Abstract

We explore a general type of stable Bessel beams in graded index media. The proposed axially symmetric medium is characterized by an “α” index profile. Explicit solutions for the radial envelope of the field E(r) are derived in terms of generalized Bessel functions. Emphasis is given on illustrating how far the conditions of the proposed modified structure permit only a Bessel function of the first kind to be uniquely retained in the solution. This paper considers both the optical and mathematical aspects. Some numerical examples corroborating our theoretical results are included, showing the stability, propagation, and diffraction of such Bessel beams.

© 2011 Optical Society of America

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References

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  1. J. C. Zapata-Rodriguez and J. J. Miret, “Diffraction-free beams in thin films,” J. Opt. Soc. Am. A 27, 663–670 (2010).
    [CrossRef]
  2. A. P. Sergey, W. Huang and M. Cada, “Dark and anti-dark diffraction free beams,” Opt. Lett. 32, 2508–2510 (2007).
    [CrossRef]
  3. Y. Z. Yu and W. B. Dou, “Vector analyses of nondiffracting Bessel beams,” Progr. Electromagn. Res. Lett. 5, 57–71 (2008).
    [CrossRef]
  4. J. Canning, “Diffraction free mode generation and propagation in optical waveguides,” Opt. Commun. 207, 35–39 (2002).
    [CrossRef]
  5. G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 6th ed. (Elsevier, 2005).
  6. F. E. Relton, Applied Bessel Functions (Blackie & Son, 1946).
  7. R. E. Collin, Field Theory of Guided Waves, 2nd ed. (IEEE, 1991).
  8. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, 1982).
  9. M. Bronstein and S. Lafaille, “Solutions of linear ordinary differential equations in terms of special functions,” Proceedings of the International Symposium on Algorithms and Computation, ISAAC ’2002 (ACM, 2002), pp. 23–28.
  10. R. Debeerst, M. Hoeij, and W. Koepf, Solving Differential Equations in Terms of Bessel Functions (ACM, 2008).
  11. Z. X. Wang and D. R. Guo, Special Functions (World Scientific, 1989).
  12. F. Bowman, Introduction to Bessel Functions (Dover, 1958).
  13. M. Shalaby, “Transformation of the three dimensional beam propagation method to two dimensions for cylindrically symmetric structures based on the Hankel transform,” Pure Appl. Opt. 5, 997–1004 (1996).
    [CrossRef]

2010 (1)

2008 (1)

Y. Z. Yu and W. B. Dou, “Vector analyses of nondiffracting Bessel beams,” Progr. Electromagn. Res. Lett. 5, 57–71 (2008).
[CrossRef]

2007 (1)

2002 (1)

J. Canning, “Diffraction free mode generation and propagation in optical waveguides,” Opt. Commun. 207, 35–39 (2002).
[CrossRef]

1996 (1)

M. Shalaby, “Transformation of the three dimensional beam propagation method to two dimensions for cylindrically symmetric structures based on the Hankel transform,” Pure Appl. Opt. 5, 997–1004 (1996).
[CrossRef]

Arfken, G. B.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 6th ed. (Elsevier, 2005).

Bowman, F.

F. Bowman, Introduction to Bessel Functions (Dover, 1958).

Bronstein, M.

M. Bronstein and S. Lafaille, “Solutions of linear ordinary differential equations in terms of special functions,” Proceedings of the International Symposium on Algorithms and Computation, ISAAC ’2002 (ACM, 2002), pp. 23–28.

Cada, M.

Canning, J.

J. Canning, “Diffraction free mode generation and propagation in optical waveguides,” Opt. Commun. 207, 35–39 (2002).
[CrossRef]

Collin, R. E.

R. E. Collin, Field Theory of Guided Waves, 2nd ed. (IEEE, 1991).

Debeerst, R.

R. Debeerst, M. Hoeij, and W. Koepf, Solving Differential Equations in Terms of Bessel Functions (ACM, 2008).

Dou, W. B.

Y. Z. Yu and W. B. Dou, “Vector analyses of nondiffracting Bessel beams,” Progr. Electromagn. Res. Lett. 5, 57–71 (2008).
[CrossRef]

Guo, D. R.

Z. X. Wang and D. R. Guo, Special Functions (World Scientific, 1989).

Hoeij, M.

R. Debeerst, M. Hoeij, and W. Koepf, Solving Differential Equations in Terms of Bessel Functions (ACM, 2008).

Huang, W.

Koepf, W.

R. Debeerst, M. Hoeij, and W. Koepf, Solving Differential Equations in Terms of Bessel Functions (ACM, 2008).

Lafaille, S.

M. Bronstein and S. Lafaille, “Solutions of linear ordinary differential equations in terms of special functions,” Proceedings of the International Symposium on Algorithms and Computation, ISAAC ’2002 (ACM, 2002), pp. 23–28.

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, 1982).

Miret, J. J.

Relton, F. E.

F. E. Relton, Applied Bessel Functions (Blackie & Son, 1946).

Sergey, A. P.

Shalaby, M.

M. Shalaby, “Transformation of the three dimensional beam propagation method to two dimensions for cylindrically symmetric structures based on the Hankel transform,” Pure Appl. Opt. 5, 997–1004 (1996).
[CrossRef]

Wang, Z. X.

Z. X. Wang and D. R. Guo, Special Functions (World Scientific, 1989).

Weber, H. J.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 6th ed. (Elsevier, 2005).

Yu, Y. Z.

Y. Z. Yu and W. B. Dou, “Vector analyses of nondiffracting Bessel beams,” Progr. Electromagn. Res. Lett. 5, 57–71 (2008).
[CrossRef]

Zapata-Rodriguez, J. C.

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

J. Canning, “Diffraction free mode generation and propagation in optical waveguides,” Opt. Commun. 207, 35–39 (2002).
[CrossRef]

Opt. Lett. (1)

Progr. Electromagn. Res. Lett. (1)

Y. Z. Yu and W. B. Dou, “Vector analyses of nondiffracting Bessel beams,” Progr. Electromagn. Res. Lett. 5, 57–71 (2008).
[CrossRef]

Pure Appl. Opt. (1)

M. Shalaby, “Transformation of the three dimensional beam propagation method to two dimensions for cylindrically symmetric structures based on the Hankel transform,” Pure Appl. Opt. 5, 997–1004 (1996).
[CrossRef]

Other (8)

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 6th ed. (Elsevier, 2005).

F. E. Relton, Applied Bessel Functions (Blackie & Son, 1946).

R. E. Collin, Field Theory of Guided Waves, 2nd ed. (IEEE, 1991).

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, 1982).

M. Bronstein and S. Lafaille, “Solutions of linear ordinary differential equations in terms of special functions,” Proceedings of the International Symposium on Algorithms and Computation, ISAAC ’2002 (ACM, 2002), pp. 23–28.

R. Debeerst, M. Hoeij, and W. Koepf, Solving Differential Equations in Terms of Bessel Functions (ACM, 2008).

Z. X. Wang and D. R. Guo, Special Functions (World Scientific, 1989).

F. Bowman, Introduction to Bessel Functions (Dover, 1958).

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Figures (13)

Fig. 1
Fig. 1

Change in the refractive index of the waveguide under study in example (1).

Fig. 2
Fig. 2

Beam profile at intermediate distances along the direction of propagation in example (1).

Fig. 3
Fig. 3

Input and output beam profiles in example (1).

Fig. 4
Fig. 4

Change in the refractive index of the waveguide under study in example (2).

Fig. 5
Fig. 5

Beam profile at intermediate distances along the direction of propagation in example (2).

Fig. 6
Fig. 6

Input and output beam profiles in example (2).

Fig. 7
Fig. 7

Change in the refractive index of the waveguide under study in example (3).

Fig. 8
Fig. 8

Beam profile at intermediate distances along the direction of propagation in example (3).

Fig. 9
Fig. 9

Input and output beam profiles in example (3).

Fig. 10
Fig. 10

Input (solid curve) and output (dashed curve) beam profiles (beam of order α = 0 , ordinary Bessel beam).

Fig. 11
Fig. 11

Input and output beam profiles (beam of order α = 1 ).

Fig. 12
Fig. 12

Input and output beam profiles (beam of order α = 2 ).

Fig. 13
Fig. 13

Input and output beam profiles (beam of order α = 3 ).

Equations (17)

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n ( r ) = n 0 [ 1 + ( Δ 2 / 2 ) r 2 α ] .
d 2 E d r 2 + 1 r d E d r + ( ( n 0 2 k 0 2 k z 2 ) + n 0 2 k o 2 Δ 2 r 2 α q 2 r 2 ) E = 0 ,
d 2 E d r 2 + 1 r d E d r + ( n 0 2 k o 2 Δ 2 r 2 α q 2 r 2 ) E = 0.
r 2 d 2 E d r 2 + r d E d r + ( a 2 b 2 r 2 b q 2 ) E = 0 ,
r h + 2 d 2 U d r 2 + r h + 1 d U d r + ( a 2 b 2 r 2 b ( h 2 + q 2 ) ) r h U = 0.
x 2 U + x U + ( x 2 ν 2 ) U = 0 ,
E ( r ) = r h Z ν ( a r b ) ,
E ( r ) = Z ν ( Δ n 0 k 0 α + 1 r α + 1 ) ,
d 2 E d r 2 + 1 r d E d r + ( n 0 2 k o 2 Δ 2 r 2 α ) E = 0 ,
E ( r ) = Z 0 ( Δ n 0 k 0 α + 1 r α + 1 ) .
E ( r ) = c 1 J 0 ( n 0 k Δ ( α + 1 ) r α + 1 ) + c 2 Y 0 ( n 0 k 0 Δ ( α + 1 ) r α + 1 ) ,
E ( r ) = E 0 J 0 ( n 0 k 0 Δ ( α + 1 ) r α + 1 ) , 0 r R .
E * ( r ) = c 3 I 0 ( n 0 k 0 Δ * ( α * + 1 ) r α * + 1 ) + c 4 K 0 ( n 0 k 0 Δ * ( α * + 1 ) r α * + 1 ) , r > R .
E ( r , z ) = n A n ( z ) J 0 ( P n w r ) ,
A n ( z ) = 1 V n 0 w [ E ( r , z ) J 0 ( P n w r ) ] r d r ,
A n ( z + δ z ) = A n ( z ) [ exp ( i δ z ( n 0 2 k 0 2 ( P n / w ) 2 ) 1 / 2 ) ] .
E ( r , z + δ z ) = E ( r , z ) [ exp ( i δ z χ ) ] ,

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