Abstract

We present a method for measuring the thickness of volume holograms by analyzing the variations of the diffraction efficiency as a function of angle of incidence. This method can be justified theoretically within the Born approximation for gratings with small modulation. But we prove experimentally that the method can work surprisingly well even for strongly modulated volume holograms. The principle of the method consists of the determination of angles of incidence for which the first-order diffraction efficiency takes its extreme (maxima and minima) values: these angles are related to the thickness of the grating.

© 1992 Optical Society of America

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References

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  1. D. Gabor, G. W. Stroke, “The theory of deep holograms,” Proc. R. Soc. London Ser. A 304, 275–289 (1968).
    [Crossref]
  2. J. G. van der Corput, “Zur Methode der Stationären Phase I,” Compositio Math. 1, 15–38 (1936); “Zur Methode der Stationären Phase II,” Compositio Math. 3, 328–372 (1936).
  3. L. Solymar, D. J. Cooke, Volume Holography and Volume Gratings (Academic, New York, 1981).
  4. E. A. de Oliveira, J. Frejlich, “Thickness and refractive index dispersion measurement in a thin film using the Haidinger interferometer,” Appl. Opt. 28, 1382–1386 (1989).
    [Crossref] [PubMed]

1989 (1)

1968 (1)

D. Gabor, G. W. Stroke, “The theory of deep holograms,” Proc. R. Soc. London Ser. A 304, 275–289 (1968).
[Crossref]

1936 (1)

J. G. van der Corput, “Zur Methode der Stationären Phase I,” Compositio Math. 1, 15–38 (1936); “Zur Methode der Stationären Phase II,” Compositio Math. 3, 328–372 (1936).

Cooke, D. J.

L. Solymar, D. J. Cooke, Volume Holography and Volume Gratings (Academic, New York, 1981).

de Oliveira, E. A.

Frejlich, J.

Gabor, D.

D. Gabor, G. W. Stroke, “The theory of deep holograms,” Proc. R. Soc. London Ser. A 304, 275–289 (1968).
[Crossref]

Solymar, L.

L. Solymar, D. J. Cooke, Volume Holography and Volume Gratings (Academic, New York, 1981).

Stroke, G. W.

D. Gabor, G. W. Stroke, “The theory of deep holograms,” Proc. R. Soc. London Ser. A 304, 275–289 (1968).
[Crossref]

van der Corput, J. G.

J. G. van der Corput, “Zur Methode der Stationären Phase I,” Compositio Math. 1, 15–38 (1936); “Zur Methode der Stationären Phase II,” Compositio Math. 3, 328–372 (1936).

Appl. Opt. (1)

Compositio Math. (1)

J. G. van der Corput, “Zur Methode der Stationären Phase I,” Compositio Math. 1, 15–38 (1936); “Zur Methode der Stationären Phase II,” Compositio Math. 3, 328–372 (1936).

Proc. R. Soc. London Ser. A (1)

D. Gabor, G. W. Stroke, “The theory of deep holograms,” Proc. R. Soc. London Ser. A 304, 275–289 (1968).
[Crossref]

Other (1)

L. Solymar, D. J. Cooke, Volume Holography and Volume Gratings (Academic, New York, 1981).

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Figures (5)

Fig. 1
Fig. 1

Diffraction by a phase grating.

Fig. 2
Fig. 2

Holographic grating profile analyzer setup: A.S. afocal system; G1, holographic grating; G2, goniometer; P, photodiode.

Fig. 3
Fig. 3

Structure of the modulation of two holographic phase gratings (pictures made by optical microscopy): (a) grating with small modulation amplitude; (b) grating with strong modulation. The strength of the modulation cannot be seen by optical microscopy.

Fig. 4
Fig. 4

Intensity of the light diffracted by a weakly modulated grating with respect to the scanning coordinates: (a) α = 0°; (b) α is the Bragg angle; (c) α = 34°; (d) α = 34° (order 1 only; enlarged); (e) α = 50° (order 1 only; enlarged).

Fig. 5
Fig. 5

Intensities of the light diffracted by a strongly modulated grating with respect to the scanning coordinates: (a) α = 0°; (b) α = 4°50′ (Bragg incidence); (c) α = 34° (first minimum); (d) α = 45° (first maximum).

Tables (4)

Tables Icon

Table I Gratings with Small Modulation

Tables Icon

Table II Gratings with Strong Modulation

Tables Icon

Table III Reproduced from Ref. 4

Tables Icon

Table IV Thickness by Direct Microscopic Measurement

Equations (18)

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( x , y , z ) = f ( a x + b y + c z ) ,
( x , y , z ) = f [ 2 π λ ( a x + b y + c z ) ] ,
a = λ d sin ω ,             b = 0 ,             c = λ d cos ω .
electric field : E = [ 0 , v ( x , z ) , 0 ] , magnetic field : H = ( - 1 i k 0 v z , 0 , 1 i k 0 v x ) .
D = C n = - n = + a n medium exp [ 2 i π λ ( n p + κ · r + r ) ] r d x d y d z ,
r = [ ( X - x ) 2 + ( Y - y ) 2 + ( Z - z ) 2 ] 1 / 2 .
r r 0 - cos θ z .
D = n = - + C a n x , y 0 z h × exp ( 2 i π λ { [ n p 0 ( x ) + κ x x + r 0 ( x ) ] + ( n c + κ z - cos θ ) z } ) r 0 d x d y d z ,
D = - n = - + C a n i λ 2 π x , y exp [ 2 i π λ ( n p 0 + κ x x + κ y y + r 0 ) ] - exp { 2 i π λ [ ( n p 0 + κ x x + κ y y + r 0 ) + ( n c + κ z - cos θ ) h ] } r 0 ( n c + κ z - cos θ ) d x d y .
C 0 E 0 a n · i λ [ exp ( 2 i π λ φ 0 ) r 0 Δ φ det φ 0 - exp ( 2 i π λ φ 1 ) r 0 Δ φ det φ 1 ] ,
φ 0 = n p 0 ( x ) + κ x x + r 0 ( x ; X , Y , Z ) ,             φ 1 = φ 0 + Δ φ ,
grad ϕ 0 = 0.
F = - 2 π C 0 E 0 a n r 0 [ det φ 0 ( m 0 ) ] 1 / 2 · exp { 2 i π λ [ φ 0 ( m 0 ) + 1 2 h χ ( m 0 ) ] } × sin [ π λ h χ ( m 0 ) ] π λ h χ ( m 0 )
2 π C 0 E 0 a n
exp { 2 i π λ [ φ 0 ( m 0 ) + 1 2 h χ ( m 0 ) ] } r 0 [ det φ 0 ( m 0 ) ] 1 / 2
sin [ π λ h χ ( m 0 ) ] π λ h χ ( m 0 ) .
π λ h χ 0 = 0 ,             π λ h χ 1 = π ,
sin ( π λ h x ) π λ h x ,

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