Abstract

A matrix approach is presented that allows one to describe a complex optical system by a matrix relating the field at the output plane to the field at the input one. The elements of the optical system may be all those characterized by an ABCD ray-transfer matrix, as well as any kind of film which introduces a wavefront modulation that can be described by a complex radial transmittance function. These include, as particular cases, stops and limiting apertures. No integral has to be computed. The method holds only for circularly symmetric optical systems and laser beams.

© 1989 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).
  2. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  3. S. A. Collins, “Lens–System Diffraction Integral Written in Terms of Matrix Optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970).
    [CrossRef]
  4. H. T. Yura, S. G. Hanson, “Optical Beam Wave Propagation Through Complex Optical Systems,” J. Opt. Soc. Am. A4, 1931–1948 (1987).
    [CrossRef]
  5. F. Bloisi, S. Martellucci, J. Quartieri, L. Vicari, “Diffraction Patterns of Self-Phase-Modulated Laser Beams,” Europhys. Lett., 4, 905–908 (1987).
    [CrossRef]

1987 (2)

H. T. Yura, S. G. Hanson, “Optical Beam Wave Propagation Through Complex Optical Systems,” J. Opt. Soc. Am. A4, 1931–1948 (1987).
[CrossRef]

F. Bloisi, S. Martellucci, J. Quartieri, L. Vicari, “Diffraction Patterns of Self-Phase-Modulated Laser Beams,” Europhys. Lett., 4, 905–908 (1987).
[CrossRef]

1970 (1)

Bloisi, F.

F. Bloisi, S. Martellucci, J. Quartieri, L. Vicari, “Diffraction Patterns of Self-Phase-Modulated Laser Beams,” Europhys. Lett., 4, 905–908 (1987).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).

Collins, S. A.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Hanson, S. G.

H. T. Yura, S. G. Hanson, “Optical Beam Wave Propagation Through Complex Optical Systems,” J. Opt. Soc. Am. A4, 1931–1948 (1987).
[CrossRef]

Martellucci, S.

F. Bloisi, S. Martellucci, J. Quartieri, L. Vicari, “Diffraction Patterns of Self-Phase-Modulated Laser Beams,” Europhys. Lett., 4, 905–908 (1987).
[CrossRef]

Quartieri, J.

F. Bloisi, S. Martellucci, J. Quartieri, L. Vicari, “Diffraction Patterns of Self-Phase-Modulated Laser Beams,” Europhys. Lett., 4, 905–908 (1987).
[CrossRef]

Vicari, L.

F. Bloisi, S. Martellucci, J. Quartieri, L. Vicari, “Diffraction Patterns of Self-Phase-Modulated Laser Beams,” Europhys. Lett., 4, 905–908 (1987).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).

Yura, H. T.

H. T. Yura, S. G. Hanson, “Optical Beam Wave Propagation Through Complex Optical Systems,” J. Opt. Soc. Am. A4, 1931–1948 (1987).
[CrossRef]

Europhys. Lett. (1)

F. Bloisi, S. Martellucci, J. Quartieri, L. Vicari, “Diffraction Patterns of Self-Phase-Modulated Laser Beams,” Europhys. Lett., 4, 905–908 (1987).
[CrossRef]

J. Opt. Soc. Am. (2)

S. A. Collins, “Lens–System Diffraction Integral Written in Terms of Matrix Optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970).
[CrossRef]

H. T. Yura, S. G. Hanson, “Optical Beam Wave Propagation Through Complex Optical Systems,” J. Opt. Soc. Am. A4, 1931–1948 (1987).
[CrossRef]

Other (2)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

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Figures (5)

Fig. 1
Fig. 1

Schematic representation of an optical system where ΣI is the input plane, Σo is the output plane, and E1, E2, … and EP are optical elements represented by the ABCD ray-transfer matrices. F1,F2FN are optical elements represented by complex transmittance functions. L1,L2,…LN+1 are ABCD ray transfer matrices of the optical elements between two consecutive filters. G1,G2,…GN+1 are M × M matrices representing the optical elements between two consecutive filters; and T1,T2,… TN are M × M matrices representing filters.

Fig. 2
Fig. 2

Schematic representation of the optical system used in the example. ΣI: input plane; Σo: output plane; E1,E2: lenses of focal length f1 and f2 respectively; F: filter located at the focal plane common to both lenses; Z1: distance between the input plane and the first lens; and Z2: distance between the second lens and the output plane.

Fig. 3
Fig. 3

Amplitude of the output signal for the optical system of Fig. 2, with the Gaussian limiting aperture F. Full line is obtained by the method of Yura and Hanson4, while circles are obtained by our method. The parameters of the optical system are: f1 = 0.4 m. f2 = 0.2 m, Z1 = Z2 = 0.6 m, Wfil = 100 μm. The beam waist at the input plane is Win = 500 μm, its amplitude on the optical axis (r = 0) is assumed to be unity and the wavelength is 0.5145 μm.

Fig. 4
Fig. 4

Phase of the output signal for the optical system of Fig. 2, with the Gaussian limiting aperture F. Full line is obtained by the method of Yura and Hanson4, while circles are obtained by our method. Parameters of the optical system and of the input beam are the same as for Fig. 3. The phase is assumed zero on the optical axis at the output plane.

Fig. 5
Fig. 5

Amplitude of the output signal for the optical system of Fig. 2, with the spatial filter F. Parameters for the optical system are: f1 = 0.4 m. f2 = 0.2 m, Z1 = Z2 = 0.6 m, and Wfil = 250 μm or 100 μm. The beam waist at the input plane is Win = 500 μm, its amplitude on the optical axis (r = 0) is assumed to be unity, and the wavelength is 0.5145 μm.

Equations (22)

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I 1 ( x , y ) = ( i k / 2 π C ) exp ( i k Z ) × + d x 0 + d y 0 O 0 ( x 0 , y 0 ) × exp { i k 2 C ( D x 0 2 + D y 0 2 2 x x 0 2 y y 0 + A x 2 + A y 2 ) } ,
I 1 ( x , y ) = ( i k / 2 π B 1 ) exp ( i k Z 1 ) × + d x 0 + d y 0 O 0 ( x 0 , y 0 ) × exp { i k 2 B 1 ( A 1 x 0 2 + A 1 y 0 2 2 x x 0 2 y y 0 + D 1 x 2 + D 1 y 2 ) } .
I 1 ( r , ϴ ) = ( i k / 2 π B 1 ) exp ( i k Z 1 ) × 0 + d r 0 r 0 0 2 π d ϴ 0 O 0 ( r 0 , ϴ 0 ) × exp { i k 2 B 1 ( D 1 r 2 2 r r 0 cos ( ϴ ϴ 0 ) + A 1 r 0 2 } ,
I 1 ( r ) = ( i k / B 1 ) exp ( i k Z 1 ) exp ( i k D 1 r 2 / 2 B 1 ) × 0 d r 0 r 0 O 0 ( r 0 ) exp ( i k A 1 r 0 2 / 2 B 1 ) J 0 ( k r r 0 / B 1 ) ,
J 0 ( k r r 0 / B 1 ) = 1 2 π 0 2 π d ϴ 0 exp { i k r r 0 B 1 cos ( ϴ ϴ 0 ) } .
I 1 ( r ) ( i k / B 1 ) exp ( i k Z 1 ) exp ( i k D 1 r 2 / 2 B 1 ) = 2 R 2 s = 1 c s J 0 ( j s r / R ) [ J 1 ( j s ) ] 2 ,
I 1 ( r ) = ( 2 i B 1 / k R 2 ) exp ( i k Z 1 ) exp ( i k D 1 r 2 / 2 B 1 ) × s = 1 c s J 0 ( j s r / R ) [ J 1 ( j s ) ] 2 ,
c s = ( i k / B 1 ) exp ( i k Z 1 ) × 0 R dr r I 1 ( r ) exp ( i k D 1 r 2 / 2 B 1 ) J 0 ( j s r / R ) .
I 1 ( r ) ( i B 1 / k ) exp ( i k Z 1 ) exp ( i k D 1 r 2 / 2 B 1 ) = 0 d r 0 r 0 O 0 ( r 0 ) exp ( i k A 1 r 0 2 / 2 B 1 ) J 0 ( k r r 0 / B 1 ) ,
O 0 ( r 0 ) exp ( i k A 1 r 0 2 / 2 B 1 ) = ( i B 1 / k ) exp ( i k Z 1 ) 0 R d ( k r / B 1 ) k r / B 1 × I 1 ( r ) exp ( i k D 1 r 2 / 2 B 1 ) J 0 ( k r r 0 / B 1 ) , = ( i k / B 1 ) exp ( i k Z 1 ) × 0 R dr r I 1 ( r ) exp ( i k D 1 r 2 / 2 B 1 ) J 0 ( k r r 0 / B 1 ) ,
c s = O 0 ( j s B 1 / k R ) exp ( i A 1 B 1 j s 2 / 2 k R 2 ) .
I 1 ( j m B 2 / k R ) = ( i 2 B 1 / k R 2 ) exp ( i k Z 1 ) exp ( i D 1 j m 2 B 2 2 / 2 k R 2 B 1 ) × s = 1 M exp ( i A 1 B 1 j s 2 / 2 k R 2 ) × J 0 ( j s j m B 2 / k R 2 ) [ J 1 ( j s ) ] 2 O 0 ( j s B 1 / k R ) .
O 0 , s = O 0 ( j s B 1 / k R ) for 1 s M , I 1 , m = I 1 ( j m B 2 / k R ) for 1 m M .
G 1 , m s = ( 2 i B 1 / k R 2 ) exp ( i k Z 1 ) × exp ( i D 1 j m 2 B 2 2 / 2 k R 2 B 1 ) × exp ( i A 1 B 1 j s 2 / 2 k R 2 ) J 0 ( j s j m B 2 / k R 2 ) [ J 1 ( j s ) ] 2 , for 1 m M , 1 s M .
T 1 , m s = δ m s t ( j m B 2 / k R ) for 1 m M , 1 s M ,
O 1 = T 1 × I 1 = T 1 × G 1 × O 0 .
G h , m s = ( i 2 B h / k R 2 ) exp ( i k Z h ) × exp ( i D h j m 2 B h + 1 2 / 2 k R 2 B h ) exp ( i A h B h j s 2 / 2 k R 2 ) × J 0 ( j s j m B h + 1 / k R 2 ) [ J 1 ( j s ) ] 2 , for 1 m M , 1 s M ,
T h , m s = δ m s t ( j m B h + 1 / k R ) for 1 m M , 1 s M .
G N + 1 , q m = ( 2 i B N + 1 / k R 2 ) exp ( i k Z N + 1 ) × exp ( i k D N + 1 r q 2 / 2 B N + 1 ) × exp ( i A N + 1 B N + 1 j m 2 / 2 k R 2 ) J 0 ( j m r q / R ) [ J 1 ( j m ) ] 2 , for 1 q Q , 1 m M ,
I N + 1 = G N + 1 × O N = G N + 1 × T N × I N = G N + 1 × T N × G N × O N 1 = G N + 1 × T N × G N × G 2 × T 1 × G 1 × O 0 .
I N + 1 = G × O 0 ,
G = G N + 1 × T N × G N × G 2 × T 1 × G 1 .

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