Abstract

In the mode-expansion method for modeling propagation of a diffracted beam, the beam at the aperture can be expanded as a weighted set of orthogonal modes. The parameters of the expansion modes are chosen to maximize the weighting coefficient of the lowest-order mode. As the beam propagates, its field distribution can be reconstructed from the set of weighting coefficients and the Gouy phase of the lowest-order mode. We have developed a simple procedure to implement the mode-expansion method for propagation through an arbitrary ABCD matrix, and we have demonstrated that it is accurate in comparison with direct calculations of diffraction integrals and much faster.

© 2007 Optical Society of America

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References

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  1. N. S. Petrovic and A. D. Rakic, "Modeling diffraction and imaging of laser beams by the mode-expansion method," J. Opt. Soc. Am. B 22, 556-566 (2005).
    [CrossRef]
  2. K. Tanaka, M. Shibukawa, and O. Fukumitsu, "Diffraction of a wave beam by an aperture," IEEE Trans. Microwave Theory Tech. MTT-20, 749-755 (1972).
    [CrossRef]
  3. N. S. Petrovic and A. D. Rakic, "Modeling diffraction in free-space optical interconnects by the mode expansion method," Appl. Opt. 42, 5308-5318 (2003).
    [CrossRef] [PubMed]
  4. Y. Li, "Oscillations and discontinuity in the focal shift of Gaussian laser beams," J. Opt. Soc. Am. A 3, 1761-1765 (1986).
    [CrossRef]
  5. N. Saga, K. Tanaka, and O. Fukumitsu, "Diffraction of a Gaussian beam through a finite aperture lens and the resulting heterodyne efficiency," Appl. Opt. 20, 2827-2831 (1981).
    [CrossRef] [PubMed]
  6. H. Kogelnik and T. Li, "Laser beams and resonators," Appl. Opt. 5, 1550-1567 (1966).
    [CrossRef] [PubMed]
  7. A. E. Siegman, in Lasers (University Science, 1986), pp. 646-648.
  8. M. F. Erden and H. M. Ozaktas, "Accumulated Gouy phase shift in Gaussian beam propagation through first-order systems," J. Opt. Soc. Am. A 14, 2190-2194 (1997).
    [CrossRef]
  9. J. E. Harvey, C. L. Vernold, A. Krywonos, and P. L. Thompson, "Diffracted radiance: a fundamental quantity in nonparaxial scalar diffraction theory," Appl. Opt. 38, 6469-6481 (1999).
    [CrossRef]

2005 (1)

2003 (1)

1999 (1)

1997 (1)

1986 (2)

1981 (1)

1972 (1)

K. Tanaka, M. Shibukawa, and O. Fukumitsu, "Diffraction of a wave beam by an aperture," IEEE Trans. Microwave Theory Tech. MTT-20, 749-755 (1972).
[CrossRef]

1966 (1)

Erden, M. F.

Fukumitsu, O.

N. Saga, K. Tanaka, and O. Fukumitsu, "Diffraction of a Gaussian beam through a finite aperture lens and the resulting heterodyne efficiency," Appl. Opt. 20, 2827-2831 (1981).
[CrossRef] [PubMed]

K. Tanaka, M. Shibukawa, and O. Fukumitsu, "Diffraction of a wave beam by an aperture," IEEE Trans. Microwave Theory Tech. MTT-20, 749-755 (1972).
[CrossRef]

Harvey, J. E.

Kogelnik, H.

Krywonos, A.

Li, T.

Li, Y.

Ozaktas, H. M.

Petrovic, N. S.

Rakic, A. D.

Saga, N.

Shibukawa, M.

K. Tanaka, M. Shibukawa, and O. Fukumitsu, "Diffraction of a wave beam by an aperture," IEEE Trans. Microwave Theory Tech. MTT-20, 749-755 (1972).
[CrossRef]

Siegman, A. E.

A. E. Siegman, in Lasers (University Science, 1986), pp. 646-648.

Tanaka, K.

N. Saga, K. Tanaka, and O. Fukumitsu, "Diffraction of a Gaussian beam through a finite aperture lens and the resulting heterodyne efficiency," Appl. Opt. 20, 2827-2831 (1981).
[CrossRef] [PubMed]

K. Tanaka, M. Shibukawa, and O. Fukumitsu, "Diffraction of a wave beam by an aperture," IEEE Trans. Microwave Theory Tech. MTT-20, 749-755 (1972).
[CrossRef]

Thompson, P. L.

Vernold, C. L.

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Figures (8)

Fig. 1
Fig. 1

Optical layout for the example described in the text. An aperture with a thin ideal lens located at z = 0 is illuminated by a Gaussian beam with waist at z 0 . The field of the diffracted beam is calculated at the measurement plane at z m . The caret accent distinguishes the parameters of the expansion modes of the diffracted beam from those of the incident beam.

Fig. 2
Fig. 2

Overlap integral for example 1. The horizontal axis is the radius of the Gaussian core (mm), and the vertical axis is C 0 (a.u.). The maximum is at 0.0688   mm .

Fig. 3
Fig. 3

Expansion coefficients C n for n = 0 , … ,  30 .

Fig. 4
Fig. 4

Irradiance distribution at the measurement plane for Example 1 as calculated (a) by the MEM and (b) from the diffraction integral. The horizontal axis is the radius in mm, r, and the vertical axis is the irradiance (log scale).

Fig. 5
Fig. 5

Irradiance distribution at the measurement plane for Example 2 as calculated (a) by the MEM and (b) by calculation of the diffraction integral. The horizontal axis is the radius in mm, r, and the vertical axis is the irradiance (log scale).

Fig. 6
Fig. 6

Overlap integral for Example 3. The horizontal axis is the radius of the Gaussian core (mm) and the vertical axis is C 0 (a.u.).

Fig. 7
Fig. 7

Expansion coefficients C n for n = 0 , … ,  140 .

Fig. 8
Fig. 8

Axial field distribution for Example 3. The horizontal axis is the displacement along the z axis from the focal plane, normalized to the lens focal length, and the vertical axis is the irradiance (log scale), normalized to its value at the focal plane.

Equations (23)

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ψ ( r , z ) = 1 w ( z ) 2 π exp [ j π r 2 λ R ( z ) ] exp [ j γ ( z ) ] × exp [ j 2 π ( z z 0 ) λ ] exp [ r 2 w ( z ) 2 ] ,
γ ( z ) = arctan [ λ ( z z 0 ) π w 0 2 ]
w ( z ) = w 0 1 + [ λ ( z z 0 ) π w 0 2 ] 2 ,
R ( z ) = ( z z 0 ) { 1 + [ π w 0 2 λ ( z z 0 ) ] } .
U ( r , z ) = n = 0 C n ( z ) Ψ n ( r , z ) ,
C n ( z ) = 0 2 π 0 U ( r , z ) Ψ * ( r , z ) r d r d θ = 2 π 0 U ( r , z ) Ψ * ( r , z ) r d r .
Ψ n ( r , z ) = 1 w ^ ( z ) 2 π L n [ 2 r 2 w ^ ( z ) 2 ] exp [ j π r 2 λ R ^ ( z ) ] × exp [ j ( 2 n + 1 ) γ ^ ( z ) ] exp [ j 2 π ( z z ^ 0 ) λ j φ ] × exp [ r 2 w ^ ( z ) 2 ] ,
C n = 2 π 0 a ψ ( r , z = 0 ) Ψ n * ( r , z = 0 ) r d r ,
U ( r , z ) n = 0 N C n Ψ n ( r , z ) .
C 0 = 4 w w ^ 0 a exp { j [ π r 2 λ R ( 0 ) π r 2 λ R ^ ( 0 ) ] } × exp [ j arctan ( λ z 0 π w 0 2 ) + j arctan ( λ z ^ 0 π w ^ 0 ) + j 2 π λ ( z 0 z ^ 0 ) + j φ ] exp [ ( r 2 w 2 + r 2 w ^ 2 ) ] r d r ,
C 0 = 4 w w ^ 0 a exp [ ( r 2 w 2 + r 2 w ^ 2 ) ] r d r = 2 w 2 w ^ 2 w 2 + w ^ 2 [ 1 exp ( a 2 w 2 a 2 w ^ 2 ) ] .
1 R 2 = 1 R 1 1 f .
w ^ 0 = w ^ / 1 + ( π w ^ 2 λ R ^ ) 2 ,
z ^ 0 = R ^ / [ 1 + ( λ R ^ π w ^ 2 ) 2 ] ,
Δ γ ( z ) = arctan [ λ ( z z ^ 0 ) π w ^ 0 2 ] arctan [ λ ( z ^ 0 ) π w ^ 0 2 ] .
Δ γ = tan 1 [ B λ π w ^ 2 ( A + B / R ^ ) ] ,
| A B C D | = | 1 0 1 / f 1 | | 1 d 0 1 | = | 1 d / f d 1 / f 1 | ,
C n = 4 w w ^ 0 a L n ( 2 r 2 w ^ 2 ) exp [ ( r 2 w 2 + r 2 w ^ 2 ) ] r d r .
U ( r , z m ) 1 w ^ m 2 π exp ( r 2 w ^ m 2 ) n = 0 N C n L n ( 2 r 2 w ^ m 2 ) × exp [ j ( 2 n + 1 ) Δ γ ( z m z 0 ) ] ,
Ψ n ( r , z m ) = 1 w ^ m 2 π L n ( 2 r 2 w ^ m 2 ) exp [ j ( 2 n + 1 ) Δ γ ( z m z 0 ) ] × exp ( r 2 w ^ m 2 ) ,
F a 2 λ f ,
C 0 = 2 2 π w ^ 0 0 a exp ( r 2 w ^ 0 2 ) r d r = 2 π w ^ 0 [ 1 exp ( a 2 w ^ 2 ) ] ,
C n = 2 2 π w ^ 0 0 a L n ( 2 r 2 w ^ 2 ) exp ( r 2 w ^ 0 2 ) r d r

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