Abstract

Recently reported experimental results on the rotation sensitivity of Lau fringes to the spatial coherence of the source have been theoretically analyzed and explained on the basis of coherence theory. A theoretical plot of the rotation angle required for the Lau fringes to vanish is obtained as a function of the coherence length of the illumination used in the Lau experiment. The theoretical results compare well with the experimental observations. The analysis as well as the experiment could form the basis for a simple and easy measurement of the coherence length of the illumination in a plane.

© 1990 Optical Society of America

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References

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  1. E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys. 6, 417–423 (1948).
    [CrossRef]
  2. J. O. Castaneda, E. E. Sicre, “Theta Modulation Decoder Based on the Lau effect,” Opt. Commun. 59, 87–91 (1986).
    [CrossRef]
  3. S. Chitralekha, K. V. Avudainayagam, S. V. Pappu, “Rotation Sensitivity of Lau fringes: an Analysis Based on Coherence Theory,” Opt. Laser Technol., in press.
  4. S. Chitralekha, K. V. Avudainayagam, S. V. Pappu, “Role of Spatial Coherence on the Rotation Sensitivity of Lau Fringes: an Experimental Study,” Appl. Opt. 28, 345–349 (1989).
    [CrossRef] [PubMed]
  5. J. W. Goodman, Statistical Optics (Academic, New York, 1985), pp. 207–222.

1989 (1)

1986 (1)

J. O. Castaneda, E. E. Sicre, “Theta Modulation Decoder Based on the Lau effect,” Opt. Commun. 59, 87–91 (1986).
[CrossRef]

1948 (1)

E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys. 6, 417–423 (1948).
[CrossRef]

Avudainayagam, K. V.

S. Chitralekha, K. V. Avudainayagam, S. V. Pappu, “Role of Spatial Coherence on the Rotation Sensitivity of Lau Fringes: an Experimental Study,” Appl. Opt. 28, 345–349 (1989).
[CrossRef] [PubMed]

S. Chitralekha, K. V. Avudainayagam, S. V. Pappu, “Rotation Sensitivity of Lau fringes: an Analysis Based on Coherence Theory,” Opt. Laser Technol., in press.

Castaneda, J. O.

J. O. Castaneda, E. E. Sicre, “Theta Modulation Decoder Based on the Lau effect,” Opt. Commun. 59, 87–91 (1986).
[CrossRef]

Chitralekha, S.

S. Chitralekha, K. V. Avudainayagam, S. V. Pappu, “Role of Spatial Coherence on the Rotation Sensitivity of Lau Fringes: an Experimental Study,” Appl. Opt. 28, 345–349 (1989).
[CrossRef] [PubMed]

S. Chitralekha, K. V. Avudainayagam, S. V. Pappu, “Rotation Sensitivity of Lau fringes: an Analysis Based on Coherence Theory,” Opt. Laser Technol., in press.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Academic, New York, 1985), pp. 207–222.

Lau, E.

E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys. 6, 417–423 (1948).
[CrossRef]

Pappu, S. V.

S. Chitralekha, K. V. Avudainayagam, S. V. Pappu, “Role of Spatial Coherence on the Rotation Sensitivity of Lau Fringes: an Experimental Study,” Appl. Opt. 28, 345–349 (1989).
[CrossRef] [PubMed]

S. Chitralekha, K. V. Avudainayagam, S. V. Pappu, “Rotation Sensitivity of Lau fringes: an Analysis Based on Coherence Theory,” Opt. Laser Technol., in press.

Sicre, E. E.

J. O. Castaneda, E. E. Sicre, “Theta Modulation Decoder Based on the Lau effect,” Opt. Commun. 59, 87–91 (1986).
[CrossRef]

Ann. Phys. (1)

E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys. 6, 417–423 (1948).
[CrossRef]

Appl. Opt. (1)

Opt. Commun. (1)

J. O. Castaneda, E. E. Sicre, “Theta Modulation Decoder Based on the Lau effect,” Opt. Commun. 59, 87–91 (1986).
[CrossRef]

Other (2)

S. Chitralekha, K. V. Avudainayagam, S. V. Pappu, “Rotation Sensitivity of Lau fringes: an Analysis Based on Coherence Theory,” Opt. Laser Technol., in press.

J. W. Goodman, Statistical Optics (Academic, New York, 1985), pp. 207–222.

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Figures (4)

Fig. 1
Fig. 1

Schematic diagram of the conventional Lau experiment: EWLS, extended white light source; G1,G2, gratings; P1,P2, planes in which G1 and G2 are located; Z0, separation distance between the gratings; L, lens; f, focal length of lens L; P3 observation plane for Lau fringes.

Fig. 2
Fig. 2

Variation of the rotation angle θv with transverse coherence length lc. Curve 1, experimental (nonuniform illumination); curve 2, experimental (uniform illumination); curve 3, theoretical (uniform illumination).

Fig. 3
Fig. 3

Schematic diagrams for setups used in carrying out (a) the Lau experiment and (b) Young’s double slit experiment: HNLA, He–Ne laser; RD, rotating diffuser; SD, static diffuser; G1,G2, gratings; P1,P2, planes in which G1 and G2 are located; L, lens; f, focal length of lens L; P3, observation plane for Lau fringes; D, distance between the diffusers; Z0, separation distance between the gratings; d0, separation distance between G2 and lens L; P, plane in which the double slit is placed (this plane corresponds to the plane of grating G1); S, observation plane for Young’s fringes.

Fig. 4
Fig. 4

Variation of contrast coefficient with rotation angle θ for the coherence lengths: (a) lc ≅ 50 μm and (b) lc ≅ 800 μm.

Tables (3)

Tables Icon

Table I Experimental Data on the Variation of Rotation Angle (θv) with Transverse Coherence Length (lc)

Tables Icon

Table II Theoretical Data on the Variation of Rotation Angle (θv) with Transverse Coherence Length (lc)

Tables Icon

Table III Theoretical Data on Variation of the Contrast Coefficient with Rotation Angle (θ) for Two Different Coherence Lengths

Equations (20)

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W G 1 - ( x ¯ , y ¯ , Δ x , Δ y ) = I w ( x ¯ , y ¯ ) μ ( Δ x , Δ y ) = I w exp [ - a ( Δ x 2 + Δ y 2 ) ] ,
W G 1 + ( x ¯ , y ¯ , Δ x , Δ y ) = g ( x 1 ) g * ( x 2 ) W G 1 - ( x ¯ , y ¯ , Δ x , Δ y ) ,
g ( x 1 ) g * ( x 2 ) = g ( x 1 ) when Δ x = α p ; α = 0 , 1 , 2 = 0 otherwise ,
W G 1 + ( x ¯ , y ¯ , Δ x , Δ y ) = g ( x ¯ - Δ x / 2 ) I w exp [ - a ( Δ x 2 + Δ y 2 ) ] .
W G 2 - ( u ¯ , v ¯ , Δ u , Δ v ) = [ 1 / ( λ z 0 ) 2 ] exp [ - ( j 2 π / λ z 0 ) ( u ¯ Δ u + v ¯ Δ v ) ] × m = - α = - c m ( - 1 ) m α ( π / a ) 1 / 2 I w × exp [ - a α 2 p 2 ] exp [ j 2 π α p u ¯ / ( λ z 0 ) ] × exp { - π 2 v ¯ 2 / [ a ( λ z 0 ) 2 ] } × δ ( Δ v ) δ [ ( m λ z 0 / p ) - Δ u ] ,
W G 2 + ( u ¯ , v ¯ , Δ u , Δ v , θ ) = W G 2 - ( u ¯ , v ¯ , Δ u , Δ v ) t ( u ¯ - Δ u / 2 , v ¯ - Δ v / 2 , θ ) × t * ( u ¯ + Δ u / 2 , v ¯ + Δ v / 2 , θ ) ,
t ( u ¯ - Δ u / 2 , v ¯ - Δ v / 2 , θ ) t * ( u ¯ + Δ u / 2 , v ¯ + Δ v / 2 , θ ) δ ( Δ v ) = t ( u ¯ - Δ u / 2 , v ¯ , θ ) if Δ u = β p / cos θ ; β = 0 , 1 , 2 = 0 otherwise ,
W G 2 + ( u ¯ , v ¯ , Δ u , Δ v , θ ) = m = - α = - C m ( - 1 ) m α ( π / a ) 1 / 2 I w × exp ( - a α 2 p 2 ) exp ( - j 2 π m u ¯ / p ) × exp ( j 2 π α p u ¯ / λ z 0 ) exp [ - ( π v / λ z 0 ) 2 / a ] × t ( u ¯ - Δ u / 2 , v , θ ) δ ( Δ v ) δ [ ( m λ z 0 / p ) - Δ u ] ,
Δ u = m λ z 0 / p = β p / cos θ .
W G 2 + ( u ¯ , v ¯ , Δ u , Δ v , θ ) = m = - α = - n = - × C m C n ( - 1 ) m α ( - 1 ) n β ( π / a ) 1 / 2 I w × exp [ - a ( α p ) 2 ] exp { - j 2 π u ¯ [ m / p - ( α p / λ z 0 ) ] } × exp [ - ( π v / λ z 0 ) 2 / a ] × exp [ - ( j 2 π n / p ) ( u ¯ cos θ + v sin θ ) ] × δ ( Δ v ) δ [ ( m λ z 0 / p ) - Δ u ] .
W f ( r 1 , s 1 , r 2 , s 2 ) = exp ( j Φ ) ( λ f ) 2 - W G 2 + ( u ¯ , v , Δ u , Δ v , θ ) × exp [ - ( j 2 π / λ f ) ( u 1 r 1 + v 1 s 1 ) ] × exp [ ( j 2 π / λ f ) ( u 2 r 2 + v 2 s 2 ) ] d u 1 d u 2 d v 1 d v 2 ,
Φ = ( π / λ f ) ( 1 - d 0 / f ) [ ( r 1 2 - r 2 2 ) + ( s 1 2 - s 2 2 ) ] .
I f ( r , s ) = - W G 2 + ( u ¯ , v , Δ u , Δ v , θ ) exp [ - j 2 π Δ u r / ( λ f ) ] × exp [ - j 2 π Δ v s / λ f ] d u ¯ d v d Δ u d Δ v .
I f = m = - α = - n = - C m C n ( - 1 ) m α + n β [ N z 0 p / ( λ f 2 ) ] I w × exp [ - a ( α p ) 2 ] sinc { π N [ α p 2 / ( λ z 0 ) - ( m + n cos θ ) ] } × exp [ - a ( n sin θ z 0 λ / p ) 2 ] exp [ - j 2 π m z 0 r / p f ] ,
I f = m = - α = - n = - C m C n ( - 1 ) m ( α + n ) ( N z 0 p / λ f 2 ) × I w exp [ - α p 2 ( α 2 + n 2 tan 2 θ ) ] × sinc { π N [ m + ( n - α ) cos θ ] } exp [ - j 2 π m z 0 r / ( p f ) ] .
I f = m = - α = - n = - [ z 0 N p / ( λ f 2 ) ] C m C n ( - 1 ) m ( α + n ) × I w exp [ - a p 2 ( α 2 + ( n tan θ ) 2 ] × sinc { π N m ( 1 - cos θ ) } exp { - j 2 π m z 0 r / ( p f ) } .
dc z 0 N p I w C 0 2 / ( λ f 2 )
exp [ - a α 2 p 2 ( 1 + tan 2 θ ) ] = 1 for α = 0 = negligible otherwise .
I f K C 0 2 - 2 K sinc [ π N ( 1 - cos = θ ) ] × { C 0 C 1 exp ( - a p 2 ) + C 1 2 exp ( - a p 2 tan 2 θ ) + C 1 C 2 exp [ - a p 2 ( 1 + 4 tan 2 θ ) ] } cos ( 2 π m z 0 r / p f ) ,
I f K C 0 2 - 2 K C 1 2 sinc [ π N ( 1 - cos θ ) ] × exp [ - 4 ( p 2 / l c 2 ) tan 2 θ ] cos ( 2 π z 0 r / p f ) ,

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