Abstract

A set of orthogonal functions, convenient for the aberration analysis of the quasi-gaussian beams such as the laser TEM00 mode is derived. It is a set of hypergeometric functions. We can expand the aberration function of a quasi-gaussian beam in terms of it, and can calculate the diffraction patterns of aberrations. This set of functions has some characteristics of orthogonality to each other and relations to the Bessel functions. It bears a resemblance to the Zernike’s circle polynomials, but has some different characteristics. It is useful in holography, optical communications, and optical measurement.

© 1974 Optical Society of America

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References

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  1. H. Kogelnik, Bell Syst. Tech. J. 43, 334 (1964).
    [Crossref]
  2. T. S. Chu, Bell Syst. Tech. J. 45, 287 (1966).
    [Crossref]
  3. F. Zernike, Physica 1, 689 (1934).
    [Crossref]
  4. D. Slepian, Bell Syst. Tech. J. 43, 3009 (1964).
    [Crossref]
  5. G. Boyd and J. Gordon, Bell Syst. Tech. J. 40, 489 (1961).
    [Crossref]
  6. H. Statz and C. Tang, J. Appl. Phys. 36, 1816 (1965).
    [Crossref]
  7. G. Szerö, Orthogonal Polynomials (American Mathematical Society, Providence, R. I., 1939), p. 62.
  8. E. Linfoot, Recent Advances in Optics (Clarendon, Oxford, 1955), p. 48.
  9. B. Nijboer, Thesis (Groningen University, 1942).

1966 (1)

T. S. Chu, Bell Syst. Tech. J. 45, 287 (1966).
[Crossref]

1965 (1)

H. Statz and C. Tang, J. Appl. Phys. 36, 1816 (1965).
[Crossref]

1964 (2)

H. Kogelnik, Bell Syst. Tech. J. 43, 334 (1964).
[Crossref]

D. Slepian, Bell Syst. Tech. J. 43, 3009 (1964).
[Crossref]

1961 (1)

G. Boyd and J. Gordon, Bell Syst. Tech. J. 40, 489 (1961).
[Crossref]

1934 (1)

F. Zernike, Physica 1, 689 (1934).
[Crossref]

Boyd, G.

G. Boyd and J. Gordon, Bell Syst. Tech. J. 40, 489 (1961).
[Crossref]

Chu, T. S.

T. S. Chu, Bell Syst. Tech. J. 45, 287 (1966).
[Crossref]

Gordon, J.

G. Boyd and J. Gordon, Bell Syst. Tech. J. 40, 489 (1961).
[Crossref]

Kogelnik, H.

H. Kogelnik, Bell Syst. Tech. J. 43, 334 (1964).
[Crossref]

Linfoot, E.

E. Linfoot, Recent Advances in Optics (Clarendon, Oxford, 1955), p. 48.

Nijboer, B.

B. Nijboer, Thesis (Groningen University, 1942).

Slepian, D.

D. Slepian, Bell Syst. Tech. J. 43, 3009 (1964).
[Crossref]

Statz, H.

H. Statz and C. Tang, J. Appl. Phys. 36, 1816 (1965).
[Crossref]

Szerö, G.

G. Szerö, Orthogonal Polynomials (American Mathematical Society, Providence, R. I., 1939), p. 62.

Tang, C.

H. Statz and C. Tang, J. Appl. Phys. 36, 1816 (1965).
[Crossref]

Zernike, F.

F. Zernike, Physica 1, 689 (1934).
[Crossref]

Bell Syst. Tech. J. (4)

D. Slepian, Bell Syst. Tech. J. 43, 3009 (1964).
[Crossref]

G. Boyd and J. Gordon, Bell Syst. Tech. J. 40, 489 (1961).
[Crossref]

H. Kogelnik, Bell Syst. Tech. J. 43, 334 (1964).
[Crossref]

T. S. Chu, Bell Syst. Tech. J. 45, 287 (1966).
[Crossref]

J. Appl. Phys. (1)

H. Statz and C. Tang, J. Appl. Phys. 36, 1816 (1965).
[Crossref]

Physica (1)

F. Zernike, Physica 1, 689 (1934).
[Crossref]

Other (3)

G. Szerö, Orthogonal Polynomials (American Mathematical Society, Providence, R. I., 1939), p. 62.

E. Linfoot, Recent Advances in Optics (Clarendon, Oxford, 1955), p. 48.

B. Nijboer, Thesis (Groningen University, 1942).

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Figures (4)

Fig. 1
Fig. 1

Profile of the laser beam.

Fig. 2
Fig. 2

Laser-beam optical system.

Fig. 3
Fig. 3

Useful range of G20 (solid curve). Results of direct calculations (dashes).

Fig. 4
Fig. 4

Useful range of G40 (solid curve). Results of direct calculations (dashes).

Tables (1)

Tables Icon

Table I Laser-beam polynomials (LBP).

Equations (45)

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( 1 - r 2 / q ) ( q - 1 ) ,             ( 7 q 5 q r 0 )
Δ U + α ( x / x + y / y ) 2 U + β ( x / x + y / y ) U + γ U = 0.
r ( 1 + α r 2 ) 2 U / r 2 + [ 1 + ( α + β ) r 2 ] U / r + [ γ r + ( 1 / r ) 2 / φ 2 ] U = 0.
U ( r , φ ) = G ( r ) cos sin m φ .
r ( 1 + α r 2 ) d 2 G / d r 2 + [ 1 + ( α + β ) r 2 ] d G / d r + ( γ r - m 2 / r ) G = 0.
d d r [ r ( 1 - r 2 / q ) q d G d r ] + ( γ r - m 2 r ) ( 1 - r 2 q ) ( q - 1 ) G = 0.
4 ( 1 - p / q ) p d 2 G / d p 2 + 4 [ 1 - ( q + 1 ) p / q ] d G / d p + ( γ - m 2 / p ) G = 0 ,
p ( 1 - p / q ) d 2 F / d p 2 + [ 1 + m - p ( m + 1 + q ) / q ] d F / d p + 0.25 × [ γ - ( m 2 + 2 m q ) / q ] F = 0.
F = F 1 1 ( ( k + m ) / 2 + q ,             - ( k - m ) / 2 ,             m + 1 p / q ) .
F = 1 ( ( k + m ) / 2 ( k - m ) / 2 ) r 2 m ( 1 - r 2 / q ) ( q - 1 ) [ d d ( r 2 ) ] ( k - m ) / 2 · [ ( r 2 ) ( k + m ) / 2 ( 1 - r 2 / q ) { ( k - m ) / 2 + q - 1 } ] ,
( x y ) = x ! y ! ( x - y ) ! .
G k m = r m 1 F 1 ( ( k + m ) / 2 + q , - ( k - m ) / 2 , m + 1 r 2 / q ) .
( G k m , G k m ) = 0 q ( 1 - r 2 / q ( q - 1 ) G k m G k m r d r = 1 2 q ( m + 1 ) ( k + q ) ( ( k + m ) / 2 ( k - m ) / 2 ) ( ( k + m ) / 2 + q - 1 ( k - m ) / 2 + q - 1 ) ( when k = k ) = 0 ( when k k ) .
G k m = 1 ( ( k + m ) / 2 ( k - m ) / 2 ) r m ( 1 - r 2 / q ) ( q - 1 ) [ d d ( r 2 ) ] ( k - m ) / 2 · [ ( r 2 ) ( k + m ) / 2 ( 1 - r 2 / q ) { ( k - m ) / 2 + q - 1 } ] .
G k m = n = 0 ( k - m ) / 2 C n r m + 2 n ,
( G k m , G k m ) = n = 0 ( k - m ) / 2 C n 0 q ( 1 - r 2 / q ) ( q - 1 ) r m + 2 n · 1 ( ( k + m ) / 2 ( k - m ) / 2 ) r m ( 1 - r 2 / q ) ( q - 1 ) [ d d ( r 2 ) ] ( k - m ) / 2 · [ ( r 2 ) ( k + m ) / 2 ( 1 - r 2 / q ) { ( k - m ) / 2 + q - 1 } ] r d r = n = 0 ( k - m ) / 2 [ C n / ( ( k + m ) / 2 ( k - m ) / 2 ) ] × 0 q r 2 n ( d d ( r 2 ) ) ( k - m ) / 2 · [ ( r 2 ) ( k + m ) / 2 ( 1 - r 2 / q ) { ( k - m ) / 2 + q - 1 } ] r d r = n = 0 ( k - m ) / 2 C n Q ( n , ( k - m ) / 2 , ( k + m ) / 2 ) / ( ( k + m ) / 2 ( k - m ) / 2 ) ,
Q ( n , ( k - m ) / 2 , ( k + m ) / 2 ) = 0 q r 2 n [ d d ( r 2 ) ] ( k - m ) / 2 · [ ( r 2 ) ( k + m ) / 2 ( 1 - r 2 / q ) { ( k - m ) / 2 + q - 1 } ] r d r .
Q ( n , s , p ) = 0 q r 2 n [ d d ( r 2 ) ] s × [ ( r 2 ) p ( 1 - r 2 / q ) ( s + q - 1 ) ] r d r .
Q ( n , s , p ) = 0 q X n [ d d X ] s [ X p ( 1 - X / q ) ( s + q - 1 ) ] d X / 2 = 0.5 × { X n [ d d X ] ( s - 1 ) [ X p ( 1 - X / q ) ( s + q - 1 ) ] 0 q - n 0 q X ( n - 1 ) [ d d X ] ( s - 1 ) × [ X p ( 1 - X / q ) ( s + q - 1 ) ] d X } .
Q ( n , s , p ) = ( - n ) Q ( n - 1 , s - 1 , p ) .
Q ( n , s , p ) = ( - n ) ( - n + 1 ) ( - 1 ) Q ( 0 , s - n , p ) = ( - ) n n ! 0 q [ d d X ] ( s - n ) [ X p ( 1 - X / q ) ( s + q - 1 ) ] d X = ( - ) n n ! [ d d X ] ( s - n - 1 ) × [ X p ( 1 - X / q ) ( s + q - 1 ) ] 0 q = 0.
Q ( n , s , p ) = ( - ) n n ! 0 q X p ( 1 - X / q ) ( n + q - 1 ) d X = ( - ) n n ! q ( n + 1 ) · p ! ( n + q - 1 ) ! ( p + n + q ) ! .
Q ( n , ( k - m ) / 2 , ( k + m ) / 2 ) = ( - ) ( k - m ) / 2 q [ ( k + m ) / 2 + 1 ] · [ ( k - m ) / 2 ] ! [ ( k + m ) / 2 ] ! [ ( k - m ) / 2 + q - 1 ] ! ( k + q ) ! if k = k , = 0 if k k .
( G k m , G k m ) = C ( k - m ) / 2 · ( - ) ( k - m ) / 2 2 ( ( k + m ) / 2 ( k - m ) / 2 ) q [ ( k + m ) / 2 + 1 ] · [ ( k + m ) / 2 ] ! [ ( k - m ) / 2 + q - 1 ] ! ( k + q ) ! = q ( m + 1 ) 2 ( k + q ) ( ( k + m ) / 2 ( k - m ) / 2 ) ( ( k + m ) / 2 + q - 1 ( k - m ) / 2 + q - 1 ) if             k = k = 0 if             k k ,
C ( k - m ) / 2 = ( - ) ( k - m ) / 2 m ! ( q + k - 1 ) ! [ ( m - k ) / q 2 ] [ ( k + m ) / 2 ] ! [ ( k + m ) / 2 + q - 1 ] ! .
0 q ( 1 - r 2 / q ) ( q - 1 ) G k m ( r ) J m ( ρ r ) r d r = ( ρ / 2 ) m ( ( k + m ) / 2 ( k - m ) / 2 ) s ( - ) s ( ρ / 2 ) 2 s s ! ( s + m ) ! × 0 q r ( 2 s + 1 ) [ d d ( r 2 ) ] ( k - m ) / 2 · [ ( r 2 ) ( k + m ) / 2 ( 1 - r 2 / q ) { ( k - m ) / 2 + q - 1 } ] d r = ( ρ / 2 ) m ( ( k + m ) / 2 ( k - m ) / 2 ) s = 0 ( - ) s ( ρ / 2 ) 2 s s ! ( s + m ) ! × Q ( s , ( k - m ) / 2 , ( k + m ) / 2 ) = ( ρ / 2 ) m ( ( k + m ) / 2 ( k - m ) / 2 ) s = 0 ( - ) s ( ρ / 2 ) 2 s s ! ( - ) ( k - m ) / 2 s ! ( s + m ) ! [ s - ( k - m ) / 2 ] ! × Q ( s - ( k - m ) / 2 , 0 , ( k + m ) / 2 ) = q ( m / 2 + 1 ) [ k ( k - m ) / 2 + q - 1 ] ! J k + q ( q ρ ) 2 · ( ( k + m ) / 2 ( k - m ) / 2 ) ( q ρ / 2 ) q .
W = l n m a l n m Θ ( 2 l + m ) r n cos m φ .
W = k m b k m R k m ( r ) cos m φ .
W = k m C k m G k m ( r ) cos m φ .
I ( 0 ) = | 0 q 0 2 π ( 1 - r 2 / q ) ( q - 1 ) e i k ˜ W r d r d φ | 2 ,
e i k ˜ W = 1 + i k ˜ W - k ˜ 2 W 2 2
I ( 0 ) = | 0 q 0 2 π ( 1 - r 2 / q ) ( q - 1 ) ( 1 + i k ˜ W - k ˜ 2 W 2 2 ) r d r d φ | 2 = | π - k ˜ 2 2 0 q 0 2 π ( 1 - r 2 / q ) ( q - 1 ) · C k m C k m · G k m G k m cos m φ cos m φ r d r d φ | 2 = | π - k ˜ 2 π 2 m 0 0 q 0 2 π ( 1 - r 2 / q ) ( q - 1 ) C k m C k m · G k m G k m r d r - k ˜ 2 π 0 q ( 1 - r 2 / q ) ( q - 1 ) C k 0 C k 0 G k 0 G k 0 r d r | 2 = | π - k ˜ 2 π 4 m 0 C k m 2 q ( m + 1 ) ( k + q ) ( ( k + m ) / 2 ( k - m ) / 2 ) ( ( k + m ) / 2 + q - 1 ( k - m ) / 2 + q - 1 ) - k ˜ 2 π 2 C k 0 2 q ( k + q ) | 2 ,
( 1 - r 2 / q ) ( q - 1 ) e i k ˜ W ,
d ( ρ , θ ) = 0 q 0 2 π ( 1 - r 2 / q ) ( q - 1 ) ) e i k ˜ W e - i ρ r cos ( θ - φ ) r d r d φ = 0 q 0 2 π ( 1 - r 2 / q ) ( q - 1 ) ( 1 + i k ˜ W - k ˜ 2 W 2 2 + ) · e - i ρ r cos ( θ - φ ) r d r d φ = π q ! ( q ρ / 2 ) q J q ( q ρ ) + i k ˜ ( 1 - r 2 / q ) ( q - 1 ) × W e - i ρ r cos ( θ - φ ) r d r d φ + 2 ( i k ˜ ) n n ! ( 1 - r 2 / q ) ( q - 1 ) × W n e - i ρ r cos ( θ - φ ) r d r d φ .
i k ˜ 0 q 0 2 π ( 1 - r 2 / q ) ( q - 1 ) W e - i ρ r cos ( θ - φ ) r d r d φ = k m i k ˜ 0 q 0 2 π ( 1 - r 2 / q ) ( q - 1 ) C k m G k m ( r ) × cos m φ e - i ρ r cos ( θ - φ ) r d r d φ = i k ˜ ( i ) m cos m θ C k m 2 π 0 q ( 1 - r 2 / q ) ( q - 1 ) × G k m ( r ) J m ( ρ r ) r d r .
i k ˜ 2 k m ( i ) m cos m θ × C k m [ ( k - m ) / 2 + q - 1 ] ! q ( m / 2 + 1 ) J k + q ( q ρ ) × 2 π ( ( k + m ) / 2 ( k - m ) / 2 ) ( q ρ / 2 ) q .
{ G k m } 2 = A ˜ 0 G 0 0 + A ˜ 2 G 2 0 + + A ˜ 2 k G 2 k 0 = B ˜ 2 m G 2 m 2 m + B ˜ 2 m + 2 G m + 2 2 m + + B ˜ 2 k G 2 k 2 m ,
{ G k m } 3 = C ˜ m G m m + C ˜ m + 2 2 G m + 2 2 + + C ˜ 3 k G 3 k m = D ˜ 3 m G 3 m 3 m + D ˜ 3 m + 2 G 3 m + 2 3 m + + D ˜ 3 k G 3 k 3 m ,
{ G k m } 4 = E ˜ 0 G 0 0 + E ˜ 2 G 2 0 + + E ˜ 4 k G 4 k 0 = F ˜ 2 m G 2 m 2 m + F ˜ 2 m + 2 G m + 2 2 m + + F ˜ 4 k G 4 k 2 m = H ˜ 4 m G 4 m 4 m + H ˜ 4 m + 2 G 4 m + 2 G m + 2 4 m + + H ˜ 4 k G 4 k 4 m ,
d ( ρ , θ ) = π q ! ( q ρ / 2 ) q J q ( q ρ ) + i k ˜ 2 k m ( i ) m cos m θ × C k m ( k - m / 2 + q - 1 ) ! q ( m / 2 + 1 ) × 2 π ( ( k + m ) / 2 ( k - m ) / 2 ) ( q ρ / 2 ) q + n = 2 ( i k ˜ ) n n ! k m ( C k m ) n 0 q 0 2 π ( 1 - r 2 / q ) ( q - 1 ) × { G k m } n cos n m θ e - i ρ r cos ( θ - φ ) r d r d φ = π q ! ( q ρ / 2 ) q J q ( q ρ ) + i k ˜ 2 k m ( i ) m cos m θ × C k m [ ( k - m ) / 2 + q - 1 ] ! q ( m / 2 + 1 ) × 2 π ( ( k + m ) / 2 ( k - m ) / 2 ) ( q ρ / 2 ) q + n = 2 ( i k ˜ ) n n ! k m ( C k m ) n s t A s t ( i ) s cos ( s θ ) × [ ( t - s ) / 2 + q - 1 ] ! q ( s / 2 + 1 ) ( ( t + s ) / 2 ( t - s ) / 2 ) ( q ρ / 2 ) q J s + q ( q ρ ) × π ,
{ G k m } n cos n m θ = s t A s t G t s cos s θ
W = k m C k m G k m cos m φ .
W = k m C k m G k m ( r ) cos m φ + k m D k m G k m ( r ) sin m φ .
I ( 0 ) normalized = 1 - ( 5 / 14 ) ( k ˜ C 20 ) 2 2
I ( 0 ) normalized = 1 - ( 5 / 18 ) k ˜ C 40 2 2 .