Abstract

A heterodyne interferometric system based on optical polarization is described. The method uses a simple rotating analyzer as the frequency shifter. The theory and the potential errors of the system are analyzed.

© 1983 Optical Society of America

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References

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  1. N. A. Massie, R. D. Nelson, S. Holly, Appl. Opt. 18, 1797 (1979).
    [CrossRef] [PubMed]
  2. R. Crane, Appl. Opt. 8, 538 (1969).
  3. G. E. Sommargren, J. Opt. Soc. Am. 65, 960 (1975).
    [CrossRef]
  4. R. N. Shagam, J. C. Wyant, Appl. Opt. 17, 3034 (1978).
    [CrossRef] [PubMed]
  5. G. E. Sommargren, Appl. Opt. 20, 610 (1981).
    [CrossRef] [PubMed]
  6. M. Balkanski, J. J. Hopfield, Phys. Status Solidi 2, 623 (1962).
    [CrossRef]
  7. J. C. Suits, Rev. Sci. Instrum. 42, 19 (1971).
    [CrossRef]
  8. We take the orthonormal Jones vectors [exp (iπ4)exp (-iπ4)] and [exp (-iπ4)exp (iπ4)] as basis vectors, which correspond to the right- and left-circular polarizations.
  9. R. C. Jones, J. Opt. Soc. Am. 32, 486 (1942).
    [CrossRef]
  10. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

1981 (1)

1979 (1)

1978 (1)

1975 (1)

1971 (1)

J. C. Suits, Rev. Sci. Instrum. 42, 19 (1971).
[CrossRef]

1969 (1)

R. Crane, Appl. Opt. 8, 538 (1969).

1962 (1)

M. Balkanski, J. J. Hopfield, Phys. Status Solidi 2, 623 (1962).
[CrossRef]

1942 (1)

Azzam, R. M. A.

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

Balkanski, M.

M. Balkanski, J. J. Hopfield, Phys. Status Solidi 2, 623 (1962).
[CrossRef]

Bashara, N. M.

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

Crane, R.

R. Crane, Appl. Opt. 8, 538 (1969).

Holly, S.

Hopfield, J. J.

M. Balkanski, J. J. Hopfield, Phys. Status Solidi 2, 623 (1962).
[CrossRef]

Jones, R. C.

Massie, N. A.

Nelson, R. D.

Shagam, R. N.

Sommargren, G. E.

Suits, J. C.

J. C. Suits, Rev. Sci. Instrum. 42, 19 (1971).
[CrossRef]

Wyant, J. C.

Appl. Opt. (4)

J. Opt. Soc. Am. (2)

Phys. Status Solidi (1)

M. Balkanski, J. J. Hopfield, Phys. Status Solidi 2, 623 (1962).
[CrossRef]

Rev. Sci. Instrum. (1)

J. C. Suits, Rev. Sci. Instrum. 42, 19 (1971).
[CrossRef]

Other (2)

We take the orthonormal Jones vectors [exp (iπ4)exp (-iπ4)] and [exp (-iπ4)exp (iπ4)] as basis vectors, which correspond to the right- and left-circular polarizations.

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

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Figures (2)

Fig. 1
Fig. 1

Heterodyne interferometer system based on polarization.

Fig. 2
Fig. 2

2J1(dR)/pR as a function of pR.

Equations (50)

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[ a o ] exp [ i ( ω o t + α ) ] test beam + [ o b ] exp ( i ω o t ) ref . beam ,
1 / 2 [ l - i - i l ] [ a o ] exp [ i ( ω o t + α ) ] = a 2 exp [ i ( ω o t + α - π 4 ) ] [ exp ( i π 4 ) exp ( - i π 4 ) ] right circularly polarized - test beam ,
1 / 2 [ l - i - i l ] [ o b ] exp ( i ω o t ) = b 2 exp [ i ( ω 0 t - π 4 ) ] [ exp ( - i π 4 ) exp ( i π 4 ) ] left circularly polarized - reference beam
test beam vector = [ cos 2 θ 1 2 sin 2 θ 1 2 sin 2 θ sin 2 θ ] [ exp ( i π 4 ) exp ( - i π 4 ) ] × a 2 exp [ i ( ω o t + α - π 4 ) ] ,
reference beam vector = [ cos 2 θ 1 2 sin 2 θ 1 2 sin 2 θ sin 2 θ ] [ exp ( - i π 4 ) exp ( i π 4 ) ] × b 2 exp [ i ( ω o t - π 4 ) ] ,
test beam vector = a 2 [ cos ω t sin ω t ] exp { i [ ( ω o - ω ) t + α ] } ,
reference beam vector = b 2 [ cos ω t sin ω t ] exp { i [ ( ω o + ω ) t - π 2 ] } .
[ E x E y ] = a 2 [ cos ω t sin ω t ] exp [ ( ω o - ω ) t + α ] + b 2 [ cos ω t sin ω t ] exp { i [ ( ω o + ω ) t - π 2 ] } , = 1 / 2 [ cos ω t sin ω t ] { a exp [ i ( - ω t + α ) ] + b exp [ i ( ω t - π 2 ) ] } exp ( i ω o t ) .
a exp [ i ( - ω t + α ) ] + b exp [ i ( ω t - π 2 ) ] = a 2 + b 2 + 2 a b sin ( 2 ω t - α ) exp ( i Θ ) ,
Θ = tan - 1 - a sin ( ω t - α ) - b cos ω t a cos ( ω t - α ) + b sin ω t ,
[ E x E y ] = 1 / 2 [ cos ω t sin ω t ] × a 2 + b 2 + 2 a b sin ( 2 ω t - α ) exp [ i ( ω o t + Θ ) ] .
I = a 2 + b 2 2 + a b sin ( 2 ω t - α ) .
[ cos δ 2 - i sin δ 2 - i sin δ 2 cos δ 2 ] .
δ = π 2 + γ ,
[ E x E y ] = [ cos δ 2 - i sin δ 2 - i sin δ 2 cos δ 2 ] [ a exp ( i g a ) b ] = [ a cos α cos δ 2 + i a sin α cos δ 2 - i b sin δ 2 a sin α sin δ 2 + b cos δ 2 - i a cos α sin δ 2 ] .
[ E r E l ] = 1 2 [ exp ( - i π 4 ) exp ( i π 4 ) exp ( i π 4 ) exp ( - i π 4 ) ] [ E x E y ] ,
[ E r E l ] and [ E x E y ]
[ E r E l ] = 1 2 exp ( - i π 4 ) [ a cos α ( sin δ 2 + cos δ 2 ) + i a sin α ( sin δ 2 + cos δ 2 ) - i b ( sin δ 2 - cos δ 2 ) a sin α ( sin δ 2 - cos δ 2 ) + b ( sin δ 2 + cos δ 2 ) - i a cos α ( sin δ 2 - cos δ 2 ) ]
E r E l = 2 a b cos α - i ( a 2 cos δ - 2 a b sin α sin δ - b 2 cos δ ) [ a sin α ( sin δ 2 - cos δ 2 ) + b ( sin δ 2 + cos δ 2 ) ] 2 + [ a cos α ( sin δ 2 - cos δ 2 ) ] 2 ,
tan α = tan α sin δ + cos δ cos α c tan 2 β ,
β = tan - 1 a b .
sin δ 1 - γ 2 2 ,
cos δ - γ .
tan α = tan α - γ ctan 2 β cos α - γ 2 2 tan α .
Δ α = α - α .
tan α = tan ( α + Δ ) tan α + Δ α cos 2 α ,
Δ α ( 1 ) = - γ cos α ctan 2 β             first - order error ,
Δ α ( 2 ) = - γ 2 4 sin 2 α             second - order error .
Δ max ( 2 ) = γ 2 2 .
[ 0 a ] and [ b 0 ]
[ cos δ 2 - 1 sin δ 2 - i sin δ 2 cos δ 2 ] [ 0 a ] exp ( i α ) = a exp ( i α ) [ - i sin δ 2 cos δ 2 ] ,
[ cos δ 2 i sin δ 2 - i sin δ 2 cos δ 2 ] [ b 0 ] = b [ cos δ 2 - i sin δ 2 ] .
Δ α ( 1 ) = - γ 1 cos α ctan 2 β             first - order error ,
Δ α ( 2 ) = - γ 1 2 sin 2 α             second - order error .
Δ α max ( 2 ) = - 2 γ 1 2 .
1 2 [ 1 - i sin 2 ɛ - i cos 2 ɛ - 1 cos 2 ɛ 1 + i sin 2 ɛ ] = 1 2 [ 1 - i 2 ɛ - i - i 1 + i 2 ɛ ] ( up to the first order ) .
[ - exp ( i α ) 0 ] and [ 0 b ]
[ E x E y ] = 1 2 [ 1 - i 2 ɛ - i - i 1 + i 2 ɛ ] [ a exp ( i α ) b ] = 1 2 [ a cos α + 2 ɛ a sin α - i ( b + 2 ɛ a cos α - a sin α ) a sin α + b - ( a cos α - 2 ɛ b ) ] .
[ E r E l ] = 2 exp ( - i π 4 ) [ a cos α + ɛ a sin α - ɛ b + i a ( sin α - ɛ cos α ) b + ɛ a cos α + i ɛ ( a sin α + b ) ] .
Δ α ( 1 ) = - 2 ɛ ( 1 + sin α cot 2 β ) ,
Δ α ( 1 ) = - 2 ɛ .
Δ α max ( 1 ) = 4 ɛ cot 2 β .
Δ α ( 1 ) max = { 4 ɛ 2 - ɛ 1 if a = b 4 ɛ 2 b 2 - ɛ 1 a 2 / a b if a b ,
I = ½ { a 2 + b 2 + 2 a b sin [ 2 ω t - α ( x , y ) ] } d x d y ,
α ( x , y ) = α o + p x ,
F = a b exp [ i ( 2 ω t - α o + π 2 ) ] × exp ( - i p x ) d x d y + ½ ( a 2 + b 2 ) π R 2 .
x = r cos η ,     y = r sin η             0 r R , 0 η 2 π ,
exp ( - i p x ) d x d y = o 2 π o R exp ( - i p r cos η ) × r d r d η = 2 π p R J 1 ( p R ) ,
I = π R 2 2 { a 2 + b 2 + 2 a b [ 2 J 1 ( p R ) p R ] · sin ( 2 ω t - α o ) } .
p R < 2.2 rad .

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