Abstract

We present a collimation technique based on a double grating system to locate with high accuracy an emitter in the focal plane of a lens. Talbot self-images are projected onto the second grating producing moiré interferences. By means of two photodetectors positioned just behind the second grating, it is possible to determine the optimal position of the light source for collimation by measuring the phase shift between the signals over the two photodetectors. We obtain mathematical expressions of the signal in terms of defocus. This allows us to perform an automated technique for collimation. In addition, a simple and accurate visual criterion for collimating a light source using a lens is proposed. Experimental results that corroborate the proposed technique are also presented.

© 2010 Optical Society of America

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  1. J. Jahns and A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination)” Opt. Commun. 28, 263–267 (1979).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  7. Ch. Siegel, F. Loewenthal, and J. E. Balmer, “A wavefront sensor based on the fractional Talbot effect,” Opt. Commun. 194, 265–275 (2001).
    [CrossRef]
  8. C. L. Hou and J. Bai, “Wavefront measurement for long focal large aperture lens based on Talbot effect of Ronchi grating,” J. Phys.: Conf. Ser. 48, 1037–1041 (2006).
    [CrossRef]
  9. S. Chang, “Geometrical aberrations of self.imaged line gratings,” Optik (Jena) 116, 379–389 (2005).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  15. S. Rana, S. Prakash, and S. Prakash, “Automated collimation testing in Lau interferometry using phase shifting technique,” Opt. Lasers Eng. 47, 656–661 (2009).
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    [CrossRef]
  19. L. M. Sanchez-Brea, J. Saenz-Landete, J. Aloso, and E. Bernabeu, “Invariant grating pseudo-imaging using polychromatic light and finite extension source,” Appl. Opt. 47, 1470–1477 (2008).
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2009

S. Rana, S. Prakash, and S. Prakash, “Automated collimation testing in Lau interferometry using phase shifting technique,” Opt. Lasers Eng. 47, 656–661 (2009).
[CrossRef]

2008

2006

C. L. Hou and J. Bai, “Wavefront measurement for long focal large aperture lens based on Talbot effect of Ronchi grating,” J. Phys.: Conf. Ser. 48, 1037–1041 (2006).
[CrossRef]

2005

S. Chang, “Geometrical aberrations of self.imaged line gratings,” Optik (Jena) 116, 379–389 (2005).
[CrossRef]

2001

Ch. Siegel, F. Loewenthal, and J. E. Balmer, “A wavefront sensor based on the fractional Talbot effect,” Opt. Commun. 194, 265–275 (2001).
[CrossRef]

1997

1992

J. P. Bétend-Bon, L. Wosinski, and M. Breidne, “Double grating phase stepping interferometry for testing aspherics,” Pure Appl. Opt. 1, 55–69 (1992).
[CrossRef]

1991

1990

S. C. Som and A. Satpathi, “The generalized Lau effect,” J. Mod. Opt. 37, 1215–1226 (1990).
[CrossRef]

1989

K. Patorski, “The self-imaging phenomenon and its applications,” Prog. Opt. 27, 3–108 (1989).

1988

1983

1982

1979

S. Szapiel and K. Patorski, “Fresnel diffraction images of periodic objects under Gaussian beam illumination,” Opt. Acta 26, 439–446 (1979).
[CrossRef]

J. Jahns and A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination)” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

1976

1975

Aggarwal, A. K.

Aloso, J.

Avudainayagam, K. V.

Bai, J.

C. L. Hou and J. Bai, “Wavefront measurement for long focal large aperture lens based on Talbot effect of Ronchi grating,” J. Phys.: Conf. Ser. 48, 1037–1041 (2006).
[CrossRef]

Balmer, J. E.

Ch. Siegel, F. Loewenthal, and J. E. Balmer, “A wavefront sensor based on the fractional Talbot effect,” Opt. Commun. 194, 265–275 (2001).
[CrossRef]

Bernabeu, E.

Bétend-Bon, J. P.

J. P. Bétend-Bon, L. Wosinski, and M. Breidne, “Double grating phase stepping interferometry for testing aspherics,” Pure Appl. Opt. 1, 55–69 (1992).
[CrossRef]

Bhattacharya, J. C.

Breidne, M.

J. P. Bétend-Bon, L. Wosinski, and M. Breidne, “Double grating phase stepping interferometry for testing aspherics,” Pure Appl. Opt. 1, 55–69 (1992).
[CrossRef]

Chang, S.

S. Chang and S. I. Lee, “First-order aberration of a misfocused self-imaging system,” Optik (Jena) 119, 742–748 (2008).
[CrossRef]

S. Chang, “Geometrical aberrations of self.imaged line gratings,” Optik (Jena) 116, 379–389 (2005).
[CrossRef]

Chitralekha, S.

K. V. Avudainayagam and S. Chitralekha, “Lau effect and beam collimation,” J. Mod. Opt. 44, 175–178 (1997).
[CrossRef]

Cohen-Sabban, Y.

Hou, C. L.

C. L. Hou and J. Bai, “Wavefront measurement for long focal large aperture lens based on Talbot effect of Ronchi grating,” J. Phys.: Conf. Ser. 48, 1037–1041 (2006).
[CrossRef]

Jahns, J.

J. Jahns and A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination)” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

Joeux, D.

Joyeux, D.

Lee, S. I.

S. Chang and S. I. Lee, “First-order aberration of a misfocused self-imaging system,” Optik (Jena) 119, 742–748 (2008).
[CrossRef]

Liu, L.

Loewenthal, F.

Ch. Siegel, F. Loewenthal, and J. E. Balmer, “A wavefront sensor based on the fractional Talbot effect,” Opt. Commun. 194, 265–275 (2001).
[CrossRef]

Lohmann, A. W.

J. Jahns and A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination)” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

Nayagam, S. Chitra

Ohnishi, K.

Patorski, K.

K. Patorski, “The self-imaging phenomenon and its applications,” Prog. Opt. 27, 3–108 (1989).

S. Szapiel and K. Patorski, “Fresnel diffraction images of periodic objects under Gaussian beam illumination,” Opt. Acta 26, 439–446 (1979).
[CrossRef]

K. Patorski, S. Yokozeki, and T. Suzuki, “Collimation test by double grating shearing interferometer,” Appl. Opt. 15, 1234–1240, (1976).
[CrossRef] [PubMed]

Prakash, S.

S. Rana, S. Prakash, and S. Prakash, “Automated collimation testing in Lau interferometry using phase shifting technique,” Opt. Lasers Eng. 47, 656–661 (2009).
[CrossRef]

S. Rana, S. Prakash, and S. Prakash, “Automated collimation testing in Lau interferometry using phase shifting technique,” Opt. Lasers Eng. 47, 656–661 (2009).
[CrossRef]

Rana, S.

S. Rana, S. Prakash, and S. Prakash, “Automated collimation testing in Lau interferometry using phase shifting technique,” Opt. Lasers Eng. 47, 656–661 (2009).
[CrossRef]

Saenz-Landete, J.

Sanchez-Brea, L. M.

Satpathi, A.

S. C. Som and A. Satpathi, “The generalized Lau effect,” J. Mod. Opt. 37, 1215–1226 (1990).
[CrossRef]

Siegel, Ch.

Ch. Siegel, F. Loewenthal, and J. E. Balmer, “A wavefront sensor based on the fractional Talbot effect,” Opt. Commun. 194, 265–275 (2001).
[CrossRef]

Som, S. C.

S. C. Som and A. Satpathi, “The generalized Lau effect,” J. Mod. Opt. 37, 1215–1226 (1990).
[CrossRef]

Suzuki, T.

Szapiel, S.

S. Szapiel and K. Patorski, “Fresnel diffraction images of periodic objects under Gaussian beam illumination,” Opt. Acta 26, 439–446 (1979).
[CrossRef]

Wosinski, L.

J. P. Bétend-Bon, L. Wosinski, and M. Breidne, “Double grating phase stepping interferometry for testing aspherics,” Pure Appl. Opt. 1, 55–69 (1992).
[CrossRef]

Yokozeki, S.

Appl. Opt.

J. Mod. Opt.

S. C. Som and A. Satpathi, “The generalized Lau effect,” J. Mod. Opt. 37, 1215–1226 (1990).
[CrossRef]

K. V. Avudainayagam and S. Chitralekha, “Lau effect and beam collimation,” J. Mod. Opt. 44, 175–178 (1997).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys.: Conf. Ser.

C. L. Hou and J. Bai, “Wavefront measurement for long focal large aperture lens based on Talbot effect of Ronchi grating,” J. Phys.: Conf. Ser. 48, 1037–1041 (2006).
[CrossRef]

Opt. Acta

S. Szapiel and K. Patorski, “Fresnel diffraction images of periodic objects under Gaussian beam illumination,” Opt. Acta 26, 439–446 (1979).
[CrossRef]

Opt. Commun.

Ch. Siegel, F. Loewenthal, and J. E. Balmer, “A wavefront sensor based on the fractional Talbot effect,” Opt. Commun. 194, 265–275 (2001).
[CrossRef]

J. Jahns and A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination)” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

Opt. Lasers Eng.

S. Rana, S. Prakash, and S. Prakash, “Automated collimation testing in Lau interferometry using phase shifting technique,” Opt. Lasers Eng. 47, 656–661 (2009).
[CrossRef]

Optik (Jena)

S. Chang and S. I. Lee, “First-order aberration of a misfocused self-imaging system,” Optik (Jena) 119, 742–748 (2008).
[CrossRef]

S. Chang, “Geometrical aberrations of self.imaged line gratings,” Optik (Jena) 116, 379–389 (2005).
[CrossRef]

Pure Appl. Opt.

J. P. Bétend-Bon, L. Wosinski, and M. Breidne, “Double grating phase stepping interferometry for testing aspherics,” Pure Appl. Opt. 1, 55–69 (1992).
[CrossRef]

Other

K. Patorski, “The self-imaging phenomenon and its applications,” Prog. Opt. 27, 3–108 (1989).

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Figures (8)

Fig. 1
Fig. 1

Double grating collimating system with the light source (a) located at the focal point of lens z 0 = f and (b) placed at z 0 = f ± Δ z .

Fig. 2
Fig. 2

Effect of defocusing: optical intensity over detector. The light source travels from z 0 = f 20 μm to z 0 = f + 20 μm . Parameters of the analysis are p = 20 μm , λ = 650 nm , w = 1 mm , x a = 4 mm , f = 10 mm , z 1 = 1 mm , and z 2 = z T = p 2 / λ . The intensity remains almost constant, showing the validity of our approach.

Fig. 3
Fig. 3

Theoretical signals S A (solid curve) and S B (dashed curve) using n = n = 1 , 0, 1 orders. The parameters for analysis are the same as in Fig. 2: (a) Δ z = 1 μm and (b) Δ z = 10 μm .

Fig. 4
Fig. 4

Theoretical phase shift angle between S A and S B when the light source travels from z 0 = f 25 μm to z 0 = f + 25 μm . The parameters for the analysis are the same as in Fig. 2.

Fig. 5
Fig. 5

Theoretical Lissajous curves using n = n = 1 , 0, 1 orders for different light source locations: (a) Δ z = 1 μm , (b) Δ z = 10 μm , (c) Δ z = 1 μm , (d) Δ z = 10 μm . The rest of the parameters are the same as in Fig. 2.

Fig. 6
Fig. 6

Schematic of the experimental setup.

Fig. 7
Fig. 7

Experimental Lissajous curves obtained for two different defocus distances Δ z : (a) Δ z = 1 μm and (b) Δ z = 10 μm . Crosses, experimental data; solid curve, theoretical prediction, Eq. (10).

Fig. 8
Fig. 8

Experimental phase shift angle between signals S A and S B when the light source is not at focus but is at a certain distance Δ z : circles, experimental data; solid curve, theoretical prediction.

Equations (11)

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t 2 ( x ) = n a n exp [ i q n ( x Δ x 2 ) ] , t 3 ( x ) = m b m exp [ i q m ( x Δ x 3 ) ] ,
U 1 ( x 1 ) = U 0 e i k ( f + Δ z ) f + Δ z e i k 2 ( f + Δ z ) x 1 2 ,
U 1 ( x 1 ) U 0 e i k ( f + Δ z ) f exp ( i k Δ z 2 f 2 x 1 2 ) .
U 2 ( x 2 ) = e i k z 1 i λ z U 1 ( x 1 ) e i k 2 z 1 ( x 2 x 1 ) 2 d x 1 U 0 e i k ( f + Δ z + z 1 ) i f e i k α x 2 2 / 2 ,
U 3 ( x 3 ) = e i k z 3 i λ z 3 U 2 ( x 2 ) e i k 2 z 2 ( x 3 x 2 ) 2 d x 2 = U 0 e i k ( Δ z + z 1 + z 2 ) i f e i k 2 z 2 x 3 2 1 1 + 1 / α z 2 n a n e i q n Δ x 2 e i q 2 n 2 2 k z 2 1 + α z 2 e i q n x 3 1 + α z 2 .
I 3 ( x 3 ) = U 3 ( x 3 ) U 3 * ( x 3 ) = I 0 f n n a n a n e i q ( n n ) Δ x 2 e i q 1 + α z 2 ( n n ) x 3 e i q 2 2 k ( n 2 n 2 ) z 2 1 + α z 2 ,
I 3 ( x 3 ) = I 0 f n n m a n a n * b m e i q [ ( n n ) Δ x 2 + m Δ x 3 ] e i q 2 2 k ( n 2 n 2 ) z 2 1 + α z 2 e i q [ ( n n ) 1 + α z 2 + m ] x 3 .
S A ( Δ x 2 ) = x a w / 2 x a + w / 2 I 3 ( x 3 ) d x 3 , S B ( Δ x 2 ) = x a w / 2 x a + w / 2 I 3 ( x 3 ) d x 3 ,
S A , B ( Δ x 2 ) = I 0 f n n m a n a n * b m exp [ i q ( n n ) Δ x 2 ] e i q m Δ x 3 A , B e i q 2 2 k ( n 2 n 2 ) z 2 1 + α z 2 e ± i q x a [ ( n n ) 1 + α z 2 + m ] sinc [ q w 2 ( ( n n ) 1 + α z 2 + m ) ] ,
S A , B ( Δ x 2 ) I 0 f n n a n a n * b ( n n ) e i q ( n n ) ( Δ x 2 Δ x 3 A , B ) e i q 2 2 k ( n 2 n 2 ) z 2 e i q x a α z 2 ( n n ) .
corr ( f , g ) = f ( x ) g ( x + u ) d u ,

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