Abstract

Many ray-optics models have been proposed to describe the propagation of paraxial Gaussian beam. However, those paraxial ray-optics models are inapplicable to the beams that violate the paraxial condition. In this paper, we present a skew line ray (SLR) based model to represent the propagation properties of nonparaxial Gaussian beam under the oblate spheroidal coordinates. The free-space evolution of complex wavefront of the light beam including amplitude and phase is derived via this model. Our analysis demonstrates that the SLR model is available for both nonparaxial and paraxial conditions, and can be used to precisely describe the propagation of complex wavefront. Furthermore, this model changes the transverse density of rays while propagating. The behavior influences the transverse intensity distribution and makes the optical rays become concentrated towards the center. We believe that this ray-optics model can be further developed to describe other kind of structured beams such as Laguerre-Gauss and Bessel-Gauss beams.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref] [PubMed]
  27. D. A. McNamara, C. W. I. Pistorius, and J. A. G. Malherbe, Introduction to the Uniform Geometrical Theory of Diffraction (ARTECH HOUSE, 1990), Chap. 2.
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2017 (1)

2014 (2)

2013 (1)

B. Shi, Z. Meng, L. Wang, and T. Liu, “Monte Carlo modeling of human tooth optical coherence tomography imaging,” J. Opt. 15(7), 075304 (2013).
[Crossref]

2010 (2)

Y. Kaganovsky and E. Heyman, “Wave analysis of Airy beams,” Opt. Express 18(8), 8440–8452 (2010).
[Crossref] [PubMed]

D. J. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light forces the pace: optical manipulation for biophotonics,” J. Biomed. Opt. 15(4), 041503 (2010).
[Crossref] [PubMed]

2008 (2)

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
[Crossref] [PubMed]

M. V. Berry and K. T. McDonald, “Exact and geometrical optics energy trajectories in twisted beams,” J. Opt. A, Pure Appl. Opt. 10(3), 035005 (2008).
[Crossref]

2007 (1)

B. Z. Wang, Z. G. Zhao, B. D. Lü, and K. L. Duan, “Vectorial Hermite–Laguerre–Gaussian beams beyond the paraxial approximation,” Chin. Phys. 16(1), 143–147 (2007).
[Crossref]

2005 (1)

2004 (1)

2002 (4)

2000 (1)

1999 (1)

Y. A. Kravtsov, G. W. Frobes, and A. A. Asatryan, “Theory and applications of complex rays,” Prog. Opt. 39, 1–67 (1999).
[Crossref]

1998 (2)

L. Novotny, E. J. Sanchez, and X. S. Xie, “Near-field optical imaging using metal tips illuminated by higher-order Hermite—Gaussian beams,” Ultramicroscopy 71(1-4), 21–29 (1998).
[Crossref]

H. Laabs, “Propagation of Hermite-Gaussian-beams beyond the paraxial approximation,” Opt. Commun. 147(1-3), 1–4 (1998).
[Crossref]

1992 (1)

A. Ashkin, “Forces of a Single-Beam Gradient Laser Trap on a Dielectric Sphere in the Ray Optics Regime,” Biophys. J. 61(2), 569–582 (1992).
[Crossref] [PubMed]

1989 (1)

1988 (1)

1985 (2)

1979 (1)

1975 (1)

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11(4), 1365–1370 (1975).
[Crossref]

1964 (1)

L. A. Vainshtein, “Open resonators with spherical mirrors,” Sov. Phys. JETP 18, 471–479 (1964).

Agrawal, G. P.

Alonso, M. A.

Andersen, P. E.

Arnaud, J.

Asatryan, A. A.

Y. A. Kravtsov, G. W. Frobes, and A. A. Asatryan, “Theory and applications of complex rays,” Prog. Opt. 39, 1–67 (1999).
[Crossref]

Ashkin, A.

A. Ashkin, “Forces of a Single-Beam Gradient Laser Trap on a Dielectric Sphere in the Ray Optics Regime,” Biophys. J. 61(2), 569–582 (1992).
[Crossref] [PubMed]

Bandres, M. A.

Barrett, H. H.

Berry, M. V.

M. V. Berry and K. T. McDonald, “Exact and geometrical optics energy trajectories in twisted beams,” J. Opt. A, Pure Appl. Opt. 10(3), 035005 (2008).
[Crossref]

Broky, J.

Chávez-Cerda, S.

Chen, C. G.

Christodoulides, D. N.

Davidson, F. M.

Dennis, M. R.

Dholakia, K.

D. J. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light forces the pace: optical manipulation for biophotonics,” J. Biomed. Opt. 15(4), 041503 (2010).
[Crossref] [PubMed]

Dogariu, A.

Duan, K.

Duan, K. L.

B. Z. Wang, Z. G. Zhao, B. D. Lü, and K. L. Duan, “Vectorial Hermite–Laguerre–Gaussian beams beyond the paraxial approximation,” Chin. Phys. 16(1), 143–147 (2007).
[Crossref]

Ferrera, J.

Frobes, G. W.

Y. A. Kravtsov, G. W. Frobes, and A. A. Asatryan, “Theory and applications of complex rays,” Prog. Opt. 39, 1–67 (1999).
[Crossref]

Fukumitsu, O.

Gunn-Moore, F.

D. J. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light forces the pace: optical manipulation for biophotonics,” J. Biomed. Opt. 15(4), 041503 (2010).
[Crossref] [PubMed]

Heilmann, R. K.

Heyman, E.

Jono, T.

Jørgensen, T. M.

Kaganovsky, Y.

Konkola, P. T.

Kravtsov, Y. A.

Y. A. Kravtsov, G. W. Frobes, and A. A. Asatryan, “Theory and applications of complex rays,” Prog. Opt. 39, 1–67 (1999).
[Crossref]

Laabs, H.

H. Laabs, “Propagation of Hermite-Gaussian-beams beyond the paraxial approximation,” Opt. Commun. 147(1-3), 1–4 (1998).
[Crossref]

Landesman, B. T.

Lax, M.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11(4), 1365–1370 (1975).
[Crossref]

Liu, T.

B. Shi, Z. Meng, L. Wang, and T. Liu, “Monte Carlo modeling of human tooth optical coherence tomography imaging,” J. Opt. 15(7), 075304 (2013).
[Crossref]

Louisell, W. H.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11(4), 1365–1370 (1975).
[Crossref]

Lü, B.

Lü, B. D.

B. Z. Wang, Z. G. Zhao, B. D. Lü, and K. L. Duan, “Vectorial Hermite–Laguerre–Gaussian beams beyond the paraxial approximation,” Chin. Phys. 16(1), 143–147 (2007).
[Crossref]

Ludlow, I. K.

McDonald, K. T.

M. V. Berry and K. T. McDonald, “Exact and geometrical optics energy trajectories in twisted beams,” J. Opt. A, Pure Appl. Opt. 10(3), 035005 (2008).
[Crossref]

McKnight, W. B.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11(4), 1365–1370 (1975).
[Crossref]

Meng, Z.

B. Shi, Z. Meng, L. Wang, and T. Liu, “Monte Carlo modeling of human tooth optical coherence tomography imaging,” J. Opt. 15(7), 075304 (2013).
[Crossref]

Nakagawa, K.

Novotny, L.

L. Novotny, E. J. Sanchez, and X. S. Xie, “Near-field optical imaging using metal tips illuminated by higher-order Hermite—Gaussian beams,” Ultramicroscopy 71(1-4), 21–29 (1998).
[Crossref]

Pattanayak, D. N.

Ricklin, J. C.

Rodríguez-Morales, G.

Sanchez, E. J.

L. Novotny, E. J. Sanchez, and X. S. Xie, “Near-field optical imaging using metal tips illuminated by higher-order Hermite—Gaussian beams,” Ultramicroscopy 71(1-4), 21–29 (1998).
[Crossref]

Schattenburg, M. L.

Shi, B.

B. Shi, Z. Meng, L. Wang, and T. Liu, “Monte Carlo modeling of human tooth optical coherence tomography imaging,” J. Opt. 15(7), 075304 (2013).
[Crossref]

Siviloglou, G. A.

Stevenson, D. J.

D. J. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light forces the pace: optical manipulation for biophotonics,” J. Biomed. Opt. 15(4), 041503 (2010).
[Crossref] [PubMed]

Takenaka, T.

Toyoshima, M.

Tycho, A.

Ulanowski, Z.

Vainshtein, L. A.

L. A. Vainshtein, “Open resonators with spherical mirrors,” Sov. Phys. JETP 18, 471–479 (1964).

Wang, B.

Wang, B. Z.

B. Z. Wang, Z. G. Zhao, B. D. Lü, and K. L. Duan, “Vectorial Hermite–Laguerre–Gaussian beams beyond the paraxial approximation,” Chin. Phys. 16(1), 143–147 (2007).
[Crossref]

Wang, L.

B. Shi, Z. Meng, L. Wang, and T. Liu, “Monte Carlo modeling of human tooth optical coherence tomography imaging,” J. Opt. 15(7), 075304 (2013).
[Crossref]

Xie, X. S.

L. Novotny, E. J. Sanchez, and X. S. Xie, “Near-field optical imaging using metal tips illuminated by higher-order Hermite—Gaussian beams,” Ultramicroscopy 71(1-4), 21–29 (1998).
[Crossref]

Yamamoto, A.

Yokota, M.

Yura, H. T.

Zhao, Z. G.

B. Z. Wang, Z. G. Zhao, B. D. Lü, and K. L. Duan, “Vectorial Hermite–Laguerre–Gaussian beams beyond the paraxial approximation,” Chin. Phys. 16(1), 143–147 (2007).
[Crossref]

Appl. Opt. (2)

Biophys. J. (1)

A. Ashkin, “Forces of a Single-Beam Gradient Laser Trap on a Dielectric Sphere in the Ray Optics Regime,” Biophys. J. 61(2), 569–582 (1992).
[Crossref] [PubMed]

Chin. Phys. (1)

B. Z. Wang, Z. G. Zhao, B. D. Lü, and K. L. Duan, “Vectorial Hermite–Laguerre–Gaussian beams beyond the paraxial approximation,” Chin. Phys. 16(1), 143–147 (2007).
[Crossref]

J. Biomed. Opt. (1)

D. J. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light forces the pace: optical manipulation for biophotonics,” J. Biomed. Opt. 15(4), 041503 (2010).
[Crossref] [PubMed]

J. Opt. (1)

B. Shi, Z. Meng, L. Wang, and T. Liu, “Monte Carlo modeling of human tooth optical coherence tomography imaging,” J. Opt. 15(7), 075304 (2013).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

M. V. Berry and K. T. McDonald, “Exact and geometrical optics energy trajectories in twisted beams,” J. Opt. A, Pure Appl. Opt. 10(3), 035005 (2008).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (7)

Opt. Commun. (1)

H. Laabs, “Propagation of Hermite-Gaussian-beams beyond the paraxial approximation,” Opt. Commun. 147(1-3), 1–4 (1998).
[Crossref]

Opt. Express (4)

Opt. Lett. (2)

Optica (1)

Phys. Rev. A (1)

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11(4), 1365–1370 (1975).
[Crossref]

Prog. Opt. (1)

Y. A. Kravtsov, G. W. Frobes, and A. A. Asatryan, “Theory and applications of complex rays,” Prog. Opt. 39, 1–67 (1999).
[Crossref]

Sov. Phys. JETP (1)

L. A. Vainshtein, “Open resonators with spherical mirrors,” Sov. Phys. JETP 18, 471–479 (1964).

Ultramicroscopy (1)

L. Novotny, E. J. Sanchez, and X. S. Xie, “Near-field optical imaging using metal tips illuminated by higher-order Hermite—Gaussian beams,” Ultramicroscopy 71(1-4), 21–29 (1998).
[Crossref]

Other (2)

M. Dietrich, Light Transmission Optics (Van Nostrand Reinhold, 1972), Chap. 6.

D. A. McNamara, C. W. I. Pistorius, and J. A. G. Malherbe, Introduction to the Uniform Geometrical Theory of Diffraction (ARTECH HOUSE, 1990), Chap. 2.

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Figures (9)

Fig. 1
Fig. 1 Two skew lines of point Q( x Q , y Q ,0 ). |Q S 1 |=| Q S 2 |.
Fig. 2
Fig. 2 Sketch of the rays arriving at point S outside the waist plane.
Fig. 3
Fig. 3 Wave front of SLR. The increment Δα corresponds to the segment Δ l 0 .
Fig. 4
Fig. 4 Sketch of the ray tracing along the skew line.
Fig. 5
Fig. 5 Rays’ trajectories of nonparaxial SLR model with different parameter. (a) kD=100; (b) kD=1.
Fig. 6
Fig. 6 Ray’s distribution patterns in an observation plane ( z=5 z r ). (a) Initial plane z=0; (b) and (d) are traced by paraxial SLR model for kD=100 and kD=1, respectively. (c) and (e) are traced by nonparaxial SLR model for kD=100 and kD=1, respectively.
Fig. 7
Fig. 7 Comparison of the intensity distribution in the plane of z=5 z r between nonparaxial and paraxial SLR models. (a) kD=100; (b) kD=1.
Fig. 8
Fig. 8 Rays’ trajectories with length equal to λ. (a) and (c) are traced by paraxial SLR model for kD=100 and kD=1. (b) and (d) are traced by nonparaxial SLR model for kD=100 and kD=1, respectively.
Fig. 9
Fig. 9 Axial patterns of phase. kD=1. (a) and (b) traced by paraxial and nonparaxial SLR model. (c) and (d) are calculated by Eq. (25) and Eq. (2) respectively.

Equations (28)

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x=D ( 1+ ξ 2 )( 1 η 2 ) cosθ, x=D ( 1+ ξ 2 )( 1 η 2 ) sinθ, z=Dξη,
ψ( ξ,η,θ )= exp[ kD( 1η ) ] kD ξ 2 + η 2 exp( ikDξ )exp( iarctan ξ η ),
D= ω 0 sinε , ω 0 = 1 k 2( 1+ 1+ k 2 D 2 ) .
x 2 + y 2 D 2 ( 1 η Q 2 ) z 2 D 2 η Q 2 =1.
η Q = 1 x Q 2 + y Q 2 D 2 .
l Q + : x= x Q y Q τ D η Q , y= y Q + x Q τ D η Q , z=τ,
l Q : x= x Q + y Q τ D η Q , y= y Q x Q τ D η Q , z=τ,
ψ GO ( S )= ψ GO + ( S )+ ψ GO ( S ),
ψ GO + ( S )= 1 2 A Q 1 J Q 1 J S exp[ i( φ Q 1 + φ l ) ],
ψ GO ( S )= 1 2 A Q 2 J Q 2 J S exp[ i( φ Q 2 + φ l ) ].
J=1+ ( τ D η Q n 2 ) 2 , n=1, 2.
d φ l dl = darg( U ) dl = d [ kD ξ S arctan( ξ S / η S ) ] dl ,
ξ S = l D .
φ l =klarctan l D η Q .
α+Δα=arctan l D η Q .
ψ GO ( S )= n=1 2 1/2 A Q n 1+ [ z/ ( D η Q n 2 ) ] 2 exp[ i( klarctan l D η Q n ) ].
ψ GO ( S )= exp[ kD( 1 η S ) ] kD ξ S 2 + η S 2 exp[ i( kD ξ S arctan ξ S η S ) ].
l=| QS |=| Q S 1 |+| S 1 S |,
kD>>1, η1,
lim ε0 D= lim ε0 ω 0 sinε = π ω 0 2 λ = z r ,
R( z )=z( 1+ D 2 z 2 )z( 1+ z r 2 z 2 ),
lz+ x 2 + y 2 2R( z ) .
l Q + : x= x Q y Q τ z r , y= y Q + x Q τ z r , z=τ,
l Q : x= x Q + y Q τ z r , y= y Q x Q τ z r , z=τ.
J=1+ ( z z r ) 2 .
φ l =k[ z+ x 2 + y 2 2R( z ) ]arctan z z r .
ψ GO ( S )= 1 1+ ( z/ z r ) 2 exp( x Q 2 + y Q 2 ω 0 2 ) ×exp{ ik[ z+ x 2 + y 2 2R( z ) ] }exp( iarctan z z r ).
ψ GO ( S )= ω 0 ω( z ) exp[ x 2 + y 2 ω 2 ( z ) ] ×exp{ ik[ z+ x 2 + y 2 2R( z ) ] }exp( iarctan z z r ).

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