Abstract

A typical axially symmetric light beam on paraxial free propagation maintains the same transverse shape as at the waist plane for a certain range along its axis. We discuss a general procedure for estimating this range.

© 1996 Optical Society of America

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References

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  1. A. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  2. P. M. Mejías, H. Weber, R. Martínez-Herrero, A. González-Urena, eds., Proceedings of the Workshop on Laser Beam Characterization (Sociedad Espanola de Optica, Madrid, 1993).
  3. H. Weber, N. Reng, J. Lüdtke, P. M. Mejías, eds., Laser Beam Characterization (Festkörper-Laser-Institute, Berlin, 1994).
  4. A waist plane can be uniquely determined for any beam in paraxial propagation. 5
  5. A. Siegman, Proc. SPIE 1224, 2 (1990).
    [CrossRef]
  6. Of course, near the diffracting screen the paraxial approximation breaks down. Situations like this are then to be considered limiting cases.
  7. R. L. Phillips, L. C. Andrews, Appl. Opt. 22, 643 (1983).
    [CrossRef] [PubMed]
  8. P. A. Bélanger, Opt. Lett. 16, 196 (1991).
    [CrossRef] [PubMed]
  9. S. Flügge, Practical Quantum Mechanics (Springer-Verlag, Berlin, 1974), Problem 42.

1991 (1)

1990 (1)

A. Siegman, Proc. SPIE 1224, 2 (1990).
[CrossRef]

1983 (1)

Andrews, L. C.

Bélanger, P. A.

Flügge, S.

S. Flügge, Practical Quantum Mechanics (Springer-Verlag, Berlin, 1974), Problem 42.

Phillips, R. L.

Siegman, A.

A. Siegman, Proc. SPIE 1224, 2 (1990).
[CrossRef]

A. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

Appl. Opt. (1)

Opt. Lett. (1)

Proc. SPIE (1)

A. Siegman, Proc. SPIE 1224, 2 (1990).
[CrossRef]

Other (6)

Of course, near the diffracting screen the paraxial approximation breaks down. Situations like this are then to be considered limiting cases.

S. Flügge, Practical Quantum Mechanics (Springer-Verlag, Berlin, 1974), Problem 42.

A. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

P. M. Mejías, H. Weber, R. Martínez-Herrero, A. González-Urena, eds., Proceedings of the Workshop on Laser Beam Characterization (Sociedad Espanola de Optica, Madrid, 1993).

H. Weber, N. Reng, J. Lüdtke, P. M. Mejías, eds., Laser Beam Characterization (Festkörper-Laser-Institute, Berlin, 1994).

A waist plane can be uniquely determined for any beam in paraxial propagation. 5

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Equations (19)

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V ( r , z ) = [ 2 π υ 2 ( z ) ] 1 / 2 exp [ i k z i Φ ( z ) ] × exp { [ i k 2 R ( z ) 1 υ 2 ( z ) ] r 2 } × n = 0 c n exp [ 2 i n Φ ( z ) ] L n [ 2 r 2 υ 2 ( z ) ] .
υ ( z ) = υ 0 [ 1 + ( λ z π υ 0 2 ) 2 ] 1 / 2 , R ( z ) = z [ 1 + ( π υ 0 2 λ z ) 2 ] , tan Φ ( z ) = λ z π υ 0 2 ,
n = 0 | c n | 2 = 1 .
2 Φ ( z ) ( n max n min ) < φ
Δ r z 2 = Δ r 0 2 [ 1 + ( M 2 λ z 2 π Δ r 0 2 ) 2 ] ,
υ 0 = Δ r 0 2 M .
Δ r 0 2 = 2 m + 1 2 w 0 2 , M 2 = 2 m + 1 .
V r ( r , z ) = ( 2 π υ 0 2 ) 1 / 2 exp ( r 2 υ 0 2 ) n = 0 c n × exp [ 2 i ( n n ¯ ) Φ ( z ) ] L n ( 2 r 2 υ 0 2 ) ,
n ¯ = n = 0 n | c n | 2 .
( z ) = [ 2 π 0 | V r ( r , z ) V ( r , 0 ) | 2 r d r ] 1 / 2 .
( z ) = ( 2 n = 0 | c n | 2 { 1 cos [ 2 ( n n ¯ ) Φ ( z ) ] } ) 1 / 2 .
1 cos ( 2 α ) = 2 sin 2 α 2 α 2 ,
( z ) 2 Φ ( z ) [ n = 0 | c n | 2 ( n n ¯ ) 2 ] 1 / 2 = 2 Φ ( z ) Δ n ,
Δ n = n 2 ¯ n ¯ 2 , n 2 ¯ = n = 0 n 2 | c n | 2 .
2 Φ ( ζ ) Δ n = max .
ζ = π υ 0 2 λ tan ( max 2 Δ n ) .
H ^ L n ( 2 r 2 υ 0 2 ) exp ( r 2 υ 0 2 ) = n L n ( 2 r 2 υ 0 2 ) exp ( r 2 υ 0 2 ) ,
H ^ = 1 2 [ υ 0 2 4 1 r d d r ( r d d r ) + r 2 υ 0 2 1 ] .
n ¯ = 2 π 0 V * ( r , 0 ) H ^ V ( r , 0 ) r d r , n 2 ¯ = 2 π 0 V * ( r , 0 ) H ^ 2 V ( r , 0 ) r d r ,

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