Abstract

A novel family of paraxial laser beams called elegant Hermite–Laguerre–Gaussian beams (EHLGBs) are presented. The EHLGBs, a unity of elegant Hermite–Gaussian beams (EHGBs) and elegant Laguerre–Gaussian beams (ELGBs), constitute the exact and continuous transition modes between EHGBs and ELGBs when an additional parameter continuously changes. The virtual source for generation of the EHLGBs is identified. From the spectral representation of the elegant Hermite–Lauguerre–Gaussian waves, we derive the first three orders of nonparaxial corrections for the corresponding paraxial EHLGBs.

© 2008 Optical Society of America

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References

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  1. A. E. Siegman, J. Opt. Soc. Am. 63, 1093 (1973).
    [CrossRef]
  2. A. E. Siegman, Lasers (University Science, 1986), Chap. 16, pp. 644-652.
  3. M. A. Bandres and J. C. Gutiérrez-Vega, Opt. Lett. 29, 144 (2004).
    [CrossRef] [PubMed]
  4. M. A. Bandres, Opt. Lett. 29, 1724 (2004).
    [CrossRef] [PubMed]
  5. T. Takenaka, M. Yokota, and O. Fukumitsu, J. Opt. Soc. Am. A 2, 826 (1985).
    [CrossRef]
  6. M. A. Bandres and J. C. Gutiérrez-Vega, Opt. Lett. 29, 2213 (2004).
    [CrossRef] [PubMed]
  7. G. A. Deschamps, Electron. Lett. 7, 684 (1971).
    [CrossRef]
  8. L. B. Felsen, J. Opt. Soc. Am. 66, 751 (1976).
    [CrossRef]
  9. S. Y. Shin and L. B. Felsen, J. Opt. Soc. Am. 67, 699 (1977).
    [CrossRef]
  10. A. A. Tovar, J. Opt. Soc. Am. A 15, 2705 (1998).
    [CrossRef]
  11. S. R. Seshadri, Opt. Lett. 27, 998 (2002).
    [CrossRef]
  12. S. R. Seshadri, Opt. Lett. 27, 1872 (2002).
    [CrossRef]
  13. S. R. Seshadri, Opt. Lett. 28, 595 (2003).
    [CrossRef] [PubMed]
  14. R. Borghi and M. Santarsiero, J. Opt. Soc. Am. A 21, 2029 (2004).
    [CrossRef]
  15. D. M. Deng, J. Opt. Soc. Am. B 23, 1228 (2006).
    [CrossRef]
  16. D. M. Deng, Q. Guo, L. J. Wu, and X. B. Yang, J. Opt. Soc. Am. B 24, 636 (2007).
    [CrossRef]
  17. D. M. Deng, Q. Guo, S. Lan, and X. Yang, J. Opt. Soc. Am. A 24, 3317 (2007).
    [CrossRef]
  18. S. R. Seshadri, J. Opt. Soc. Am. A 19, 2134 (2002).
    [CrossRef]
  19. R. Borghi and M. Santarsiero, Opt. Lett. 28, 774 (2003).
    [CrossRef] [PubMed]
  20. Y. C. Zhang, Y. J. Song, Z. R. Chen, J. H. Ji, and Z. X. Shi, Opt. Lett. 32, 292 (2007).
    [CrossRef] [PubMed]
  21. E. G. Abramochkin and V. G. Volostnikov, J. Opt. A, Pure Appl. Opt. 6, S157 (2004).
    [CrossRef]
  22. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1970).

2007 (3)

2006 (1)

2004 (5)

2003 (2)

2002 (3)

1998 (1)

1985 (1)

1977 (1)

1976 (1)

1973 (1)

1971 (1)

G. A. Deschamps, Electron. Lett. 7, 684 (1971).
[CrossRef]

Abramochkin, E. G.

E. G. Abramochkin and V. G. Volostnikov, J. Opt. A, Pure Appl. Opt. 6, S157 (2004).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1970).

Bandres, M. A.

Borghi, R.

Chen, Z. R.

Deng, D. M.

Deschamps, G. A.

G. A. Deschamps, Electron. Lett. 7, 684 (1971).
[CrossRef]

Felsen, L. B.

Fukumitsu, O.

Guo, Q.

Gutiérrez-Vega, J. C.

Ji, J. H.

Lan, S.

Santarsiero, M.

Seshadri, S. R.

Shi, Z. X.

Shin, S. Y.

Siegman, A. E.

A. E. Siegman, J. Opt. Soc. Am. 63, 1093 (1973).
[CrossRef]

A. E. Siegman, Lasers (University Science, 1986), Chap. 16, pp. 644-652.

Song, Y. J.

Stegun, I.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1970).

Takenaka, T.

Tovar, A. A.

Volostnikov, V. G.

E. G. Abramochkin and V. G. Volostnikov, J. Opt. A, Pure Appl. Opt. 6, S157 (2004).
[CrossRef]

Wu, L. J.

Yang, X.

Yang, X. B.

Yokota, M.

Zhang, Y. C.

Electron. Lett. (1)

G. A. Deschamps, Electron. Lett. 7, 684 (1971).
[CrossRef]

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

E. G. Abramochkin and V. G. Volostnikov, J. Opt. A, Pure Appl. Opt. 6, S157 (2004).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (5)

J. Opt. Soc. Am. B (2)

Opt. Lett. (8)

Other (2)

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1970).

A. E. Siegman, Lasers (University Science, 1986), Chap. 16, pp. 644-652.

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Figures (1)

Fig. 1
Fig. 1

Propagation dynamics of the EHLGB in free space. Transverse normalized intensity distributions at (a)–(c) z = 0 plane and (d)–(f) z = z R plane. Phase distributions at (g)–(i) z = 0 plane and (j)–(l) z = z R plane. The different columns represent the different θ given in the top of the figure. The parameters are chosen as n = 5 , m = 3 ; z R = k w 0 2 2 is the Rayleigh range.

Equations (24)

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Δ E + k 2 E = 0 ,
2 Φ x 2 + 2 Φ y 2 + 2 i k Φ z = 0 .
Φ n , m ( x , y , z θ ) = T n m ( x , y , θ ) E 0 ( x , y , z ) = [ c ( z ) ] n + m E 0 ( x , y , z ) s = 0 n + m i s cos n s θ sin m s θ P s ( n s , m s ) ( cos 2 θ ) H n + m s [ c ( z ) x ] H s [ c ( z ) y ] ,
T n m ( x , y , θ ) = ( x cos θ + i y sin θ ) n ( x sin θ i y cos θ ) m ,
Φ n , m ( x , y , z 0 ) = [ c ( z ) ] n + m E 0 ( x , y , z ) ( i ) m H n [ c ( z ) x ] H m [ c ( z ) y ] ,
Φ n , m ( x , y , z π 4 ) = [ c ( z ) ] n + m E 0 ( x , y , z ) ( 1 ) min 2 max × min ! [ c ( z ) r ] n m L min n m [ c 2 ( z ) r 2 ] × exp [ i n m φ ] ,
( Δ + k 2 ) E n , m = S cs T n m ( x , y , θ ) δ ( x ) δ ( y ) δ ( z z cs ) .
E n , m ( x , y , z ) = + d p x d p y E ̃ n , m ( p x , p y , z ) exp ( i 2 π v ) ,
E ̃ n , m ( p x , p y , z ) = + E n , m ( x , y , z ) exp ( i 2 π v ) d x d y ,
E n , m ( x , y , z ) = i 2 S cs ( i 2 π ) n + m + d p x d p y exp ( i 2 π v ) T n m ( p x , p y , θ ) exp [ i ζ ( z z cs ) ] ζ ,
E n , m , p ( x , y , z ) = i 2 k exp [ i k ( z z cs ) ] ( i 2 π ) n + m S cs × + d p x d p y exp [ i 2 π v ] × T n m ( p x , p y , θ ) exp [ i k 2 π 2 p 2 ( z z cs ) ] .
E n , m , p ( x , y , z ) = exp ( i k z ) S cs T n m ( x , y , θ ) × { exp ( i k z cs ) 4 π ( z z cs ) exp [ i k ( x 2 + y 2 ) 2 ( z z cs ) ] } .
z cs = i k w 0 2 2 = i z R ,
S cs = 4 i π z R w 0 n + m exp ( k z R ) .
E n , m ( x , y , z ) = 2 π z R ( i 2 π w 0 ) n + m exp ( k z R ) × + d p x d p y 1 ζ exp ( i 2 π v ) × T n m ( p x , p y , θ ) exp [ i ζ ( z i z R ) ] .
( Δ + k 2 ) G ( r , z ) = S cs δ ( x ) δ ( y ) δ ( z z cs )
G ( r , z ) = S cs exp ( i k R ) ( 4 π R ) ,
E n , m ( x , y , z ) = i z R w 0 n + m exp ( k z R ) T n m ( x , y , θ ) [ exp ( i k R ) R ] .
E n , m ( x , y , z ) = π w 0 n + m + 2 exp ( i k z ) + d p x d p y exp ( i 2 π v ) T n m ( p x , p y , θ ) × exp [ i 2 π 2 p 2 ( z z e x ) k ] j = 0 3 G ̃ ( 2 j ) ( 4 π 2 w 0 2 p 2 , z ) ( k w 0 ) 2 j ,
G ̃ ( 2 ) ( t , z ) = a 2 , 1 t a 2 , 2 t 2 q 2 ( z ) ,
G ̃ ( 4 ) ( t , z ) = a 4 , 2 t 2 a 4 , 3 t 3 q 2 ( z ) + a 4 , 4 t 4 q 4 ( z ) ,
G ̃ ( 6 ) ( t , z ) = a 6 , 3 t 3 a 6 , 4 t 4 q 2 ( z ) + a 6 , 5 t 5 q 4 ( z ) a 6 , 6 t 6 q 6 ( z ) ,
E n , m ( x , y , z ) = exp ( i k z ) [ c ( z ) ] n + m E 0 ( x , y , z ) s = 0 n + m i s × cos n s θ sin m s θ P s ( n s , m s ) ( cos 2 θ ) × j = 0 3 ( k w 0 ) 2 j ( 1 ) j q 2 j ( z ) f n , m ( 2 j ) ( x , y , z ) ,
f n , m ( 2 j ) ( x , y , z ) = t = j 2 j a 2 j , t l = 0 t t ! l ! ( t l ) ! H n + m s + 2 ( t l ) [ c ( z ) x ] H s + 2 l [ c ( z ) y ] .

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