Abstract

A simple and novel interferometric setup for wave-front testing that uses a cross slit is described. In this method, the test beam illuminates a cross slit placed at the front focal plane of a Fourier lens. It selects two orthogonal slices from the test beam, and the interference of these two beams is observed at a slightly defocused plane near the back focal plane. Fringes of different conical forms (circular, elliptical, or hyperbolic) so obtained can be used for testing a coherent wave front in general. The theory, supported by some experimental results, is presented. An application of the method to the study of the nature of asymmetry in the beam profile of a semiconductor diode laser beam is demonstrated.

© 2000 Optical Society of America

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References

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  1. R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers,” in Proceedings of the ICO Conference on Optical Methods in Scientific and Industrial Measurements (Jpn. J. Appl. Phys.14, Suppl. 1, 351–353 (1975).
  2. T. W. Cole, “Quasi-optical techniques of radio astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1977), Vol. XV, pp. 231–232.
  3. P. V. Avizonis, J. S. Fender, R. R. Butts, “Interference of diffracted multiple wave fronts in the geometrical shadow of Fresnel region,” Appl. Opt. 28, 163–172 (1989).
    [CrossRef] [PubMed]
  4. J. Xie, Y. Qiu, H. Ming, “Double exposure hologram of hyperbolic fringes and its use in the measurement of optical coherence,” Appl. Opt. 29, 4221–4224 (1990).
    [CrossRef] [PubMed]
  5. R. N. Chakraborty, S. K. Sarkar, A. Basuray, “Optical method of determination of stress at a point,” Opt. Lett. 22, 427–429 (1997).
    [CrossRef] [PubMed]
  6. K. Nitta, K. Itaya, M. Ishikawa, Y. Watanabe, G. Hatakoshi, Y. Ulematsu, “Astigmatism of ridge stripe InGaAlP laser diodes,” presented at the Twenty-First Conference on Solid State Devices and Material, Tokyo, 28–30 August 1989.
  7. J. Alda, D. Vazquez, E. Bernabeu, “Wavefront and amplitude profile for astigmatic beams in semiconductor lasers: analytical and graphical treatment,” J. Opt. (Paris) 19, 201–206 (1988).
    [CrossRef]
  8. M. Lang, “Correcting astigmatism in diode lasers,” Laser Optron. 8(9) , 51–55 (1989).
  9. W. T. Welford, Geometrical Optics: Optical Instruments (North-Holland, Amsterdam, 1962), p. 74.

1997 (1)

1990 (1)

1989 (2)

1988 (1)

J. Alda, D. Vazquez, E. Bernabeu, “Wavefront and amplitude profile for astigmatic beams in semiconductor lasers: analytical and graphical treatment,” J. Opt. (Paris) 19, 201–206 (1988).
[CrossRef]

Alda, J.

J. Alda, D. Vazquez, E. Bernabeu, “Wavefront and amplitude profile for astigmatic beams in semiconductor lasers: analytical and graphical treatment,” J. Opt. (Paris) 19, 201–206 (1988).
[CrossRef]

Avizonis, P. V.

Basuray, A.

Bernabeu, E.

J. Alda, D. Vazquez, E. Bernabeu, “Wavefront and amplitude profile for astigmatic beams in semiconductor lasers: analytical and graphical treatment,” J. Opt. (Paris) 19, 201–206 (1988).
[CrossRef]

Butts, R. R.

Chakraborty, R. N.

Cole, T. W.

T. W. Cole, “Quasi-optical techniques of radio astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1977), Vol. XV, pp. 231–232.

Fender, J. S.

Hatakoshi, G.

K. Nitta, K. Itaya, M. Ishikawa, Y. Watanabe, G. Hatakoshi, Y. Ulematsu, “Astigmatism of ridge stripe InGaAlP laser diodes,” presented at the Twenty-First Conference on Solid State Devices and Material, Tokyo, 28–30 August 1989.

Ishikawa, M.

K. Nitta, K. Itaya, M. Ishikawa, Y. Watanabe, G. Hatakoshi, Y. Ulematsu, “Astigmatism of ridge stripe InGaAlP laser diodes,” presented at the Twenty-First Conference on Solid State Devices and Material, Tokyo, 28–30 August 1989.

Itaya, K.

K. Nitta, K. Itaya, M. Ishikawa, Y. Watanabe, G. Hatakoshi, Y. Ulematsu, “Astigmatism of ridge stripe InGaAlP laser diodes,” presented at the Twenty-First Conference on Solid State Devices and Material, Tokyo, 28–30 August 1989.

Lang, M.

M. Lang, “Correcting astigmatism in diode lasers,” Laser Optron. 8(9) , 51–55 (1989).

Ming, H.

Nitta, K.

K. Nitta, K. Itaya, M. Ishikawa, Y. Watanabe, G. Hatakoshi, Y. Ulematsu, “Astigmatism of ridge stripe InGaAlP laser diodes,” presented at the Twenty-First Conference on Solid State Devices and Material, Tokyo, 28–30 August 1989.

Qiu, Y.

Sarkar, S. K.

Smartt, R. N.

R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers,” in Proceedings of the ICO Conference on Optical Methods in Scientific and Industrial Measurements (Jpn. J. Appl. Phys.14, Suppl. 1, 351–353 (1975).

Steel, W. H.

R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers,” in Proceedings of the ICO Conference on Optical Methods in Scientific and Industrial Measurements (Jpn. J. Appl. Phys.14, Suppl. 1, 351–353 (1975).

Ulematsu, Y.

K. Nitta, K. Itaya, M. Ishikawa, Y. Watanabe, G. Hatakoshi, Y. Ulematsu, “Astigmatism of ridge stripe InGaAlP laser diodes,” presented at the Twenty-First Conference on Solid State Devices and Material, Tokyo, 28–30 August 1989.

Vazquez, D.

J. Alda, D. Vazquez, E. Bernabeu, “Wavefront and amplitude profile for astigmatic beams in semiconductor lasers: analytical and graphical treatment,” J. Opt. (Paris) 19, 201–206 (1988).
[CrossRef]

Watanabe, Y.

K. Nitta, K. Itaya, M. Ishikawa, Y. Watanabe, G. Hatakoshi, Y. Ulematsu, “Astigmatism of ridge stripe InGaAlP laser diodes,” presented at the Twenty-First Conference on Solid State Devices and Material, Tokyo, 28–30 August 1989.

Welford, W. T.

W. T. Welford, Geometrical Optics: Optical Instruments (North-Holland, Amsterdam, 1962), p. 74.

Xie, J.

Appl. Opt. (2)

J. Opt. (Paris) (1)

J. Alda, D. Vazquez, E. Bernabeu, “Wavefront and amplitude profile for astigmatic beams in semiconductor lasers: analytical and graphical treatment,” J. Opt. (Paris) 19, 201–206 (1988).
[CrossRef]

Laser Optron. (1)

M. Lang, “Correcting astigmatism in diode lasers,” Laser Optron. 8(9) , 51–55 (1989).

Opt. Lett. (1)

Other (4)

K. Nitta, K. Itaya, M. Ishikawa, Y. Watanabe, G. Hatakoshi, Y. Ulematsu, “Astigmatism of ridge stripe InGaAlP laser diodes,” presented at the Twenty-First Conference on Solid State Devices and Material, Tokyo, 28–30 August 1989.

W. T. Welford, Geometrical Optics: Optical Instruments (North-Holland, Amsterdam, 1962), p. 74.

R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers,” in Proceedings of the ICO Conference on Optical Methods in Scientific and Industrial Measurements (Jpn. J. Appl. Phys.14, Suppl. 1, 351–353 (1975).

T. W. Cole, “Quasi-optical techniques of radio astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1977), Vol. XV, pp. 231–232.

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Figures (5)

Fig. 1
Fig. 1

Schematic optical layout: L, He–Ne laser; MO, microscope objective; PH, pinhole; CS, cross slit; FL, Fourier lens; BFP, back focal plane; OP, observation plane. Photographs a and b are described in the text.

Fig. 2
Fig. 2

Schematic layout depicting the photographs (a–g) that correspond to different locations of the observation plane and photographs (h and k) that correspond to cross-slit rotation by 45° (the discontinuity at the left indicates the same components up to the collimating lens as in Fig. 1). CL, cylindrical lens; CS, cross slit; FL, Fourier lens; BFP, back focal plane.

Fig. 3
Fig. 3

Fringe patterns observed for the experiment performed with the semiconductor diode laser. (a) Fringe pattern corresponding to the maximum divergence position for diode laser I. (b) Fringe pattern corresponding to 45° rotation of the cross slit. (c), (d) Same as (a) and (b), respectively, for diode laser II.

Fig. 4
Fig. 4

Fringe analysis for obtaining the optical path difference profile. (a) Thresholded version of Fig. 3(a). (b) Line profile (pixel position versus gray level) along the maximum divergence direction in (a). (c) Line profile along the direction orthogonal to that of (b).

Fig. 5
Fig. 5

Optical path difference (OPD) profile of a diode laser along (a) the maximum divergence direction and (b) the orthogonal direction. (The effective separation between the two foci is 2.5 µm.)

Equations (20)

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Au, v=Rectu/2pRectv/2q+Rectu/2qRectv/2p-Rectu/2pRectv/2p,
ax, y=K -- Au, vexp-i2πux+vydudv,
ax, y=K1sinc2πpxsinc2πqy+sinc2πqxsinc2πpy-p/qsinc2πpxsinc2πpy,
ax, y=K1sinc2πpxsinc2πqy+sinc2πqxsinc2πpy.
ax, y=k1sinc2πpxsinc2πqy+sinc2πqxsinc2πpy* 1z expi 2πλzx2+y2.
ax, y=Kzexpik y2z+expik x2z,
Ix, y=K2z2+2 coskx22z-ky22z.
kx22z-ky22z=2nπ
1m2x2-y2=1,
m2=2nλz.
ϕx, y=ax+by.
ϕx, y=ax2+bxy+cy2.
ϕx, y=ax3+bx2y+cxy2+dy3.
kx22z1-ky22z2=2nπ
x2m2-y2l2=1,
m2=2nλz1,  l2=2nλz2.
m2=2nλz1=2nλz0+Δz,
l2=2nλz2=2nλz0-Δz.
m2l2=z0+Δzz0-Δz.
ml1+Δzz0.

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