Abstract

An ac ellipsometer has been designed and built to study the feasibility of ellipsometric measurement of gradient-index profiles. Faraday effect polarization modulation and a phase-locking detection system are used to monitor spatial changes in index of refraction while absolute index of refraction is determined through simple ellipsometric procedures. This technique is found to be viable for measurement of commonly encountered refractive-index gradients in both glass and infrared transmitting materials.

© 1984 Optical Society of America

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References

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  1. L. G. Atkinson, S. N. Houde-Walter, D. T. Moore, D. P. Ryan, J. M. Stagaman, Appl. Opt. 21, 993 (1982).
    [CrossRef] [PubMed]
  2. J. D. Forer, S. N. Houde-Walter, J. J. Miceli, D. T. Moore, M. J. Nadeau, D. P. Ryan, J. M. Stagaman, N. J. Sullo, Appl. Opt. 22, 407, (1983).
    [CrossRef] [PubMed]
  3. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1977).
  4. J. Monin, G. A. Boutry, Nouv. Rev. Opt. 4, 159 (1973).
    [CrossRef]
  5. H. J. Mathieu, D. E. McClure, R. H. Muller, Rev. Sci. Instrum. 45, 798 (1974).
    [CrossRef]
  6. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), pp. 619–620.
  7. W. A. Shurcliff, Polarized Light (Harvard U.P.Cambridge, 1962).
  8. D. C. Leiner, D. T. Moore, Rev. Sci. Instrum. 49, 1701 (1978).
    [CrossRef]
  9. G. W. Johnson, D. C. Leiner, D. T. Moore, Opt. Eng. 18, 46 (1979).
    [CrossRef]

1983 (1)

1982 (1)

1979 (1)

G. W. Johnson, D. C. Leiner, D. T. Moore, Opt. Eng. 18, 46 (1979).
[CrossRef]

1978 (1)

D. C. Leiner, D. T. Moore, Rev. Sci. Instrum. 49, 1701 (1978).
[CrossRef]

1974 (1)

H. J. Mathieu, D. E. McClure, R. H. Muller, Rev. Sci. Instrum. 45, 798 (1974).
[CrossRef]

1973 (1)

J. Monin, G. A. Boutry, Nouv. Rev. Opt. 4, 159 (1973).
[CrossRef]

Atkinson, L. G.

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1977).

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1977).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), pp. 619–620.

Boutry, G. A.

J. Monin, G. A. Boutry, Nouv. Rev. Opt. 4, 159 (1973).
[CrossRef]

Forer, J. D.

Houde-Walter, S. N.

Johnson, G. W.

G. W. Johnson, D. C. Leiner, D. T. Moore, Opt. Eng. 18, 46 (1979).
[CrossRef]

Leiner, D. C.

G. W. Johnson, D. C. Leiner, D. T. Moore, Opt. Eng. 18, 46 (1979).
[CrossRef]

D. C. Leiner, D. T. Moore, Rev. Sci. Instrum. 49, 1701 (1978).
[CrossRef]

Mathieu, H. J.

H. J. Mathieu, D. E. McClure, R. H. Muller, Rev. Sci. Instrum. 45, 798 (1974).
[CrossRef]

McClure, D. E.

H. J. Mathieu, D. E. McClure, R. H. Muller, Rev. Sci. Instrum. 45, 798 (1974).
[CrossRef]

Miceli, J. J.

Monin, J.

J. Monin, G. A. Boutry, Nouv. Rev. Opt. 4, 159 (1973).
[CrossRef]

Moore, D. T.

Muller, R. H.

H. J. Mathieu, D. E. McClure, R. H. Muller, Rev. Sci. Instrum. 45, 798 (1974).
[CrossRef]

Nadeau, M. J.

Ryan, D. P.

Shurcliff, W. A.

W. A. Shurcliff, Polarized Light (Harvard U.P.Cambridge, 1962).

Stagaman, J. M.

Sullo, N. J.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), pp. 619–620.

Appl. Opt. (2)

Nouv. Rev. Opt. (1)

J. Monin, G. A. Boutry, Nouv. Rev. Opt. 4, 159 (1973).
[CrossRef]

Opt. Eng. (1)

G. W. Johnson, D. C. Leiner, D. T. Moore, Opt. Eng. 18, 46 (1979).
[CrossRef]

Rev. Sci. Instrum. (2)

D. C. Leiner, D. T. Moore, Rev. Sci. Instrum. 49, 1701 (1978).
[CrossRef]

H. J. Mathieu, D. E. McClure, R. H. Muller, Rev. Sci. Instrum. 45, 798 (1974).
[CrossRef]

Other (3)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), pp. 619–620.

W. A. Shurcliff, Polarized Light (Harvard U.P.Cambridge, 1962).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1977).

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Figures (11)

Fig. 1
Fig. 1

Amplitude of ac signal at frequency ω (PSMA); n = 1.5, θi = 45°, α = 45°.

Fig. 2
Fig. 2

Amplitude of ac signal at frequency ω (PMSA); n = 1.5, θi = 45°, α = 45°.

Fig. 3
Fig. 3

Ratio of output modulation amplitude to input modulation amplitude: n = 1.5.

Fig. 4
Fig. 4

Schematic of optical system.

Fig. 5
Fig. 5

Block diagram of electronics.

Fig. 6
Fig. 6

Profile of ZnSe–ZnS gradient, position 1.

Fig. 7
Fig. 7

Profile of ZnSe–ZnS gradient, position 2.

Fig. 8
Fig. 8

Repeatability of ZnSe–ZnS profile, position 1.

Fig. 9
Fig. 9

Repeatability of ZnSe–ZnS profile, position 2.

Fig. 10
Fig. 10

Profile of Corning perform gradient.

Fig. 11
Fig. 11

Profile of University of Rochester radial gradient.

Tables (6)

Tables Icon

Table I Effect of Error in θi on n; (θi = 45°)

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Table II Effect of Error in α and ψr on n; (θi = 45°, α = 45°)

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Table III Predicted Error in Relative Index

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Table IV Absolute Index Measurements; (θi = 45°)

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Table V Absolute Index Measurements (Calibrated θi)

Tables Icon

Table VI Predicted Error in Absolute Index

Equations (30)

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r p = n 2 cos θ i - n 1 cos θ t n 2 cos θ i + n 1 cos θ t = tan ( θ i - θ t ) tan ) θ i + θ t ) , r s = n 1 cos θ i - n 2 cos θ t n 1 cos θ i + n 2 cos θ t = - sin ( θ i - θ t ) sin ( θ i + θ t ) ,
tan ψ = r s r p ,
Δ = δ p - δ s ,
tan ψ = r s exp ( - i δ s ) r p exp ( - i δ p ) , or             tan ψ - ρ exp ( i Δ ) .
1 = n 2 ( 1 - κ 2 ) = sin 2 θ i + sin 2 θ i tan 2 θ i ( cos 2 2 ψ - sin 2 2 ψ sin 2 Δ ) ( 1 + sin 2 ψ cos Δ ) 2 , 2 = 2 n 2 κ = sin 2 θ i tan 2 θ i ( 2 sin 2 ψ cos 2 ψ sin Δ ) ( 1 + sin 2 ψ cos Δ ) 2 .
n 2 = sin 2 θ i + sin 2 θ i tan 2 θ i ( cos 2 2 ψ ) ( 1 sin 2 ψ ) 2 ,
( tan ψ ) exp ( - i Δ ) = r s r p + E r s / E i s E r p / E i p ,
tan ψ = tan ψ r exp ( i Δ ) tan α .
θ = V H l ,
[ 1 0 0 0 0 cos δ sin δ 0 0 ± sin δ cos δ 0 0 0 0 1 ] ,
[ ( r p 2 + r s 2 ) ( r p 2 - r s 2 ) 0 0 ( r p 2 - r s 2 ) ( r p 2 + r s 2 ) 0 0 0 0 2 r p r s 0 0 0 0 2 r p r s ] reflection matrix             · [ 1 cos 2 α sin 2 α 0 ] linearly polarized light at α = [ r p 2 + r s 2 + cos 2 α ( r p 2 - r s 2 ) r p 2 - r s 2 + cos 2 α ( r p 2 + r s 2 ) 2 r p r s sin 2 α 0 ] .
[ 1 0 0 0 0 cos δ - sin δ 0 0 sin δ cos δ 0 0 0 0 1 ] Faraday rotator matrix · [ r p 2 + r s 2 + cos 2 α ( r p 2 - r s 2 ) r p 2 - r s 2 + cos 2 α ( r p 2 + r s 2 ) 2 r p r s sin 2 α 0 ] = [ r p 2 + r s 2 + cos 2 α ( r p 2 - r s 2 ) cos δ [ r p 2 - r s 2 + cos 2 α ( r p 2 + r s 2 ) ] - 2 r p r s sin δ sin 2 α sin δ [ r p 2 - r s 2 + cos 2 α ( r p 2 + r s 2 ) ] + 2 r p r s cos δ sin 2 α 0 ] .
tan 2 β = sin δ [ r p 2 - r s 2 + cos 2 α ( r p 2 + r s 2 ) ] + 2 r p r s cos δ sin 2 α cos δ [ r p 2 - r s 2 + cos 2 α ( r p 2 + r s 2 ) ] - 2 r p r s sin δ sin 2 α .
[ 1 cos 2 θ sin 2 θ 0 cos 2 θ cos 2 2 θ cos 2 θ sin 2 θ 0 sin 2 θ cos 2 θ sin 2 θ sin 2 2 θ 0 0 0 0 0 ] analyzer at azimuth θ · [ r p 2 + r s 2 + cos 2 α ( r p 2 - r s 2 ) cos δ [ r p 2 - r s 2 + cos 2 α ( r p 2 + r s 2 ) ] - 2 r p r s sin δ sin 2 α sin δ [ r p 2 - r s 2 + cos 2 α ( r p 2 + r s 2 ) ] + 2 r p r s cos δ sin 2 α ] 0 ] .
I = r p 2 + r s 2 + cos 2 α ( r p 2 - r s 2 ) + cos 2 θ { cos δ [ r p 2 - r s 2 + cos 2 α ( r p 2 + r s 2 ) ] - 2 r p r s sin δ sin 2 α } + sin 2 θ { sin δ [ r p 2 - r s 2 + cos 2 α ( r p 2 + r s 2 ) ] + 2 r p r s cos δ sin 2 α } .
I = r p 2 + r s 2 + cos 2 α ( r p 2 - r s 2 ) + cos 2 θ { cos ( A sin ω t ) [ r p 2 - r s 2 + cos 2 α ( r p 2 + r s 2 ) ] - 2 r p r s sin 2 α sin ( A sin ω t ) } + sin 2 θ { sin ( A sin ω t ) [ r p 2 - r s 2 + cos 2 α ( r p 2 + r s 2 ) ] + 2 r p r s sin 2 α cos ( A sin ω t ) } .
I = r p 2 + r s 2 + cos 2 α ( r p 2 - r s 2 ) + cos 2 θ { [ J 0 ( A ) + 2 J 2 ( A ) cos 2 ω t ] [ r p 2 - r s 2 + cos 2 α ( r p 2 + r s 2 ) ] - 2 r p r s sin 2 α [ 2 J 1 ( A ) sin ω t ] } + sin 2 θ { [ 2 J 1 ( A ) sin ω t ] [ r p 2 - r s 2 + cos 2 α ( r p 2 + r s 2 ) ] + 2 r p r s sin 2 α [ J 0 ( A ) + 2 J 2 ( A ) cos 2 ω t ] } .
- 4 r p r s sin 2 α cos 2 θ J 1 ( A ) + 2 [ r p 2 - r s 2 + cos 2 α ( r p 2 + r s 2 ) ] sin 2 θ J 1 ( A ) .
2 r p r s sin 2 α cos 2 θ = [ r p 2 - r s 2 + cos 2 α ( r p 2 + r s 2 ) ] sin 2 θ
tan 2 θ = 2 r p r s tan θ r p 2 - r s 2 tan 2 α , θ = tan - 1 [ r s r p tan α ] or θ = tan - 1 [ - r p r s tan α ] ,
cos 2 θ 2 J 2 ( A ) [ r p 2 - r s 2 + cos 2 α ( r p 2 + r s 2 ) ] + sin 2 θ 2 J 2 ( A ) 2 r p r s sin 2 α .
2 J 2 ( A ) [ 2 r p 4 cos 2 α + 2 r s 4 sin 2 α tan 2 α + 4 r p 2 r s 2 sin 2 α r p 2 + r s 2 tan 2 α ] ,
2 J 2 ( A ) r p 2 + r s 2 tan 2 α [ - 2 r p 4 cos 2 α - 2 r s 4 sin 2 α tan 2 α - 4 r p 2 r s 2 sin 2 α ] .
- ( r p 2 - r s 2 ) 2 J 1 ( A ) sin 2 α - cos 2 θ ( r p 2 + r s 2 ) 2 J 1 ( A ) sin 2 α + sin 2 θ ( 2 r p r s ) 2 J 1 ( A ) cos 2 α .
2 r p r s sin 2 θ cos 2 α - ( r p 2 + r s 2 ) cos 2 θ sin 2 α - ( r p 2 - r s 2 ) sin 2 α = 0.
θ = tan - 1 ( - r p r s tan α ) .
d ψ r d α = ± tan ψ cos 2 ψ r cos 2 α .
n 2 = sin 2 θ i + sin 2 θ i tan 2 θ i cos 2 ( 2 ψ ) ( 1 + sin 2 ψ ) 2 ,
d n = | cos 2 ( 2 ψ ) n [ 1 + sin 2 ψ ] 2 [ sin 2 θ i tan θ i + tan 3 θ i ] + sin θ i cos θ i | d θ i + | 2 sin 2 θ i tan 2 θ i n [ - cos 2 ψ sin 2 ψ [ 1 + sin 2 ψ ] - cos 3 ( 2 ψ ) [ 1 + sin 2 ψ ] 3 ] | d ψ .
tan ψ = tan ψ r tan α , d ψ = | 1 + tan 2 ψ tan 2 α sec 2 ψ tan α | d ψ r + | sin ψ cos ψ sin α cos α | d α .

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