Abstract

From diffraction theory, an expression is derived for the radius of a gaussian beam, in an output plane of a single lens, on-axis optical system, as a function of input waist radius, lens focal length, input waist to lens spacing, and lens to output plane spacing. Several special cases are discussed and plots of the important cases are included. The choice of the plots, and their scaling, was made on the basis of providing useful information to those people involved with typical problems in laser scanning systems, such as the location and size of the focused spot. The derivation includes a discussion of truncation effects.

© 1970 Optical Society of America

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References

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  1. A. L. Bloom, Appl. Opt. 8, 716 (1969).
    [CrossRef] [PubMed]
  2. H. Kogelnik, Bell System Tech. J. 44, 455 (1965).
  3. H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
    [CrossRef] [PubMed]
  4. G. Goubau, F. Schwering, IRE Trans. Antennas Propagation 9, 248 (1961).
    [CrossRef]
  5. A. Boivin, Theorie et Calcul des Figures de Diffraction de Revolution (Les Presses de l’Universite Laval, Quebec, 1964).
  6. H. Arsenault, A. Boivin, J. Opt. Soc. Amer. 58, 1490 (1968).
    [CrossRef]
  7. H. Arsenault, A. Boivin, J. Opt. Soc. Amer. 38, 3988 (1967).
  8. M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1965).

1969 (1)

1968 (1)

H. Arsenault, A. Boivin, J. Opt. Soc. Amer. 58, 1490 (1968).
[CrossRef]

1967 (1)

H. Arsenault, A. Boivin, J. Opt. Soc. Amer. 38, 3988 (1967).

1966 (1)

1965 (1)

H. Kogelnik, Bell System Tech. J. 44, 455 (1965).

1961 (1)

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

Arsenault, H.

H. Arsenault, A. Boivin, J. Opt. Soc. Amer. 58, 1490 (1968).
[CrossRef]

H. Arsenault, A. Boivin, J. Opt. Soc. Amer. 38, 3988 (1967).

Bloom, A. L.

Boivin, A.

H. Arsenault, A. Boivin, J. Opt. Soc. Amer. 58, 1490 (1968).
[CrossRef]

H. Arsenault, A. Boivin, J. Opt. Soc. Amer. 38, 3988 (1967).

A. Boivin, Theorie et Calcul des Figures de Diffraction de Revolution (Les Presses de l’Universite Laval, Quebec, 1964).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1965).

Goubau, G.

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

Kogelnik, H.

H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
[CrossRef] [PubMed]

H. Kogelnik, Bell System Tech. J. 44, 455 (1965).

Li, T.

Schwering, F.

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

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1965).

Appl. Opt. (2)

Bell System Tech. J. (1)

H. Kogelnik, Bell System Tech. J. 44, 455 (1965).

IRE Trans. Antennas Propagation (1)

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

J. Opt. Soc. Amer. (2)

H. Arsenault, A. Boivin, J. Opt. Soc. Amer. 58, 1490 (1968).
[CrossRef]

H. Arsenault, A. Boivin, J. Opt. Soc. Amer. 38, 3988 (1967).

Other (2)

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1965).

A. Boivin, Theorie et Calcul des Figures de Diffraction de Revolution (Les Presses de l’Universite Laval, Quebec, 1964).

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Figures (10)

Fig. 1
Fig. 1

Small angle diffraction parameters for a coherently illuminated aperture: ρ = radius of curvature of beam incident on lens aperture; l = thin lens; r = radial distance in aperture; s = distance from lens aperture to observation plane; f = lens focal length; P = observation plane; σ = radial distance in observation plane; z = (2π/λ) (σ/s); λ = wavelength of the light.

Fig. 2
Fig. 2

Focal plane intensity patterns for nontruncated and truncated (at r = r0) gaussian plane waves at the lens. Normalized plots of focal plane patterns for nontruncated and truncated beams for r0 = 1. ····, untruncated gaussian; ————, truncated gaussian (truncation occurs at 1/e2 radius); I0 = intensity at center of beam at focal plane.

Fig. 3
Fig. 3

Propagation of a gaussian beam along the axis of a single lens, nontruncating, diffraction limited optical system.

Fig. 4
Fig. 4

Plot of beam radius as a function of distance from source waist for a gaussian beam.

Fig. 5
Fig. 5

Expanded Fig. 4.

Fig. 6
Fig. 6

Plot of beam waists near the focal plane of a lens for gaussian beams of various f/fF ratios.

Fig. 7
Fig. 7

Plot of the ratio of waist radius to focal plane radius vs the ratio of lens focal length to the parameter fF.

Fig. 8
Fig. 8

Plot of the ratio of effective focal length to the lens focal length vs the ratio of lens focal length to the parameter fF.

Fig. 9
Fig. 9

Output waist to lens spacing as a function of input waist to lens spacing.

Fig. 10
Fig. 10

Output waist radius as a function of input waist to lens spacing.

Equations (64)

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A ( z , s , f , ρ ) = C 0 s 0 1 F ( r 2 ) exp i π λ ( 1 s 1 f + 1 ρ ) r 2 J 0 ( z r ) r d r ,
F ( r 2 ) = exp [ ( r / r 0 ) 2 ] ,
A ( z , s , f , ρ ) = C 0 s 0 1 exp [ ( r r 0 ) 2 + i π λ ( 1 s 1 f + 1 ρ ) r 2 ] J 0 ( z r ) r d r ,
A ( z , k ) = C 0 s 0 1 exp ( k r 2 ) J 0 ( z r ) r d r ,
k = 1 r 0 2 i π λ ( 1 s 1 f + 1 ρ ) .
A ( z , k ) n = C 0 s 0 1 ( 1 ) n z 2 n r 2 n 2 2 n ( n ! ) 2 exp ( k r 2 ) r d r .
υ = r 2 , d υ = 2 r d r , s C 0 A ( z , k ) n = ( 1 ) n z 2 n 2 2 n + 1 ( n ! ) 2 0 1 υ n c k υ d υ .
s C 0 A ( z , k ) n = ( 1 ) n z 2 n 2 2 n + 1 ( n ! ) 2 [ e k ( k ) ( 1 + n k + n ( n 1 ) k 2 + + n ! k n ) + n ! k n + 1 ]
A ( z , k ) = C 0 2 k s ( n = 0 [ 1 k ( z 2 ) 2 ] n ! n e k n = 0 { [ ( z / 2 ) 2 ] n n ! i = 0 n 1 k n i i ! } ) .
A ( z , k ) = C 0 2 k s ( exp [ 1 k ( z 2 ) 2 ] e k n = 0 × { [ ( z / 2 ) 2 ] n n ! i = 0 n 1 k n i i ! } ) ,
k = 1 r 0 2 i π λ ( 1 s 1 f + 1 ρ ) .
A ( z , k ) = ( C 0 / 2 k s ) exp [ ( 1 / k ) ( z / 2 ) 2 ] .
I ( z , k ) = A A * = ( C 0 C 0 * / 4 s 2 k k * ) exp [ ( 2 r 0 2 / k k * ) ( z / 2 ) 2 ] ,
I ( z , s , f , ρ ) = C 1 s 2 [ 1 r 0 4 + π 2 λ 2 ( 1 s 1 f + 1 ρ ) 2 ] 1 × exp { 1 2 ( z r 0 ) / [ 1 r 0 4 + π 2 λ 2 ( 1 s 1 f + 1 ρ ) 2 ] } .
I ( z ) = B ( s , f , ρ , r 0 ) exp [ 2 ( z / z 0 ) 2 ] ,
z 0 = 2 r 0 [ 1 r 0 4 + π 2 λ 2 ( 1 s 1 f + 1 ρ ) 2 ] 1 2
I = B exp { 2 [ ( π / λ f ) r 0 σ ] 2 } .
σ = λ f / π r 0 .
A ( 0 , s , f , ρ ) = C 0 s 0 1 exp [ ( r r 0 ) 2 + i ω r 2 ] r d r ,
ω = π λ ( 1 s 1 f + 1 ρ ) .
A ( 0 , s , f , ρ ) = C s 0 1 exp [ ( i ω 1 r 0 2 ) υ ] d υ ,
A = C s r 0 2 { 1 exp [ i ω ( 1 / r 0 2 ) ] 1 i r 0 2 ω } .
I = C 2 s 2 r 0 4 [ 1 ( e i ω + e i ω ) exp ( 1 / r 0 2 ) + exp ( 2 / r 0 2 ) 1 + r 0 4 ω 2 ]
I = C 2 s 2 r 0 4 [ 1 2 exp ( 1 / r 0 2 ) cos ω + exp ( 2 / r 0 2 ) 1 + r 0 4 ω 2 ] .
I ( P ) = I ( 0 , ω ) = ( C 2 / s 2 ) r 0 4 [ 1 / ( 1 + r 0 4 ω 2 ) ] ,
ω = π λ ( 1 s 1 f + 1 ρ ) ,
r 0 2 / r P 2 = I ( P ) / I ( 0 ) .
I ( 0 ) = C 2 λ 2 / π 2 ;
r P 2 = r 0 2 I ( 0 ) / I ( P ) = r 0 2 ( C 2 λ 2 / π 2 ) ( s 2 / C 2 r 0 4 ) ( 1 + r 0 4 ω 2 )
r P = ( s λ / π r 0 ) ( 1 + r 0 4 ω 2 ) 1 2
r P = s λ π r 0 [ 1 + r 0 4 π 2 λ 2 ( 1 s 1 f + 1 ρ ) 2 ] 1 2 .
r 2 = r 2 ( r 0 , d 1 , d 2 , f ) for a given λ .
r P = s λ / π r 0 [ 1 + ( r 0 4 π 2 / s 2 λ 2 ) ] 1 2 ,
r d 1 = d 1 λ / π r 0 [ 1 + ( r 0 4 π 2 / d 1 2 λ 2 ) ] 1 2 .
ρ = r d r / d s ,
ρ s = s [ 1 + ( π r 0 2 / λ s ) 2 ]
ρ d 1 = d 1 [ 1 + ( π r 0 2 / λ d 1 ) ] .
r d 2 = d 2 λ π r d 1 ( 1 + r d 1 4 π 2 λ 2 { 1 d 2 1 f + 1 d 1 [ 1 + ( π r 0 2 / λ d 1 ) 2 ] } 2 ) 1 2
r 2 = d 2 λ π r 1 ( 1 + r 1 4 π 2 λ 2 { 1 d 2 1 f + 1 d 1 [ 1 + ( π r 0 2 / λ d 1 ) 2 ] } 2 ) 1 2 .
r 2 = r 0 d 2 d 1 [ 1 + ( f F 2 / d 1 2 ) ] 1 2 ( 1 + d 1 4 f F 2 ( 1 + f F 2 d 1 2 ) 2 × { 1 d 2 1 f + 1 d 1 [ 1 + ( f F 2 / d 1 2 ) ] } 2 ) 1 2 ,
r 2 = r 0 ( d 2 / f F ) [ 1 + ( f F 2 / d 2 2 ) ] 1 2
r 2 = r 0 [ 1 + ( d 2 2 / f F 2 ) ] 1 2 .
r 2 = r 0 d 2 f F [ 1 + f F 2 ( 1 d 2 1 f ) 2 ] 1 2 .
r 2 r f = d 2 f [ 1 + ( f F f ) 2 ( 1 f d 2 ) 2 ] 1 2 .
r 2 min = r f [ 1 + ( f / f F ) 2 ] 1 2
d 2 min = f [ 1 + ( f f F ) 2 ] 1 .
1 / d 2 min = ( 1 / f ) + ( 1 / f F ) ( f / f F ) .
1 d 2 min = 1 f + 1 f F , when f = f F .
r 2 = f λ / π r 0 .
d 2 min f = ( d 1 f ) f 2 / [ ( d 1 f ) 2 + f F 2 ] .
d 2 min f f f F f = ( d 1 f ) / f F 1 + [ ( d 1 f ) / f F ] 2 .
f F 1.25 m .
d 2 = f ( d 1 f ) 2 + f F 2 ( d 1 ( d 1 f ) + f F 2 ± { [ d 1 ( d 1 f ) + f F 2 ] 2 [ ( d 1 f ) 2 + f F 2 ] · [ d 1 2 + f F 2 ( 1 r 2 2 r 0 2 ) ] } 1 2 ) .
d 2 min = f ( d 1 f ) 2 + f F 2 [ d 1 ( d 1 f ) + f F 2 ] .
r 2 min = r 0 f [ ( d 1 f ) 2 + f F 2 ] 1 2 .
( r 2 min / r 0 ) ( f F / f ) = { 1 + [ ( d 1 f ) / f F ] 2 } 1 2 .
r 2 = r 0 d 2 d 1 [ 1 + ( f F 2 / d 1 2 ) ] 1 2 ( 1 + d 1 4 f F 2 ( 1 + f F 2 d 1 2 ) 2 × { 1 d 2 1 f + 1 d 1 [ 1 + ( f F 2 / d 1 2 ) ] } 2 ) 1 2 ,
r 2 = r 0 [ 1 + ( d 2 2 / f F 2 ) ] 1 2 . See Figs . 4 and 5 .
r 2 = r 0 d 2 f F [ 1 + f F 2 ( 1 d 2 1 f ) 2 ] 1 2 . See Fig . 6
r 2 min = r f [ 1 + ( f / f F ) 2 ] 1 ( waist radius for d 1 = 0 ) . See Fig . 7 .
d 2 min = f [ 1 + ( f / f F ) 2 ] 1 ( waist location for d 1 = 0 ) . See Fig . 8 .
r 2 = f λ / π r 0 independent of d 1 .
d 2 min f = ( d 1 f ) f 2 / [ ( d 1 f ) 2 + f F 2 ] ( waist location ) . See Fig . 9.
r 2 min = r 0 f / [ ( d 1 f ) 2 + f F 2 ] 1 2 ( waist radius ) . See Fig . 10.

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