Abstract

Full characterization of optical systems, diffractive and geometric, is possible by using the Fresnel Gaussian Shape Invariant (FGSI) previously reported in the literature. The complex amplitude distribution in the object plane is represented by a linear superposition of complex Gaussians wavelets and then propagated through the optical system by means of the referred Gaussian invariant. This allows ray tracing through the optical system and at the same time allows calculating with high precision the complex wave-amplitude distribution at any plane of observation. This method is similar to conventional ray tracing additionally preserving the undulatory behavior of the field distribution. That is, we are propagating a linear combination of Gaussian shaped wavelets; keeping always track of both, the ray trajectory, and the wave phase of the whole complex optical field. This technique can be applied in a wide spectral range where the Fresnel diffraction integral applies including visible, X-rays, acoustic waves, etc. We describe the technique and we include one-dimensional illustrative examples.

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References

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  1. W. T. Welford, Aberrations of optical systems, (Academic Press, 1974), pp. 220.
  2. R. Kigslake, Lens design fundamentals, (Academic Press, 1978), pp. 8.
  3. W. J. Smith, Modern Optical Engineering, 3rd ed., (Mc Graw-Hill, 2000) p. 372.
  4. C.-S. Liu and P. D. Lin, “Computational method for deriving the geometric point spread function of an optical system,” Appl. Opt. 49(1), 126–136 (2010).
    [CrossRef] [PubMed]
  5. M. Herzberger, Modern Geometrical Optics, (Interscience Publishers, Inc. N.Y., 1958) pp. 383–400.
  6. M. Cywiak, A. Morales, J. M. Flores, and M. Servín, “Fresnel-Gaussian shape invariant for optical ray tracing,” Opt. Express 17(13), 10564–10572 (2009).
    [CrossRef] [PubMed]
  7. M. Cywiak, M. Servín, and F. Mendoza-Santoyo, “Wave-front propagation by Gaussian superposition,” Opt. Commun. 195(5-6), 351–359 (2001).
    [CrossRef]
  8. M. Cywiak, A. Morales, M. Servín, and R. Gómez-Medina, “A technique for calculating the amplitude distribution of propagated fields by Gaussian sampling,” Opt. Express 18(18), 19141–19155 (2010).
    [CrossRef] [PubMed]
  9. A. W. Greynolds, “Propagation of generally astigmatic Gaussian beams along skew ray paths,” Proc. SPIE 560, 33–50 (1985).
  10. M. M. Popov, “A new method of computation of wave fields using Gaussian beams,” Wave Motion 4(1), 85–97 (1982).
    [CrossRef]
  11. http://www.breault.com/software/asap.php

2010 (2)

C.-S. Liu and P. D. Lin, “Computational method for deriving the geometric point spread function of an optical system,” Appl. Opt. 49(1), 126–136 (2010).
[CrossRef] [PubMed]

M. Cywiak, A. Morales, M. Servín, and R. Gómez-Medina, “A technique for calculating the amplitude distribution of propagated fields by Gaussian sampling,” Opt. Express 18(18), 19141–19155 (2010).
[CrossRef] [PubMed]

2009 (1)

M. Cywiak, A. Morales, J. M. Flores, and M. Servín, “Fresnel-Gaussian shape invariant for optical ray tracing,” Opt. Express 17(13), 10564–10572 (2009).
[CrossRef] [PubMed]

2001 (1)

M. Cywiak, M. Servín, and F. Mendoza-Santoyo, “Wave-front propagation by Gaussian superposition,” Opt. Commun. 195(5-6), 351–359 (2001).
[CrossRef]

1985 (1)

A. W. Greynolds, “Propagation of generally astigmatic Gaussian beams along skew ray paths,” Proc. SPIE 560, 33–50 (1985).

1982 (1)

M. M. Popov, “A new method of computation of wave fields using Gaussian beams,” Wave Motion 4(1), 85–97 (1982).
[CrossRef]

Cywiak, M.

M. Cywiak, A. Morales, M. Servín, and R. Gómez-Medina, “A technique for calculating the amplitude distribution of propagated fields by Gaussian sampling,” Opt. Express 18(18), 19141–19155 (2010).
[CrossRef] [PubMed]

M. Cywiak, A. Morales, J. M. Flores, and M. Servín, “Fresnel-Gaussian shape invariant for optical ray tracing,” Opt. Express 17(13), 10564–10572 (2009).
[CrossRef] [PubMed]

M. Cywiak, M. Servín, and F. Mendoza-Santoyo, “Wave-front propagation by Gaussian superposition,” Opt. Commun. 195(5-6), 351–359 (2001).
[CrossRef]

Flores, J. M.

M. Cywiak, A. Morales, J. M. Flores, and M. Servín, “Fresnel-Gaussian shape invariant for optical ray tracing,” Opt. Express 17(13), 10564–10572 (2009).
[CrossRef] [PubMed]

Gómez-Medina, R.

M. Cywiak, A. Morales, M. Servín, and R. Gómez-Medina, “A technique for calculating the amplitude distribution of propagated fields by Gaussian sampling,” Opt. Express 18(18), 19141–19155 (2010).
[CrossRef] [PubMed]

Greynolds, A. W.

A. W. Greynolds, “Propagation of generally astigmatic Gaussian beams along skew ray paths,” Proc. SPIE 560, 33–50 (1985).

Lin, P. D.

C.-S. Liu and P. D. Lin, “Computational method for deriving the geometric point spread function of an optical system,” Appl. Opt. 49(1), 126–136 (2010).
[CrossRef] [PubMed]

Liu, C.-S.

C.-S. Liu and P. D. Lin, “Computational method for deriving the geometric point spread function of an optical system,” Appl. Opt. 49(1), 126–136 (2010).
[CrossRef] [PubMed]

Mendoza-Santoyo, F.

M. Cywiak, M. Servín, and F. Mendoza-Santoyo, “Wave-front propagation by Gaussian superposition,” Opt. Commun. 195(5-6), 351–359 (2001).
[CrossRef]

Morales, A.

M. Cywiak, A. Morales, M. Servín, and R. Gómez-Medina, “A technique for calculating the amplitude distribution of propagated fields by Gaussian sampling,” Opt. Express 18(18), 19141–19155 (2010).
[CrossRef] [PubMed]

M. Cywiak, A. Morales, J. M. Flores, and M. Servín, “Fresnel-Gaussian shape invariant for optical ray tracing,” Opt. Express 17(13), 10564–10572 (2009).
[CrossRef] [PubMed]

Popov, M. M.

M. M. Popov, “A new method of computation of wave fields using Gaussian beams,” Wave Motion 4(1), 85–97 (1982).
[CrossRef]

Servín, M.

M. Cywiak, A. Morales, M. Servín, and R. Gómez-Medina, “A technique for calculating the amplitude distribution of propagated fields by Gaussian sampling,” Opt. Express 18(18), 19141–19155 (2010).
[CrossRef] [PubMed]

M. Cywiak, A. Morales, J. M. Flores, and M. Servín, “Fresnel-Gaussian shape invariant for optical ray tracing,” Opt. Express 17(13), 10564–10572 (2009).
[CrossRef] [PubMed]

M. Cywiak, M. Servín, and F. Mendoza-Santoyo, “Wave-front propagation by Gaussian superposition,” Opt. Commun. 195(5-6), 351–359 (2001).
[CrossRef]

Appl. Opt. (1)

C.-S. Liu and P. D. Lin, “Computational method for deriving the geometric point spread function of an optical system,” Appl. Opt. 49(1), 126–136 (2010).
[CrossRef] [PubMed]

Opt. Commun. (1)

M. Cywiak, M. Servín, and F. Mendoza-Santoyo, “Wave-front propagation by Gaussian superposition,” Opt. Commun. 195(5-6), 351–359 (2001).
[CrossRef]

Opt. Express (2)

M. Cywiak, A. Morales, M. Servín, and R. Gómez-Medina, “A technique for calculating the amplitude distribution of propagated fields by Gaussian sampling,” Opt. Express 18(18), 19141–19155 (2010).
[CrossRef] [PubMed]

M. Cywiak, A. Morales, J. M. Flores, and M. Servín, “Fresnel-Gaussian shape invariant for optical ray tracing,” Opt. Express 17(13), 10564–10572 (2009).
[CrossRef] [PubMed]

Proc. SPIE (1)

A. W. Greynolds, “Propagation of generally astigmatic Gaussian beams along skew ray paths,” Proc. SPIE 560, 33–50 (1985).

Wave Motion (1)

M. M. Popov, “A new method of computation of wave fields using Gaussian beams,” Wave Motion 4(1), 85–97 (1982).
[CrossRef]

Other (5)

http://www.breault.com/software/asap.php

M. Herzberger, Modern Geometrical Optics, (Interscience Publishers, Inc. N.Y., 1958) pp. 383–400.

W. T. Welford, Aberrations of optical systems, (Academic Press, 1974), pp. 220.

R. Kigslake, Lens design fundamentals, (Academic Press, 1978), pp. 8.

W. J. Smith, Modern Optical Engineering, 3rd ed., (Mc Graw-Hill, 2000) p. 372.

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Figures (11)

Fig. 1
Fig. 1

Ray tracing through a singlet. The rays emerging from the slit are parallel to the optical axis.

Fig. 2
Fig. 2

Normalized intensity distribution at the back focal plane for the situation of Fig. 1

Fig. 3
Fig. 3

The incident beam is directed with an angle of 6°.

Fig. 4
Fig. 4

Normalized intensity distribution at the back focal plane for the situation of Fig. 3.

Fig. 5
Fig. 5

Normalized intensity distribution at the back focal plane according to Eq. (10).

Fig. 6
Fig. 6

Normalized intensity distribution at an observation plane located at f/2 to the right of the back focal plane. The incident beam is parallel to the optical axis.

Fig. 7
Fig. 7

Normalized intensity distribution at an observation plane located at f/2 to the right of the back focal plane. The incident beam is tilted 10° with respect to the optical axis.

Fig. 8
Fig. 8

Spherical mirror as described in the text for an object beam (red rays) parallel to the optical axis. The black lines represent reflected rays.

Fig. 9
Fig. 9

Normalized intensity distribution at the focal plane for the situation of Fig. 8.

Fig. 10
Fig. 10

Spherical mirror as described in the text. The object beam is tilted 8°.

Fig. 11
Fig. 11

Normalized intensity distribution at the focal plane for the situation of Fig. 10.

Equations (19)

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Ψ n ( x ) = P n exp ( i α n x ) exp ( i β n x 2 ) exp ( ( x A n ) 2 r n 2 ) exp ( i γ n [ x B n ] 2 ) ,
Ψ n + 1 ( ξ , z ) = exp ( i 2 π z / λ ) i λ z Ψ n ( x ) exp ( i π λ z ( x ξ ) 2 ) d x .
Ψ n + 1 ( ξ , z ) = P n + 1 exp ( i α n + 1 ξ ) exp ( i β n + 1 ξ 2 ) exp ( ( ξ A n + 1 ) 2 r n + 1 2 ) exp ( i γ n + 1 [ ξ B n + 1 ] 2 ) ,
P n + 1 = P n exp ( i 2 π z / λ ) i λ z π     r n 2 λ z λ z i r n 2 ( β n λ z + γ n λ z + π )     ×       exp ( i γ n ( B n ) 2 ) exp ( i λ z ( A n ) 2 r n 4 ( β n λ z + γ n λ z + π ) ) ,
α n + 1 = 0 ,     β n + 1 = π λ z ,     γ n + 1 = π 2 r n 4 D n λ z ( β n λ z + γ n λ z + π ) ,
r n + 1 = D n π r n ,
A n + 1 = A n + α n λ z 2 π γ n λ z ( B n ) π + ( β n + γ n ) λ z ( A n ) π ,
B n + 1 = α n λ z 2 π γ n λ z ( B n ) π λ 2 z 2 β n λ z + γ n λ z + π A n π r n 4 .
D n = λ 2 z 2 + r n 4 ( β n λ z + γ n λ z + π ) 2 .
α n = 2 π λ n tan ( θ n )     ,
α n + 1 = 2 π λ n + 1 tan ( θ n ) + 2 γ n + 1 B n + 1 2 ( β n + 1 + γ n + 1 ) A n + 1 .
P n + 1 = P n + 1 exp ( i α n + 1 A n + 1 ) .
Ψ T ( x ) = R ( x ) exp [ i φ ( x ) ] = m P 0 , m exp ( i α 0 , m x ) exp ( i β 0 , m x 2 ) exp ( ( x A 0 , m ) 2 r 0 2 ) × exp ( i γ 0 , m [ x A 0 , m ] 2 )     .
Ψ T ( x ) = m P 0 , m exp ( ( x A 0 , m ) 2 r 0 , m 2 ) × exp [ i { ( α 0 , m + β 0 , m A 0 , m ) A 0 , m + ( α 0 , m + 2 β 0 , m A 0 , m ) ( x A 0 , m ) + ( β 0 , m + γ 0 , m ) ( x A 0 , m ) 2 } ]     ,
φ ( x ) = { φ ( x m ) + d φ ( x m ) d x ( x x m ) } + 1 2 d 2 φ ( x m ) d x 2 ( x x m ) 2 ,
β 0 , m + γ 0 , m = 1 2 d 2 φ ( x m ) d x 2 ;     α 0 , m + 2 β 0 , m x m = d φ ( x m ) d x ;     ( α 0 , m + β 0 , m x m ) x m = φ ( x m ) .
P 0 , m = R ( x m )     ; α 0 , m = 1 x m [ 2 φ ( x m ) x m d φ ( x m ) d x ] ,
β 0 , m = 1 x m 2 [ x m d φ ( x m ) d x φ ( x m ) ]     ;     γ 0 , m = 1 x m 2 [ x m 2 d 2 φ ( x m ) d x 2 x m d φ ( x ) d x + 2 φ ( x m ) ] .
exp ( i [ 7.510 2 π λ f x 2 7.310 5 ( π λ f ) 2 x 4 ] ) .

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