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  1. H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
    [CrossRef]
  2. D. R. Thomkins, P. F. Rodney, Phys. Rev. A 12, 599 (1975).
    [CrossRef]
  3. W. H. Carter, Appl. Opt. 19, 1027 (1980).
    [CrossRef] [PubMed]
  4. R. J. Pressley, Ed., CRC Laser Handbook (CRC Press, Cleveland, 1971), pp. 421–447.
  5. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).
  6. L. J. Slater, Confluent Hypergeometric Functions (Cambridge U. P., London, 1960).
  7. W. H. Carter, Appl. Opt. 21, 7 (1982).
    [CrossRef] [PubMed]

1982 (1)

1980 (1)

1975 (1)

D. R. Thomkins, P. F. Rodney, Phys. Rev. A 12, 599 (1975).
[CrossRef]

1966 (1)

H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

Carter, W. H.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

Kogelnik, H.

H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

Li, T.

H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

Rodney, P. F.

D. R. Thomkins, P. F. Rodney, Phys. Rev. A 12, 599 (1975).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

Slater, L. J.

L. J. Slater, Confluent Hypergeometric Functions (Cambridge U. P., London, 1960).

Thomkins, D. R.

D. R. Thomkins, P. F. Rodney, Phys. Rev. A 12, 599 (1975).
[CrossRef]

Appl. Opt. (2)

Phys. Rev. A (1)

D. R. Thomkins, P. F. Rodney, Phys. Rev. A 12, 599 (1975).
[CrossRef]

Proc. IEEE (1)

H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

Other (3)

R. J. Pressley, Ed., CRC Laser Handbook (CRC Press, Cleveland, 1971), pp. 421–447.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

L. J. Slater, Confluent Hypergeometric Functions (Cambridge U. P., London, 1960).

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Tables (1)

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Table I Fraction of Beam Energy Contained Within the Spot Size as a Function of m and p for a Laguerre Gaussian Beam

Equations (20)

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W ( z ) = W 0 [ 1 + ( λ z π W 0 2 ) 2 ] 1 / 2 ,
σ r 2 ( z ) m p = 2 0 2 π 0 r 2 I ( r , ϕ , z ) r d r d ϕ 0 2 π 0 I ( r , ϕ , z ) r d r d ϕ ,
I ( r , ϕ , z ) = K ( z ) ( 2 r 2 W 2 ) m [ L p m ( 2 r 2 W 2 ) ] 2 exp ( 2 r 2 / W 2 ) { sin m ϕ cos m ϕ } 2.
σ r 2 ( z ) m p = W 2 ( z ) 0 x m + 1 [ L p m ( x ) ] 2 exp ( x ) d x 0 x m [ L p m ( x ) ] 2 exp ( x ) d x ,
0 x m [ L p m ( x ) ] 2 exp ( x ) d x = ( p + m ) ! p ! .
0 x m + 1 [ L p m ( x ) ] 2 exp ( x ) d x = ( p + m ) ! p ! ( 2 p + m + 1 ) ,
σ r ( z ) m p = W ( z ) ( 2 p + m + 1 ) 1 / 2 ,
σ r ( z ) m p = W 0 ( 2 p + m + 1 ) 1 / 2 [ 1 + ( λ z π W 0 2 ) 2 ] 1 / 2 .
π ( 2 p + m 1 2 ) / 4 ( 2 p + m + 1 ) 1 / 2 .
π ( 2 p + m 1 2 ) 4 ( 2 p + m + 1 ) 1 / 2 < ( 2 p + m + 1 ) 1 / 2
π 4 ( 2 p + m 1 2 ) < 2 p + m + 1.
I ( r ) = K ( z ) ( 2 r 2 ) m [ L p m ( 2 r 2 ) ] 2 exp ( 2 r 2 ) ,
I ( r ) = K ( z ) 2 m + 1 exp ( 2 r 2 ) L p m ( 2 r 2 ) × [ ( m 2 r 2 ) L p m ( 2 r 2 ) 4 r 2 L p 1 m + 1 ( 2 r 2 ) ] ,
L p m ( x ) ( 1 ) p p ! x p , x 1 ,
L p m ( 2 r 2 ) [ ( m 2 r 2 ) L p m ( 2 r 2 ) 4 r 2 L p 1 m + 1 ( 2 r 2 ) ] [ 2 σ r 2 ( z ) m p ] 2 p ( p ! ) 2 [ 2 p + m 2 σ r 2 ( z ) m p ] [ 2 ( 2 p + m + 1 ) ] 2 p ( p ! ) 2 ( 2 p m 1 ) ,
E m , p = 0 a x m [ L p m ( x ) ] 2 exp ( x d x ) = 1 ( p + m p ) e a n = 0 p ( p ) n n ! × k = 0 p ( p ) k ( m + n + 1 ) k ( m + 1 ) k K ! e m + n + k a ,
a = 2 ( 2 p + m + 1 ) ,
( p + m p ) = ( p + m ) ! p ! m ! ,
( b ) k = Γ ( b + k ) Γ ( b ) ,
e n x = k = 0 n x k k ! .

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