Abstract

When a freely propagating light beam passes through a limiting aperture into an interferometer a set of the interferometer modes is coupled even when the beam is matched to the interferometer. The incoming beam and the modes of the interferometer are assumed to be described by Gaussian Laguerre functions and the limiting aperture to be circular and on axis with the modes. The coupling coefficients are obtained for an arbitrary incoming mode and arbitrary expanding set. Simpler solutions are found for an incoming Gaussian beam by choosing the expanding set matched to the incoming beam parameters.

© 1976 Optical Society of America

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References

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  1. G. Goubau, F. Schwering, IRE Trans. Antennas Propag. AP-9, 248 (1961).
    [CrossRef]
  2. A short account of a few results were given by O. Andrade, C. G. Thomas, “Mode Conversion and Transmission by Circular Apertures,” in Conference in Opto-Electronics, Southampton 1969. For a full discussion see O. Andrade, Thesis, “Laser Interferometer Method for the Measurement of Small Changes in Optical Path” (Southampton University, 1971).
  3. H. Kogelink, T. Li, Appl. Opt. 5, 1550 (1966).
    [CrossRef]
  4. For the definition of the parameters of Gaussian beam waves see, for example, Ref. 3.
  5. E. D. Rainville, Special Functions (Chelsea, New York, 1960).
  6. A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 1.
  7. A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transform (McGraw-Hill, New York, 1954), Vol. 1.

1966 (1)

1961 (1)

G. Goubau, F. Schwering, IRE Trans. Antennas Propag. AP-9, 248 (1961).
[CrossRef]

Andrade, O.

A short account of a few results were given by O. Andrade, C. G. Thomas, “Mode Conversion and Transmission by Circular Apertures,” in Conference in Opto-Electronics, Southampton 1969. For a full discussion see O. Andrade, Thesis, “Laser Interferometer Method for the Measurement of Small Changes in Optical Path” (Southampton University, 1971).

Erdélyi, A.

A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 1.

A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transform (McGraw-Hill, New York, 1954), Vol. 1.

Goubau, G.

G. Goubau, F. Schwering, IRE Trans. Antennas Propag. AP-9, 248 (1961).
[CrossRef]

Kogelink, H.

Li, T.

Magnus, W.

A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transform (McGraw-Hill, New York, 1954), Vol. 1.

A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 1.

Oberhettinger, F.

A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 1.

A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transform (McGraw-Hill, New York, 1954), Vol. 1.

Rainville, E. D.

E. D. Rainville, Special Functions (Chelsea, New York, 1960).

Schwering, F.

G. Goubau, F. Schwering, IRE Trans. Antennas Propag. AP-9, 248 (1961).
[CrossRef]

Thomas, C. G.

A short account of a few results were given by O. Andrade, C. G. Thomas, “Mode Conversion and Transmission by Circular Apertures,” in Conference in Opto-Electronics, Southampton 1969. For a full discussion see O. Andrade, Thesis, “Laser Interferometer Method for the Measurement of Small Changes in Optical Path” (Southampton University, 1971).

Tricomi, F. G.

A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transform (McGraw-Hill, New York, 1954), Vol. 1.

A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 1.

Appl. Opt. (1)

IRE Trans. Antennas Propag. (1)

G. Goubau, F. Schwering, IRE Trans. Antennas Propag. AP-9, 248 (1961).
[CrossRef]

Other (5)

A short account of a few results were given by O. Andrade, C. G. Thomas, “Mode Conversion and Transmission by Circular Apertures,” in Conference in Opto-Electronics, Southampton 1969. For a full discussion see O. Andrade, Thesis, “Laser Interferometer Method for the Measurement of Small Changes in Optical Path” (Southampton University, 1971).

For the definition of the parameters of Gaussian beam waves see, for example, Ref. 3.

E. D. Rainville, Special Functions (Chelsea, New York, 1960).

A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 1.

A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transform (McGraw-Hill, New York, 1954), Vol. 1.

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Figures (1)

Fig. 1
Fig. 1

Coordinates system and beam parameters for the incoming mode (dashed parameters) and the expanding set; only the fundamental mode of the base set is shown.

Equations (49)

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ψ p l ( r , θ , z ) = A p l ( z ) N p l ( r 2 ω ) l L p l ( 2 r 2 ω 2 ) × exp ( - r 2 ω 2 - j k r 2 2 R ) cos ( l θ ) ,
A p l ( z ) = exp { j [ ( 2 p + l + 1 ) tan - 1 ( λ z π ω 0 2 ) - k z ] }
N p l = 2 ω [ π ( 1 + δ 0 l ) ] 1 / 2 [ p ! ( p + l ) ! ] 1 / 2
0 2 π d θ 0 r d r ψ p l 2 = 1.
ψ = ψ p ¯ l ¯ ( ω ¯ ) H ( a - r ) ,
H ( a - r ) = { 1 for r a , 0 for r > a ,
ψ = p , l c p ¯ l ¯ , p l ψ p l ( ω ) ,
c p ¯ l ¯ , p l = 0 2 π d θ 0 a ψ p ¯ l ¯ ( ω ¯ ) ψ p l * ( ω ) r d r .
K p ¯ p , l = c p ¯ p , l 2 .
R ¯ = R ; ω ¯ = ω ; p ¯ = p .
c p p , l = 0 2 π d θ 0 a ψ p l 2 r d r ,
K p p , l = c p p , l 2 = P T 2 .
c p ¯ p , l = A p ¯ l ( z ¯ ) A p l * ( z ) N p ¯ p , l α l + 1 0 u 0 u l L p l ( u ) × L p ¯ l ( α 2 u ) exp ( - β u ) d u ,
N p ¯ p , l = [ p ¯ ! p ! ( p ¯ + l ) ! ( p + l ) ! ] 1 / 2 ; α = ω ω ¯ ; u = 2 r 2 ω 2 ; u 0 = 2 a 2 ω 2 ; β = 1 2 ( 1 + α 2 ) - j k ω 2 4 ( 1 R ¯ - 1 R ) .
c 0 ¯ p , 0 = A 0 ¯ 0 ( z ) A p 0 * ( z ) α 0 u 0 L p ( u ) exp ( - β u ) d u ,
c 0 ¯ p , 0 = A 0 ¯ 0 ( z ¯ ) A p 0 * ( z ) α u 0 ϕ 2 [ - p , p + 1 ; 2 ; ( 1 - β ) u 0 , - β u 0 ] ,
ϕ 2 ( λ 1 , λ 2 ; c ; x 1 , x 2 ) = j , k = 0 ( λ 1 ) j ( λ 2 ) k ( c ) j + k x 1 j j ! x 2 k k ! ,
( λ ) j = ( λ + j - 1 ) ! ( λ - 1 ) ! λ > 0 , ( - λ ) j = ( - 1 ) j λ ! ( λ - j ) ! λ j > 0 , ( - λ ) j = 0 λ < j .
c 0 ¯ 0 , 0 = A 0 ¯ 0 ( z ¯ ) A 00 * ( z ) α β [ 1 - exp ( - β u 0 ) ] ,
κ 0 ¯ 0 , 0 = α 2 β 2 1 - exp ( - β u 0 ) 2 ,
c 0 ¯ 1 , 0 = A 0 ¯ 0 ( z ¯ ) A 10 * ( z ) α u 0 { exp ( - β u 0 ) [ 1 β + 1 β u 0 ( 1 β - 1 ) ] - 1 β u 0 ( 1 β - 1 ) } ,
c 0 ¯ 2 , 0 = A 0 ¯ 0 ( z ¯ ) A 20 * ( z ) α u 0 ( ( 1 - β ) 2 u 0 β 3 ( 1 + β u 0 ) 2 + exp ( - β u 0 ) { 1 + u 0 2 ( β - 2 ) - ( 1 - β ) 2 2 u 0 β 3 [ 1 + ( 1 + β u 0 ) 2 ] } ) .
ϕ 2 ( λ 1 , λ 2 ; c ; 0 , x 2 ) = k = 0 ( λ 2 ) k ( c ) k x 2 k k ! ,
c 0 ¯ p , 0 = A p u 0 ϕ ( p + 1 , 2 , - u 0 ) ,
A p = exp [ 2 j p tan - 1 ( λ z π ω 0 2 ) ] .
ϕ ( a , a + 1 , - x ) = ( a / x a ) γ ( a , x ) .
c 0 ¯ 0 , 0 = 1 - exp ( - u 0 ) ,
γ ( a + 1 , x ) = a ! [ 1 - exp ( - x ) s = 0 a x 2 s ! ] .
ϕ ( c - a , c ; - x ) = exp ( - x ) ϕ ( a , c ; x ) ,
c = 2 ; c - a = p + 1 a = 1 - p ,
ϕ ( p + 1 , 2 ; - u 0 ) = exp ( - u 0 ) ϕ ( 1 - p , 2 ; u 0 ) .
L s l ( u ) = ( l + 1 ) s s ! ϕ ( - s , l + 1 ; u 0 ) ,             s = 0 , 1 , 2 ,
ϕ ( 1 - p , 2 ; u 0 ) = ( p - 1 ) ! ( 2 ) p - 1 L p - 1 1 ( u 0 ) = 1 p L p - 1 1 ( u 0 ) ,             p 1.
c 0 ¯ p , 0 u 0 p exp ( - u 0 ) L p - 1 1 ( u 0 ) ,             p = 1 , 2 .
L p - 1 1 ( u 0 ) = j = 0 p - 1 ( p p - j - 1 ) ( - u 0 ) j j ! .
c 0 ¯ 0 , 0 = 1 - exp ( - u 0 ) ;
c 0 ¯ 1 , 0 = u 0 exp ( - u 0 ) L 0 1 ( u 0 ) = u 0 exp ( - u 0 ) ;
c 0 ¯ 2 , 0 = u 0 exp ( - u 0 ) 2 L 1 1 ( u 0 ) = u 0 exp ( - u 0 ) ( 1 - u 0 2 ) ;
c 0 ¯ 3 , 0 = u 0 exp ( - u 0 ) L 2 1 ( u 0 ) L 2 1 ( u 0 ) = u 0 exp ( - u 0 ) ( 1 - u 0 + u 0 2 6 ) ;
α ( 1 / α ) ; β ( β * / α 2 ) ; u 0 ( u 0 / α 2 ) .
c p ¯ 0 , 0 = A p ¯ 0 ( z ¯ ) A 00 * ( z ) α u 0 ϕ 2 [ - p ¯ , p ¯ + 1 ; 2 ; ( β - 1 ) u 0 , - β * u 0 ] .
L p l ( u ) = j = 0 p ( p + l p - j ) ( - u ) j j ! ,
C p ¯ p , l = A p ¯ l ( z ¯ ) A p l * ( z ) N p ¯ p , l ( α β ) l + 1 [ j = 0 p ¯ k = 0 p ( p ¯ + l p ¯ - j ) ( p + l p - k ) · ( - 1 β ) k ( - α 2 β ) j · 1 j ! k ! γ ( j + k + l + 1 , β u 0 ) ] .
c 0 ¯ 0 , l = A 0 ¯ l ( z ¯ ) A 0 l * ( z ) ( 1 / l ! ) ( α / β ) l + 1 γ ( l + 1 , β u 0 ) ,
c 0 ¯ 0 , l = ( 1 / l ! ) γ ( l + 1 , u 0 ) .
0 u 0 exp ( - β u ) L p ( u ) d u = L - 1 [ 1 s g ( s ) ] ,
g ( s ) = L [ exp ( - β u 0 ) L p ( u 0 ) ] .
g ( s ) = 1 s [ 1 - 1 - β s ] p ( 1 + β s ) - ( p + 1 )
0 u 0 exp ( - β u ) L p ( u ) d u = u 0 ϕ 2 [ - p , p + 1 ; 2 ; ( 1 - β ) u 0 , - β u 0 ] ,

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