Abstract

A generalized beam propagation method is described that uses the ABCD matrices to treat optical systems that have modest amounts of aberrations including gradient refractive-index elements. We can make calculations from any point in the near or far field to any other point by using appropriate numerical algorithms. The variation of the reduced length is discussed as a limitation to accuracy. The diffraction properties of a complex stigmatic system may be represented by those of an equivalent elementary system. This facilitates calculations using the standard diffraction operations for homogeneous media. The modified propagation technique replaces the large number of diffraction steps commonly used for the split-step solution of inhomogeneous media with one step for stigmatic media and in general no more than a few steps for aberrated media. Maxwell’s fisheye lens is discussed in detail to show application of the method.

© 1992 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 77–96.
  2. E. A. Sziklas, A. E. Siegman, “Mode calculations in unstable resonator with flowing saturable gain. Fast Fourier transform method,” Appl. Opt. 14, 1874–1889 (1975).
    [CrossRef] [PubMed]
  3. R. H. Hardin, F. D. Tappert, “Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations,” SIAM Rev. 15, 423–433 (1973).
  4. A. Korpel, H. H. Lin, D. J. Mehrl, “Convenient operator formalism for Fourier optics and inhomogeneous and nonlinear wave propagation,” J. Opt. Soc. Am. A 6, 630–635 (1989).
    [CrossRef]
  5. J. J. Gribble, J. M. Arnold, “Beam-propagation method ray equations,” Opt. Lett. 13, 611–613 (1988).
    [CrossRef] [PubMed]
  6. W. Brouwer, Matrix Methods in Optical Instrument Design (Benjamin, New York, 1964).
  7. L. Casperson, “Synthesis of Gaussian beam optical systems,” Appl. Opt. 20, 2243–2249 (1981).
    [CrossRef] [PubMed]
  8. A. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 20.
  9. H. T. Yura, S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A 4, 1931–1948 (1987).
    [CrossRef]
  10. A. E. Siegman, “A canonical formulation for analyzing multielement unstable resonators,” IEEE J. Quantum Electron. QE-12, 35–40 (1976).
    [CrossRef]
  11. E. W. Marchand, Gradient Index Optics (Academic, New York, 1978).
  12. W. T. Welford, Aberrations of Optical Systems (Hilger, Boston, 1986).
  13. S.-H. Hwang, G. Lawrence, “Physical optics analysis of gradient index optics,” in Current Developments in Optical Engineering and Commercial Optics, R. E. Fischer, H. M. Pollicove, W. J. Smith, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1168, 360–368 (1989).
  14. glad Users Manual, a physical optics propagation program (Applied Optics Research, Tucson, Ariz.).
  15. Optical Design, MIL-HDBK-141 (U.S. Government Printing Office, Washington, D.C., 1962).
  16. P. Baues, “Huygens’ principle in inhomogeneous, isotropic media and a general integral equation applicable to optical resonators,” Opto-electronics 1, 751–758 (1971).
  17. S. Eckhardt, “Beam propagation and shift-variant optics,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1990).
  18. S. Eckhardt, “Non-shift-invariant diffraction effects in simple optical systems,” to be submitted to Opt. Lett.
  19. H. Kraus, “Huygens–Fresnel–Kirchhoff wave-front diffraction formulation: spherical waves,” J. Opt. Soc. Am. A 6, 1196–1205 (1989).
    [CrossRef]
  20. M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), Chap 4, p. 81.

1989 (2)

1988 (1)

1987 (1)

1981 (1)

1976 (1)

A. E. Siegman, “A canonical formulation for analyzing multielement unstable resonators,” IEEE J. Quantum Electron. QE-12, 35–40 (1976).
[CrossRef]

1975 (1)

1973 (1)

R. H. Hardin, F. D. Tappert, “Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations,” SIAM Rev. 15, 423–433 (1973).

1971 (1)

P. Baues, “Huygens’ principle in inhomogeneous, isotropic media and a general integral equation applicable to optical resonators,” Opto-electronics 1, 751–758 (1971).

Abramowitz, M.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), Chap 4, p. 81.

Arnold, J. M.

Baues, P.

P. Baues, “Huygens’ principle in inhomogeneous, isotropic media and a general integral equation applicable to optical resonators,” Opto-electronics 1, 751–758 (1971).

Brouwer, W.

W. Brouwer, Matrix Methods in Optical Instrument Design (Benjamin, New York, 1964).

Casperson, L.

Eckhardt, S.

S. Eckhardt, “Beam propagation and shift-variant optics,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1990).

S. Eckhardt, “Non-shift-invariant diffraction effects in simple optical systems,” to be submitted to Opt. Lett.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 77–96.

Gribble, J. J.

Hanson, S. G.

Hardin, R. H.

R. H. Hardin, F. D. Tappert, “Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations,” SIAM Rev. 15, 423–433 (1973).

Hwang, S.-H.

S.-H. Hwang, G. Lawrence, “Physical optics analysis of gradient index optics,” in Current Developments in Optical Engineering and Commercial Optics, R. E. Fischer, H. M. Pollicove, W. J. Smith, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1168, 360–368 (1989).

Korpel, A.

Kraus, H.

Lawrence, G.

S.-H. Hwang, G. Lawrence, “Physical optics analysis of gradient index optics,” in Current Developments in Optical Engineering and Commercial Optics, R. E. Fischer, H. M. Pollicove, W. J. Smith, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1168, 360–368 (1989).

Lin, H. H.

Marchand, E. W.

E. W. Marchand, Gradient Index Optics (Academic, New York, 1978).

Mehrl, D. J.

Siegman, A.

A. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 20.

Siegman, A. E.

A. E. Siegman, “A canonical formulation for analyzing multielement unstable resonators,” IEEE J. Quantum Electron. QE-12, 35–40 (1976).
[CrossRef]

E. A. Sziklas, A. E. Siegman, “Mode calculations in unstable resonator with flowing saturable gain. Fast Fourier transform method,” Appl. Opt. 14, 1874–1889 (1975).
[CrossRef] [PubMed]

Stegun, I.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), Chap 4, p. 81.

Sziklas, E. A.

Tappert, F. D.

R. H. Hardin, F. D. Tappert, “Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations,” SIAM Rev. 15, 423–433 (1973).

Welford, W. T.

W. T. Welford, Aberrations of Optical Systems (Hilger, Boston, 1986).

Yura, H. T.

Appl. Opt. (2)

IEEE J. Quantum Electron. (1)

A. E. Siegman, “A canonical formulation for analyzing multielement unstable resonators,” IEEE J. Quantum Electron. QE-12, 35–40 (1976).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Lett. (1)

Opto-electronics (1)

P. Baues, “Huygens’ principle in inhomogeneous, isotropic media and a general integral equation applicable to optical resonators,” Opto-electronics 1, 751–758 (1971).

SIAM Rev. (1)

R. H. Hardin, F. D. Tappert, “Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations,” SIAM Rev. 15, 423–433 (1973).

Other (11)

W. Brouwer, Matrix Methods in Optical Instrument Design (Benjamin, New York, 1964).

A. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 20.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 77–96.

S. Eckhardt, “Beam propagation and shift-variant optics,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1990).

S. Eckhardt, “Non-shift-invariant diffraction effects in simple optical systems,” to be submitted to Opt. Lett.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), Chap 4, p. 81.

E. W. Marchand, Gradient Index Optics (Academic, New York, 1978).

W. T. Welford, Aberrations of Optical Systems (Hilger, Boston, 1986).

S.-H. Hwang, G. Lawrence, “Physical optics analysis of gradient index optics,” in Current Developments in Optical Engineering and Commercial Optics, R. E. Fischer, H. M. Pollicove, W. J. Smith, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1168, 360–368 (1989).

glad Users Manual, a physical optics propagation program (Applied Optics Research, Tucson, Ariz.).

Optical Design, MIL-HDBK-141 (U.S. Government Printing Office, Washington, D.C., 1962).

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Figures (7)

Fig. 1
Fig. 1

Propagation of a complex amplitude distribution from some initial point to an arbitrary final point represented by an ABCD propagation matrix. The elementary operators are change in magnification, change in index of refraction, a thin lens, and a thickness. Only the thickness requires significant computation time because it represents a diffraction propagation.

Fig. 2
Fig. 2

Collimated BPM (a) treating all index variations as aberrations. Large phase values are accumulated with respect to the plane reference surfaces, and the beam size changes in the array, creating difficulties in sampling. The index of refraction should be calculated at the true position of the phase front, not at the plane reference surface. Hundreds of propagation steps are often required. A more general noncollimated treatment (b) breaks the GRIN medium into a series of lenses and spaces and uses curved reference surfaces and noncollimated propagators. The noncollimated propagators maintain good sampling by having the matrix change size to match the beam approximately. Most of the phase curvature is treated by the curved reference surfaces. The generalized BPM (c) can move from an arbitrary point to any other in one step for stigmatic media by using the paraxial behavior to calculate the properties of an equivalent system.

Fig. 3
Fig. 3

Half of Maxwell’s fisheye lens focusing a collimated beam to a focus. Collimated light incident on the flat face will be perfectly focused (geometrically) at point F. An exact geometrical ray will describe a circular arc. A paraxial ray describes a parabolic arc and departs from the exact ray significantly beyond the focus point. A Gaussian beam with a waist at the flat front face will have a waist inside the paraxial focus.

Fig. 4
Fig. 4

For a large initial choice of Gaussian waist (w = 0.01) at the center of refractive symmetry (z = 0) a second waist forms before the paraxial focus and the beam diverge beyond the waist (dashed curve). For a small initial Gaussian waist (w = 0.001) the beam will not form another waist and will diverge strongly (solid curve). For the waist radius of ω 1 = ( λ f / π n 1 2 ) 1 / 2 = 0.0026 the maximum degree of collimation will be achieved (solid curve with asterisks). Conditions for the calculation were n0 = 2, λ = 632.8 nm, f = 1.

Fig. 5
Fig. 5

Gaussian beam focused by a half Maxwell’s fisheye lens (as shown in Fig. 3). The Gaussian beam has a waist of radius ω = 0.003445 cm, the wavelength is λ = 0.6328 × 10−4 cm. Plots of intensity (solid curve) and phase (dashed curve) across the beam are shown for z = 0.6 [inside the waist (a)], z = 0.8 [waist location (b)], and z = 1.0 [paraxial focus (c)]. The peak intensity occurs at the calculated waist position of z = 0.8. The phase is the most sensitive indicator and is flat at the waist position. The phase is the phase of the complex amplitude distribution and is expressed in radians. A negative phase indicates that the wave front is ahead of the reference plane.

Fig. 6
Fig. 6

Propagation of a beam in homogeneous media represented approximately by an equivalent Gaussian beam. The Rayleigh range of the surrogate Gaussian beam provides a useful point to switch between near- and far-field propagators. The four possible cases for movement from either inside or outside the Rayleigh range to any other point are shown. The dashed curve shows schematically the change in size of a computer array representing the beam. The computer array would be of a fixed size inside the Rayleigh region and would grow linearly outside.

Fig. 7
Fig. 7

Geometrical-ray trajectory as a circular arc of radius R. The ray is characterized by the ray height a at z = 0. The angle θ may be used to describe points on the ray trajectory.

Tables (4)

Tables Icon

Table 1 Some ABCD Matrix Operators for Common Conventional and Unconventional Elements

Tables Icon

Table 2 Collimated BPM

Tables Icon

Table 3 Noncollimated BPM

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Table 4 Generalized BPM by Using the ABCD Matrix

Equations (56)

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[ y ( z 2 ) u ( z 2 ) ] = [ f 11 ( z 1 , z 2 ) f 12 ( z 1 , z 2 ) f 21 ( z 1 , z 2 ) f 22 ( z 1 , z 2 ) ] [ y ( z 1 ) u ( z 1 ) ] ,
[ A B C D ] = [ 1 t 0 1 ] [ 1 0 - ϕ 1 ] [ 1 0 0 n 1 / n 2 ] [ M 0 0 1 / M ] .
[ A B C D ] = [ 1 0 - ϕ 2 1 ] [ M 0 0 1 / M ] [ 1 L eq 0 1 ] [ 1 0 - ϕ 1 1 ] ,
M = A D - B C D ,
n 1 n 2 = A D - B C ,
ϕ = - C D A D - B C = - C M ,
t = B D .
1 R 1 = 1 n 2 n 1 M 2 R 1 - ϕ .
[ A B C D ] [ 1 t 2 0 1 ] [ 1 0 - ϕ 2 1 ] [ 1 0 0 n 1 / n 2 ] × [ 1 t 1 0 1 ] [ 1 0 - ϕ 1 1 ] [ 1 0 0 n 0 / n 1 ] .
[ A B C D ] k = 1 N [ 1 t k 0 1 ] [ 1 0 - ϕ k 1 ] [ 1 0 0 n k - 1 / n k ] ,
diffraction impulse response = 1 j λ z eff exp ( j π λ z eff r 2 ) .
reduced distance = z eff = d s n ( s ) M 2 ( s ) ,
n ( x , y , z ) = n s ( x , y , z ) + n e ( x , y , z ) ,
n ( x , y , x ) n 0 1 + ( x - x 0 ) 2 + ( y - y 0 ) 2 + ( z - z 0 ) 2 f 2 .
W ( Δ s ) = 0 Δ s n e ( s ) d s ,
n ( r ) = n 0 1 + r 2 f 2 .
z 1 z 2 = - f 2 .
y = ( z 2 - f 2 ) a + z b ,
u = 2 a z + b ,
H = y 1 n u 2 - y 2 n u 1 = - n 0 f 2 ( a 1 b 2 - b 2 a 1 ) .
[ y u ] = [ A B C D ] [ y 0 u 0 ] = 1 f 2 [ - z 2 + f 2 z f 2 - 2 z f 2 ] [ y 0 u 0 ] ,
[ A B C D ] ( z 1 , z 2 ) = 1 z 1 2 + f 2 × [ 2 z 1 z 2 - z 2 2 + f 2 z 1 z 2 2 - z 2 z 1 2 + ( z 2 - z 1 ) f 2 2 ( z 1 - z 2 ) 2 z 1 z 2 - z 1 2 + f 2 ] .
M = A D - B C D = f 2 + z 2 2 2 z 1 z 2 - z 1 2 + f 2 ,
n 1 n 2 = A D - B C = f 2 + z 2 2 f 2 + z 1 2 ,
ϕ = - C D A D - B C = - C M = - 2 ( z 1 - z 2 ) ( 2 z 1 z 2 - z 1 2 + f 2 ) ( f 2 + z 2 2 ) ( f 2 + z 1 2 ) ,
t = B D = z 1 z 2 2 - z 2 z 1 2 + ( z 2 - z 1 ) f 2 2 z 1 z 2 - z 1 2 + f 2 .
[ A B C D ] = [ 0 f - 2 / f 1 ] .
1 q = 1 R - i λ π n ω 2 .
1 q 2 = C + D q 1 A + B q 1 .
A C - B D q 1 2 z 2 [ z 2 2 f 2 - 1 + 1 2 ( λ f π n 1 ω 1 2 ) 2 ] = 0.
z 2 = 0 ,
z 2 = ± f [ 1 - 1 2 ( λ f π n 1 ω 1 2 ) 2 ] 1 / 2 .
ω 1 > ( λ f π n 1 2 ) 1 / 2             waists at z 2 = 0 ,     ± f [ 1 - 1 2 ( λ f π n 1 ω 1 2 ) 2 ] 1 / 2 ,
ω 1 ( λ f π n 1 2 ) 1 / 2             the only waist is at z 1 = z 2 = 0.
ω 1 = ( λ f π n 1 2 ) 1 / 2
ω 2 2 = ω 1 2 ( A 2 + B 2 λ 2 π 2 n 1 2 ω 0 4 ) ,
ω 2 2 = ω 1 2 [ ( 2 z 1 z 2 - z 2 2 + f 2 z 1 2 + f 2 ) 2 + ( z 1 z 2 2 - z 2 z 1 2 + ( z 2 - z 1 ) f 2 z 1 2 + f 2 λ π n 0 w 1 2 ) 2 ] .
waist for equivalent system = f 1 + [ λ f π n 2 ( 4 ω 1 ) ] 2 ,
actual waist locations = ± f [ 1 - 1 2 ( λ f π n 1 ω 1 2 ) 2 ] 1 / 2 .
waist for equivalent system , Δ z = - 1 4 ( λ f π n 1 ω 1 2 ) 2 ,
actual waist locations , Δ z = - 1 4 ( λ f π n 1 ω 1 2 ) 2 .
z 2 = 0.6 , A B C D = [ 0.64 0.6 - 1.2 1.00 ] , z 2 = 0.8 , A B C D = [ 0.36 0.8 - 1.6 1.00 ] , z 2 = 1.0 , A B C D = [ 0.00 1.0 - 2.0 1.00 ] .
z w = - R 1 [ 1 + ( λ R 1 π n 1 ω 1 2 ) 2 ] ,
ω 0 = ω 1 [ 1 + ( π n 1 ω 1 2 λ R 1 ) 2 ] 1 / 2 ,
inside             ( z - z w ) z R ,
outside             ( z - z w ) z R .
II ( z 1 , z 21 , n )             inside z R to inside z R , IO ( z 1 , z 2 , n )             inside z R to outside z R , OI ( z 1 , z 2 , n )             outside z R to inside z R , OO ( z 1 , z 2 , n )             outside z R to outside z R , I I x ( z 1 , z 2 , n ) = PTP x ( z 2 - z 1 ) , IO x ( z 1 , z 2 , n ) = WTC x ( z 2 - z w x ) PTP x ( z w x - z 1 ) , OI x ( z 1 , z 2 , n ) = PTP x ( z 2 - z w x ) CTW x ( z w x - z 1 ) , OO x ( z 1 , z 2 , n ) = WTC x ( z 2 - z w x ) CTW x ( z w x - z 1 ) ,
PTP x ( Δ z , n ) = F - 1 [ T x ( Δ z , n ) F [ (     ) ] } ,             Δ x 2 = Δ x 1 ,
T x ( Δ z , n ) = exp ( - j π λ n Δ z ξ 2 ) ,
WTC x ( Δ z , n ) = 1 ( j λ n Δ z ) 1 / 2 F s { q x ( Δ z , n ) (     ) ] } , Δ x 2 = λ Δ z n Δ x 1 , q x ( Δ z , n ) = exp ( j π n x 2 λ Δ z ) ,
s = Δ z Δ z ,
CTW x ( Δ z , n ) = 1 ( j λ n Δ z ) 1 / 2 { q x ( Δ z , n ) F s [ (     ) ] } , Δ x 2 = λ Δ z n Δ x 1 ,
R = a 2 + f 2 2 a .
d s n ( s ) = 1 n 0 ( 1 + r 2 f 2 ) R d θ = 1 n 0 { 2 R 3 f 2 [ θ m - sin ( θ m ) ] + a 2 + f 2 f } ,
d s n ( s ) 1 n 0 ( 2 R 3 f 2 { [ f R + 1 6 ( f R ) 3 + ] - f R } + a 2 + f 2 f ) ,
d s n ( s ) f n 0 [ 4 3 + a 2 f 2 ] ,

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