Abstract

We demonstrate that the beam-propagation method can be employed to calculate the electric-field amplitude inside and outside a grating structure excited by an arbitrary incident field. We establish the accuracy of the method by comparing our results for constant-period gratings with results obtained from other theoretical methods. Subsequently, we analyze thick-focusing gratings with the beam-propagation method.

© 1982 Optical Society of America

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References

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  1. L. Thylén and D. Yevick, “Beam-propagation method in anisotropic media,” Appl. Opt. (to be published).
  2. J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
    [CrossRef]
  3. M. D. Feit and J. A. Fleck, “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990–3998 (1978).
    [CrossRef] [PubMed]
  4. D. Marcuse, Light Transmission Optics (Van Nostrand Reinholt, New York, 1972), pp. 61–72.
  5. M. G. Moharam and L. Young, “Criterion for Bragg and Raman–Nath diffraction regimes,” Appl. Opt. 17, 1757–1759 (1978).
    [CrossRef] [PubMed]
  6. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
    [CrossRef]
  7. D. Yevick and L. Thylén, “A numerical analysis of grating structures,” presented at the International School of Quantum Electronics Course on Integrated Optics, Erice, Italy, August 1981.
  8. L. Thylén and L. Stensland, “Lensless integrated optics spectrum analyzer,” IEEE J. Quantum Electron. QE-18, 381–385 (1982).
    [CrossRef]
  9. J. Van Roey and P. Lagasse, “Coupled wave analysis of obliquely incident waves in thin film gratings,” Appl. Opt. 20, 423–429 (1981).
    [CrossRef] [PubMed]

1982 (1)

L. Thylén and L. Stensland, “Lensless integrated optics spectrum analyzer,” IEEE J. Quantum Electron. QE-18, 381–385 (1982).
[CrossRef]

1981 (1)

1978 (2)

1976 (1)

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

Feit, M. D.

M. D. Feit and J. A. Fleck, “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990–3998 (1978).
[CrossRef] [PubMed]

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Fleck, J. A.

M. D. Feit and J. A. Fleck, “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990–3998 (1978).
[CrossRef] [PubMed]

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

Lagasse, P.

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinholt, New York, 1972), pp. 61–72.

Moharam, M. G.

Morris, J. R.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Stensland, L.

L. Thylén and L. Stensland, “Lensless integrated optics spectrum analyzer,” IEEE J. Quantum Electron. QE-18, 381–385 (1982).
[CrossRef]

Thylén, L.

L. Thylén and L. Stensland, “Lensless integrated optics spectrum analyzer,” IEEE J. Quantum Electron. QE-18, 381–385 (1982).
[CrossRef]

D. Yevick and L. Thylén, “A numerical analysis of grating structures,” presented at the International School of Quantum Electronics Course on Integrated Optics, Erice, Italy, August 1981.

L. Thylén and D. Yevick, “Beam-propagation method in anisotropic media,” Appl. Opt. (to be published).

Van Roey, J.

Yevick, D.

D. Yevick and L. Thylén, “A numerical analysis of grating structures,” presented at the International School of Quantum Electronics Course on Integrated Optics, Erice, Italy, August 1981.

L. Thylén and D. Yevick, “Beam-propagation method in anisotropic media,” Appl. Opt. (to be published).

Young, L.

Appl. Opt. (3)

Appl. Phys. (1)

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

IEEE J. Quantum Electron. (1)

L. Thylén and L. Stensland, “Lensless integrated optics spectrum analyzer,” IEEE J. Quantum Electron. QE-18, 381–385 (1982).
[CrossRef]

Other (3)

D. Yevick and L. Thylén, “A numerical analysis of grating structures,” presented at the International School of Quantum Electronics Course on Integrated Optics, Erice, Italy, August 1981.

L. Thylén and D. Yevick, “Beam-propagation method in anisotropic media,” Appl. Opt. (to be published).

D. Marcuse, Light Transmission Optics (Van Nostrand Reinholt, New York, 1972), pp. 61–72.

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Figures (7)

Fig. 1
Fig. 1

Wave-vector-space electric-field modulus distributions for a low-efficiency index grating graphed at 25-, 50-, and 75-μm propagation distances. A and B correspond to Δn = 1 × 10−3 (normal incidence) and Δn = 0.5 × 10−3 (Bragg-angle incidence), respectively.

Fig. 2
Fig. 2

Evolution of wave-vector-space electric-field modulus distribution along a grating for Bragg-angle incidence. Total energy transfer is achieved after the theoretical coupling length for the phase-matched wave-vector components. Second-order effects appear since the grating is not operated fully in the Bragg regime. The beam width in B is much larger than that in A. The grating period is 5 μm.

Fig. 3
Fig. 3

Schematic representation of a linearly chirped index grating. S and I designate conjugate points with respect to the grating; T is the thickness of the grating.

Fig. 4
Fig. 4

Spatial electric-field amplitude distribution at planes at different axial positions near the nominal focus of a focusing grating of thickness 200 μm (A) and 500 μm (B). The curves labeled 1,2, 3, and 4 correspond to propagation distances of 15, 16, 17, and 18 mm, respectively, from the grating exit plane. The graph also shows the diffracted order corresponding to a diverging beam. Here h is set equal to 50 mm and hI = 16.667 mm, corresponding to a grating focal length of 12.5 mm.

Fig. 5
Fig. 5

Increase of efficiency obtained by tilting the grating such that the Bragg condition is met at the grating midpoint. Parameters as in Fig. 4, except that the spatial window is 1200 μm. The curves labeled 1, 2, 3, and 4 correspond to propagation distances of 15, 16, 17, and 18 mm, respectively, from the grating exit plane.

Fig. 6
Fig. 6

Increase of resolution obtained by widening the Gaussian beam to 440 μm from 160 μm. The electric field is graphed at distances 1, 15 mm; 2, 17 mm; and 3, 19 mm from the grating exit face along the z axis. Note the reduction in the depth of focus.

Fig. 7
Fig. 7

The effect of using a beam with a constant intensity distribution over a finite (600-μm) window on the spatial-amplitude distribution 17 mm from the grating exit face (near the nominal focus). Note the Fresnel-type diffraction pattern of the zeroth-order beam and the sinc-type side lobes of the diffracted focused beam.

Tables (1)

Tables Icon

Table 1 Comparison of Diffraction Efficiencies as Obtained from the Beam-Propagation Method and from the Perturbation Method of Ref. 4

Equations (7)

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2 E + ω 2 c 2 n 2 ( x , y , z ) E = 0 ,
n ( x ¯ ) = n + δ n ( x ¯ ) ,
E ( x , y , Δ z ) = exp [ - i Δ z ( 2 + ( ω c ) 2 ( n + δ n ) 2 ) 1 / 2 ] E ( x , y , 0 ) .
[ 2 + ( ω c ) 2 ( n + δ n ) 2 ] 1 / 2 ~ [ 2 + ( ω c ) 2 n 2 ] 1 / 2 + ω c δ n .
( 2 + k 2 ) 1 / 2 = 2 ( 2 + k 2 ) 1 / 2 + k + k .
E ( x , y , z ) = exp ( - i k z ) ( x , y , z ) ,
( x , y , Δ z ) = exp { - i Δ z 2 [ 2 ( 2 + k 2 ) 1 / 2 + k ] } × exp ( - i Δ z k δ n n ) × exp { - i Δ z 2 [ 2 ( 2 + k 2 ) 1 / 2 + k ] } ( x , y , 0 ) + 0 ( Δ z 3 ) .