 2 n a n 2 θ i 4 1 2 n a n 3 θ i 4 ) ] ,
Δ m = 2 t λ [ ( 1 2 n s n 2 + 1 2 n a n a n ) θ i 2 + ( 1 2 n a n 2 1 2 n a n 3 ) θ i 4 ] ,
Δ m = 2 t λ [ ( 1 2 n s n a 2 n s 2 + 1 2 n a n a n ) θ i 2 + ( 1 2 n a n 2 1 2 n a n 3 ) θ i 4 ] ,
Δ m = t λ [ ( n s n a 2 n s 2 + n a 2 n a n ) θ i 2 + ( n a n 2 n a n 3 ) θ i 4 ] ,
Δ m = t λ [ ( n a n a n ) θ i 2 + ( n a n 2 n a n 3 ) θ i 4 ] ,
Δ m = [ t λ n a ( 1 n ) ] θ i 2 + [ t λ n a ( 1 n ) ] n 2 θ i 4 .
y = A x 2 + A n 2 x 4
A n 2
n 2
Δ m
θ i
n 2
n a
n s
Δ m
θ i
y = A x 2 + B x 4
B = A n 2
n 2 ( n = n air / n sample )
n sample
θ i
Δ m
θ i
< 200
Δ m = 200 400
n + 1
n sample = 1.4578
n sample = 1.4578
Δ m
θ i
Δ m = 250 400
n sample
30 ° C
6 × 10 4
n L
Δ λ = λ 2 / 2 L
10 3   nm
λ n
n L = 2 L / λ n
20   cm
± 2 × 10 3
n sample
± 0.02
Δ m
θ i
n sample = 1.4578
± 0.37   mm
30.30 ( ± 0.76 )  mm
29 .96 ( ± 0 .01 )  mm
± 0 .02%
6 × 10 4
± 1.4 %
< 0.5 %
< 0.28 % ( ± 0.004 )
n air / n sample = B / A
n sample = n air / n
θ t =
θ i =

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