Abstract

Decomposition of a general arbitrary field into a set of Gaussian beams has been one of the challenges in the Gaussian beam decomposition method for field propagation through optical systems. The most commonly used method in this regard is the Gabor expansion, which decomposes initial fields into shifted and rotated Gaussian beams in a plane. Since the Gaussian beams used have zero initial curvatures, the Gabor expansion method does not utilize the ability of the Gaussian beams to represent the quadratic behavior of the local wavefront. In this paper, we describe an alternative method of decomposing an arbitrary field with smooth wavefront into a set of Gaussian beams with non-zero initial curvatures. The individual Gaussian beams are used to represent up to the quadratic term in the Taylor expansion of the local wavefront. This significantly reduces the number of Gaussian beams required for the decomposition of the field with smooth wavefront and gives more accurate decomposition results. The proposed method directly gives the five ray sets representing the parabasal Gaussian beams, which can then be directly used for propagation of the Gaussian beams through optical systems. To demonstrate the application of the method, we have presented results for the decomposition of fields with strongly curved spherical wavefronts, a cone shaped wavefront, and a wavefront with large spherical aberration. The numerical comparison of the input field with the field reconstructed after the decomposition shows very good agreement in both amplitude and phase profiles. We also show results for the far field intensity distributions of the decomposed wavefronts by propagating in free space using the Gaussian beam propagation method.

© 2018 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  7. A. Goshtasby and D. Schonfeld, “Signal representation based on a Gaussian decomposition,” in Proceedings of the 1991 Conference Information Sciences and System (1991), pp. 1–6.
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    [Crossref]
  10. A. Rohani, A. Shishegar, and S. Safavi-Naeini, “A fast Gaussian beam tracing method for reflection and refraction of general vectorial astigmatic Gaussian beams from general curved surfaces,” Opt. Commun. 232, 1–10 (2004).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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  23. X. Li, Z. Li, and Z. Zeng, “Curvature analysis and geometric description of landforms using MATLAB,” in International Conference on Environmental Science and Information Application Technology (ESIAT) (IEEE, 2010), Vol. 1, pp. 712–715.
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  25. M. Mansuripur, “Distribution of light at and near the focus of high-numerical-aperture objectives: erratum Certain computational aspects of vector diffraction problems: erratum,” J. Opt. Soc. Am. A 10, 382–383 (1993).
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2017 (1)

2015 (2)

B. Andreas, G. Mana, and C. Palmisano, “Vectorial ray-based diffraction integral,” J. Opt. Soc. Am. A 32, 1403–1424 (2015).
[Crossref]

J. E. Harvey, R. G. Irvin, and R. N. Pfisterer, “Modeling physical optics phenomena by complex ray tracing,” Opt. Eng. 54, 035105 (2015).
[Crossref]

2013 (1)

2012 (1)

2011 (2)

2010 (1)

2004 (2)

J. Goldfeather and V. Interrante, “A novel cubic-order algorithm for approximating principal direction vectors,” ACM Trans. Graph. 23, 45–63 (2004).
[Crossref]

A. Rohani, A. Shishegar, and S. Safavi-Naeini, “A fast Gaussian beam tracing method for reflection and refraction of general vectorial astigmatic Gaussian beams from general curved surfaces,” Opt. Commun. 232, 1–10 (2004).
[Crossref]

2001 (2)

M. J. Bastiaans, “Gabor’s signal expansion based on a nonorthogonal sampling geometry,” Proc. SPIE 4392, 46–59 (2001).
[Crossref]

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

1993 (1)

1986 (4)

M. Mansuripur, “Distribution of light at and near the focus of high-numerical-aperture objectives,” J. Opt. Soc. Am. A 3, 2086–2093 (1986).
[Crossref]

A. W. Greynolds, “Vector formulation of the ray-equivalent method for general Gaussian beam propagation,” Proc. SPIE 679, 129–134 (1986).
[Crossref]

A. W. Greynolds, “Propagation of generally astigmatic Gaussian beams along skew ray paths,” Proc. SPIE 560, 33–52 (1986).
[Crossref]

P. Einziger, S. Raz, and M. Shapira, “Gabor representation and aperture theory,” J. Opt. Soc. Am. A 3, 508–522 (1986).
[Crossref]

1985 (1)

1970 (1)

J. Arnaud, “Nonorthogonal optical waveguides and resonators,” Bell Labs Tech. J. 49, 2311–2348 (1970).
[Crossref]

Andreas, B.

Arnaud, J.

J. Arnaud, “Representation of Gaussian beams by complex rays,” Appl. Opt. 24, 538–543 (1985).
[Crossref]

J. Arnaud, “Nonorthogonal optical waveguides and resonators,” Bell Labs Tech. J. 49, 2311–2348 (1970).
[Crossref]

Bastiaans, M. J.

M. J. Bastiaans, “Gabor’s signal expansion based on a nonorthogonal sampling geometry,” Proc. SPIE 4392, 46–59 (2001).
[Crossref]

M. J. Bastiaans, “The expansion of an optical signal into a discrete set of Gaussian beams,” in Erzeugung und Analyse von Bildern und Strukturen (Springer, 1980), pp. 23–32.

Broemel, A.

Bruegge, T.

B. Stone and T. Bruegge, “Practical considerations for simulating beam propagation: a comparison of three approaches,” in International Optical Design Conference (Optical Society of America, 2002), paper IWA3.

Chiaki, H.

H. T. Tanaka, M. Ikeda, and H. Chiaki, “Curvature-based face surface recognition using spherical correlation. Principal directions for curved object recognition,” in 3rd IEEE International Conference on Automatic Face and Gesture Recognition (IEEE, 1998), pp. 372–377.

Cywiak, M.

Einziger, P.

Fasshauer, G. E.

G. E. Fasshauer, Meshfree Approximation Methods with MATLAB (World Scientific, 2007).

Goldfeather, J.

J. Goldfeather and V. Interrante, “A novel cubic-order algorithm for approximating principal direction vectors,” ACM Trans. Graph. 23, 45–63 (2004).
[Crossref]

Gómez-Medina, R.

Goshtasby, A.

A. Goshtasby and D. Schonfeld, “Signal representation based on a Gaussian decomposition,” in Proceedings of the 1991 Conference Information Sciences and System (1991), pp. 1–6.

Greynolds, A. W.

A. W. Greynolds, “Propagation of generally astigmatic Gaussian beams along skew ray paths,” Proc. SPIE 560, 33–52 (1986).
[Crossref]

A. W. Greynolds, “Vector formulation of the ray-equivalent method for general Gaussian beam propagation,” Proc. SPIE 679, 129–134 (1986).
[Crossref]

A. W. Greynolds, “Fat rays revisited: a synthesis of physical and geometrical optics with Gaußlets,” in International Optical Design Conference (Optical Society of America, 2014), paper ITu1A.3.

Gross, H.

Hartung, J.

Harvey, J. E.

J. E. Harvey, R. G. Irvin, and R. N. Pfisterer, “Modeling physical optics phenomena by complex ray tracing,” Opt. Eng. 54, 035105 (2015).
[Crossref]

Ikeda, M.

H. T. Tanaka, M. Ikeda, and H. Chiaki, “Curvature-based face surface recognition using spherical correlation. Principal directions for curved object recognition,” in 3rd IEEE International Conference on Automatic Face and Gesture Recognition (IEEE, 1998), pp. 372–377.

Interrante, V.

J. Goldfeather and V. Interrante, “A novel cubic-order algorithm for approximating principal direction vectors,” ACM Trans. Graph. 23, 45–63 (2004).
[Crossref]

Irvin, R. G.

J. E. Harvey, R. G. Irvin, and R. N. Pfisterer, “Modeling physical optics phenomena by complex ray tracing,” Opt. Eng. 54, 035105 (2015).
[Crossref]

Kuhn, M.

F. Wyrowski and M. Kuhn, “Introduction to field tracing,” J. Mod. Opt. 58, 449–466 (2011).
[Crossref]

Li, X.

X. Li, Z. Li, and Z. Zeng, “Curvature analysis and geometric description of landforms using MATLAB,” in International Conference on Environmental Science and Information Application Technology (ESIAT) (IEEE, 2010), Vol. 1, pp. 712–715.

Li, Z.

X. Li, Z. Li, and Z. Zeng, “Curvature analysis and geometric description of landforms using MATLAB,” in International Conference on Environmental Science and Information Application Technology (ESIAT) (IEEE, 2010), Vol. 1, pp. 712–715.

Mana, G.

Mansuripur, M.

Morales, A.

Ochse, D.

Onural, L.

Palmisano, C.

Pfisterer, R. N.

J. E. Harvey, R. G. Irvin, and R. N. Pfisterer, “Modeling physical optics phenomena by complex ray tracing,” Opt. Eng. 54, 035105 (2015).
[Crossref]

Raz, S.

Rohani, A.

A. Rohani, A. Shishegar, and S. Safavi-Naeini, “A fast Gaussian beam tracing method for reflection and refraction of general vectorial astigmatic Gaussian beams from general curved surfaces,” Opt. Commun. 232, 1–10 (2004).
[Crossref]

Safavi-Naeini, S.

A. Rohani, A. Shishegar, and S. Safavi-Naeini, “A fast Gaussian beam tracing method for reflection and refraction of general vectorial astigmatic Gaussian beams from general curved surfaces,” Opt. Commun. 232, 1–10 (2004).
[Crossref]

Sahin, E.

Santoyo, F. M.

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

Schmidt, J. D.

J. D. Schmidt, Numerical Simulation of Optical Wave Propagation with Examples in MATLAB (SPIE, 2010).

Schonfeld, D.

A. Goshtasby and D. Schonfeld, “Signal representation based on a Gaussian decomposition,” in Proceedings of the 1991 Conference Information Sciences and System (1991), pp. 1–6.

Servín, M.

Shapira, M.

Shishegar, A.

A. Rohani, A. Shishegar, and S. Safavi-Naeini, “A fast Gaussian beam tracing method for reflection and refraction of general vectorial astigmatic Gaussian beams from general curved surfaces,” Opt. Commun. 232, 1–10 (2004).
[Crossref]

Stock, J.

Stone, B.

B. Stone and T. Bruegge, “Practical considerations for simulating beam propagation: a comparison of three approaches,” in International Optical Design Conference (Optical Society of America, 2002), paper IWA3.

Tanaka, H. T.

H. T. Tanaka, M. Ikeda, and H. Chiaki, “Curvature-based face surface recognition using spherical correlation. Principal directions for curved object recognition,” in 3rd IEEE International Conference on Automatic Face and Gesture Recognition (IEEE, 1998), pp. 372–377.

Wyrowski, F.

F. Wyrowski and M. Kuhn, “Introduction to field tracing,” J. Mod. Opt. 58, 449–466 (2011).
[Crossref]

Zeng, Z.

X. Li, Z. Li, and Z. Zeng, “Curvature analysis and geometric description of landforms using MATLAB,” in International Conference on Environmental Science and Information Application Technology (ESIAT) (IEEE, 2010), Vol. 1, pp. 712–715.

ACM Trans. Graph. (1)

J. Goldfeather and V. Interrante, “A novel cubic-order algorithm for approximating principal direction vectors,” ACM Trans. Graph. 23, 45–63 (2004).
[Crossref]

Appl. Opt. (2)

Bell Labs Tech. J. (1)

J. Arnaud, “Nonorthogonal optical waveguides and resonators,” Bell Labs Tech. J. 49, 2311–2348 (1970).
[Crossref]

J. Mod. Opt. (1)

F. Wyrowski and M. Kuhn, “Introduction to field tracing,” J. Mod. Opt. 58, 449–466 (2011).
[Crossref]

J. Opt. Soc. Am. A (6)

Opt. Commun. (2)

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

A. Rohani, A. Shishegar, and S. Safavi-Naeini, “A fast Gaussian beam tracing method for reflection and refraction of general vectorial astigmatic Gaussian beams from general curved surfaces,” Opt. Commun. 232, 1–10 (2004).
[Crossref]

Opt. Eng. (1)

J. E. Harvey, R. G. Irvin, and R. N. Pfisterer, “Modeling physical optics phenomena by complex ray tracing,” Opt. Eng. 54, 035105 (2015).
[Crossref]

Opt. Express (2)

Proc. SPIE (3)

A. W. Greynolds, “Vector formulation of the ray-equivalent method for general Gaussian beam propagation,” Proc. SPIE 679, 129–134 (1986).
[Crossref]

M. J. Bastiaans, “Gabor’s signal expansion based on a nonorthogonal sampling geometry,” Proc. SPIE 4392, 46–59 (2001).
[Crossref]

A. W. Greynolds, “Propagation of generally astigmatic Gaussian beams along skew ray paths,” Proc. SPIE 560, 33–52 (1986).
[Crossref]

Other (8)

A. W. Greynolds, “Fat rays revisited: a synthesis of physical and geometrical optics with Gaußlets,” in International Optical Design Conference (Optical Society of America, 2014), paper ITu1A.3.

B. Stone and T. Bruegge, “Practical considerations for simulating beam propagation: a comparison of three approaches,” in International Optical Design Conference (Optical Society of America, 2002), paper IWA3.

G. E. Fasshauer, Meshfree Approximation Methods with MATLAB (World Scientific, 2007).

A. Goshtasby and D. Schonfeld, “Signal representation based on a Gaussian decomposition,” in Proceedings of the 1991 Conference Information Sciences and System (1991), pp. 1–6.

M. J. Bastiaans, “The expansion of an optical signal into a discrete set of Gaussian beams,” in Erzeugung und Analyse von Bildern und Strukturen (Springer, 1980), pp. 23–32.

H. T. Tanaka, M. Ikeda, and H. Chiaki, “Curvature-based face surface recognition using spherical correlation. Principal directions for curved object recognition,” in 3rd IEEE International Conference on Automatic Face and Gesture Recognition (IEEE, 1998), pp. 372–377.

X. Li, Z. Li, and Z. Zeng, “Curvature analysis and geometric description of landforms using MATLAB,” in International Conference on Environmental Science and Information Application Technology (ESIAT) (IEEE, 2010), Vol. 1, pp. 712–715.

J. D. Schmidt, Numerical Simulation of Optical Wave Propagation with Examples in MATLAB (SPIE, 2010).

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Figures (22)

Fig. 1.
Fig. 1. Effect of the overlap factor on the amount of ripple on the top-hat amplitude profile and the slope of the edge.
Fig. 2.
Fig. 2. Rays used to represent a Gaussian beam at its waist. Here rays only in the yz cross section are shown. There exist an additional waist and divergence ray in the xz cross section. (The waist radius is not shown to scale. For a paraxial Gaussian beam the divergence angle is much smaller than depicted in the figure).
Fig. 3.
Fig. 3. Rays used to represent a parabasal Gaussian beam with initial curvature. The waist ray starts on the wavefront surface, and its direction is made perpendicular to the local wavefront. (Here rays in only one cross section are shown and the divergence angle and the waist radius are not shown to scale. For a paraxial Gaussian beam the divergence angle is much smaller than depicted on the figure).
Fig. 4.
Fig. 4. One-dimensional illustration of the idea of decomposing an arbitrary smooth wavefront into a set of Gaussian beams using an ideal wavefront shaping optical system. The wavefronts are indicated by a red curve, the Gaussian beams are indicated by blue envelopes, and their direction is shown with dark arrows.
Fig. 5.
Fig. 5. Sketch showing the surface normal, a normal plane, and the basis vectors on the tangential plane.
Fig. 6.
Fig. 6. Sketch showing different parameters of the rays representing a single Gaussian beam computed by the decomposition algorithm. Here only the yz cross section is shown for simplicity and the size of the Gaussian beam is exaggerated.
Fig. 7.
Fig. 7. Simple schematic diagram showing representation of a converging spherical wavefront (red) by the quadratic phase (blue) of a single Gaussian beam that is not at its waist.
Fig. 8.
Fig. 8. Residual phase profile of the Gaussian beam with (a) 1/e width of 0.5 mm and flat initial phase, (b) 1/e width of 0.5 mm and curved initial phase, and (c) 1/e width of 0.2 mm and curved initial phase used to represent a converging spherical wavefront with different radii of curvature. The field is computed directly on the wavefront surface, and the amplitude profile is shown with a dotted line with scale shown on the right-side y axis.
Fig. 9.
Fig. 9. Weighted RMS values of the residual wavefront error when a single Gaussian beam with different beam 1/e width w and having (a) curved and (b) flat initial phase is used to represent a spherical wavefront with different radius R. For the computation we have used 100 equidistant sampling points in the range y[3w,3w], where w is the 1/e width of the Gaussian beam. The logarithm values of the calculated results are shown in the plot, and the values on the y axis are different for the two plots.
Fig. 10.
Fig. 10. Fibonacci sampling scheme used for the distribution of the Gaussian beam for fields after circular aperture.
Fig. 11.
Fig. 11. Results for the decomposition of a spherical wavefront with NA=0.9 using 700 Gaussian beams having non-zero initial curvatures. (a) Input field amplitude profile on the wavefront surface. (b) Amplitude profile. (c) Corresponding amplitude error. (d) Residual phase error after the Gaussian beam decomposition.
Fig. 12.
Fig. 12. (a) Amplitude and (b) residual phase error of the decomposition of the spherical wavefront with radius of curvature of 11 mm using 700 Gaussian beams with zero initial curvatures.
Fig. 13.
Fig. 13. Cross-sectional view of the normalized intensity profile at the focal plane calculated using Gaussian beam decomposition with and without initial curvature compared with that calculated using the Fraunhofer propagation.
Fig. 14.
Fig. 14. (a) Simple sketch of a simple axicon system generating a cone shaped wavefront as shown in (b).
Fig. 15.
Fig. 15. (a) Maximum and (b) minimum curvature of the cone shaped wavefront together with their respective orientations indicated by the red lines.
Fig. 16.
Fig. 16. Optical invariant of the set of rays representing the Gaussian beams used to decompose the cone shaped wavefront. The green and red circles at r=1.45  mm and r=2.0  mm indicate the points where the optical invariant drops to L=0.2λ and L=0.1λ values, respectively.
Fig. 17.
Fig. 17. Results for the decomposition of a cone shaped wavefront with half cone angle of 60 deg using 4000 Gaussian beams having non-zero initial curvatures. (a) Input field amplitude profile on the wavefront surface. (b) Amplitude profile. (c) Corresponding amplitude error. (d) Residual phase error after the Gaussian beam decomposition.
Fig. 18.
Fig. 18. Normalized intensity plots calculated by propagating the cone shaped wavefront with uniform ring illumination using Gaussian beam decomposition method in free space: (a) yz cross section, (b) axial profile, and (c) transversal profile.
Fig. 19.
Fig. 19. Wavefront with strong spherical aberration to be decomposed to Gaussian beams (a) with and (b) without the spherical part.
Fig. 20.
Fig. 20. (a) Maximum and (b) minimum principal curvatures of the strongly aberrated wavefront together with their corresponding directions projected into the XY plane shown by the red lines.
Fig. 21.
Fig. 21. Results for the decomposition of a converging wavefront with spherical aberration of c9=100λ using 2400 Gaussian beams having non-zero initial curvatures. (a) Input field amplitude profile on the wavefront surface. (b) Amplitude profile. (c) Corresponding amplitude error. (d) Residual phase error after the Gaussian beam decomposition.
Fig. 22.
Fig. 22. (a) yz cross section of the intensity profile computed in the focal region of the converging wavefront with spherical aberration c9=100λ. (b) Geometric rays traced from the wavefront to the focal region.

Equations (23)

Equations on this page are rendered with MathJax. Learn more.

E(x,y)=A(x,y)exp(i2πϕ(x,y)),
OF=DbCb=2ω0Cb.
NG=W.OF2ω0,
a(xi)=k=1Nckgk(xi),
(g11g12g1Ng21g22g2NgM1gM2gMN)G(c1c2cN)c=(a1a2aM)a,
c=(GTG)1GTa.
rms=i=1MΔai2M,
ψ(r)=E0h1×h2·exp(ik(h1×r)(u2·r)(h2×r)(u1·r)2h1×h2),
h1×u2h2×u1=0.
SN=[fx,fy,1](1/D),
O=(EFFG)1I1(LMMN)II,
E=1+fx2,F=fxfy,G=1+fy2,L=fxx/D,M=fxy/D,N=fyy/D,
su=[1,0,fx],sv=[0,1,fy].
d1=k1(1)su+k1(2)sv,d2=k2(1)su+k2(2)sv,
Cp=(x,y,z),
Dpx=Dpy=Cp.
Ddx=cos(θx)Cd+sin(θx)Xl,Ddy=cos(θy)Cd+sin(θy)Yl,
Wpxt=Cp+wxXlWpyt=Cp+wyYl.
ϵ(y)=cy21+1c2y2cpy22,
ϵrms*=1A*Ig(y)(ϵ(y)ϵ¯*)2dy,
ϵrms*=1Ad*k=1NIg,k(ϵkϵ¯d*)2,
AS(x,y)=exp(x2+y2R2)m,
ϕ(x,y)=c(x2+y2)1+1c2(x2+y2)+c9(16(xn2+yn2)+6(xn2+yn2)2),

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