Abstract

The relationship between conventional optics and Gaussian beam optics is discussed. Geometrical concepts such as image and object planes, pupil plane, F-number, resolution, and depth of field are discussed for Gaussian beams and are given a precise meaning in terms of the Gaussian beam parameters. The approach is useful in the design of laser scanning systems and holographic optical memory systems.

© 1972 Optical Society of America

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References

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  1. H. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965).
  2. H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966); also, Proc. IEEE 54, 1312 (1966).
    [CrossRef] [PubMed]
  3. L. D. Dickson, Appl. Opt. 9, 1854 (1970).
    [CrossRef] [PubMed]
  4. G. E. Francis, F. M. Librecht, J. J. Engelen, Appl. Opt. 10, 1157 (1971).
    [CrossRef]
  5. G. D. Boyd, H. Kogelnik, Bell Syst. Tech. J. 41, 1347 (1962).
  6. T. C. Lee, J. D. Zook, IEEE J. Quantum Electron. QE-4, 442 (1968).
    [CrossRef]
  7. T. C. Lee, D. Gossen, Appl. Opt. 10, 961 (1971).
    [CrossRef] [PubMed]
  8. D. A. Holmes, J. E. Korka, P. V. Avizonis, Appl. Opt. 11, 565 (1972).
    [CrossRef] [PubMed]
  9. M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964), pp. 439–442.
  10. T. C. Lee, Appl. Opt. 11, 384 (1972).
    [CrossRef] [PubMed]

1972 (2)

1971 (2)

1970 (1)

1968 (1)

T. C. Lee, J. D. Zook, IEEE J. Quantum Electron. QE-4, 442 (1968).
[CrossRef]

1966 (1)

1965 (1)

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

1962 (1)

G. D. Boyd, H. Kogelnik, Bell Syst. Tech. J. 41, 1347 (1962).

Avizonis, P. V.

Born, M.

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964), pp. 439–442.

Boyd, G. D.

G. D. Boyd, H. Kogelnik, Bell Syst. Tech. J. 41, 1347 (1962).

Dickson, L. D.

Engelen, J. J.

Francis, G. E.

Gossen, D.

Holmes, D. A.

Kogelnik, H.

H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966); also, Proc. IEEE 54, 1312 (1966).
[CrossRef] [PubMed]

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

G. D. Boyd, H. Kogelnik, Bell Syst. Tech. J. 41, 1347 (1962).

Korka, J. E.

Lee, T. C.

Li, T.

Librecht, F. M.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964), pp. 439–442.

Zook, J. D.

T. C. Lee, J. D. Zook, IEEE J. Quantum Electron. QE-4, 442 (1968).
[CrossRef]

Appl. Opt. (6)

Bell Syst. Tech. J. (2)

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

G. D. Boyd, H. Kogelnik, Bell Syst. Tech. J. 41, 1347 (1962).

IEEE J. Quantum Electron. (1)

T. C. Lee, J. D. Zook, IEEE J. Quantum Electron. QE-4, 442 (1968).
[CrossRef]

Other (1)

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964), pp. 439–442.

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Figures (6)

Fig. 1
Fig. 1

Definition of Gaussian beam parameters. W0 is the beam waist diameter, and θ0 is the angle between the asymptotes of the beam diameter.

Fig. 2
Fig. 2

Conjugate planes for a Gaussian beam. The planes P1 and P2 are conjugate planes in the sense described in the text. W1 is the beam diameter at P1; θ1 is the diffraction angle associated with the diameter W1; and F1 is the f-number of the beam at P1. The definitions of W2, θ2, and F2 are similar to those of W1, θ1 and F1.

Fig. 3
Fig. 3

Effective pupil in a cumulative one-dimensional deflector system. D1, D2, … are deflectors at distances d1d2, … from the waist plane, producing deflections ϕ1, ϕ2, …, respectively. The effective pupil plane is determined by the intersection of the incoming and outgoing beams. The resolution of the deflector system is maximum at the display plane that is conjugate to the effective pupil plane.

Fig. 4
Fig. 4

Transformation of a Gaussian beam by a simple lens. The lens at P2 images its conjugate plane P1 to its other conjugate plane P2′ according to the simple lens laws. For the simple lens the planes P2 and P1′ are the same, but for a thick lens they are the principal planes of the lens.

Fig. 5
Fig. 5

Generalized Fourier-transform relationships in a simple coherent system using Gaussian beams. (a) The pupil function located at P1 is in front of the lens, and its virtual FT (apart from a quadratic phase factor) is located at P2, which is conjugate to P1. The lens images the virtual FT to a real FT at P2′. (b) The pupil function is behind the lens, so its FT is located at P2, which is the conjugate plane of P1.

Fig. 6
Fig. 6

Deflector optics in a holographic memory system. P2 is the first display plane where the deflected spots are resolved, L2 is a field lens, P1 and P3 are the first and second pupil planes, and P4 is the final display plane. The resolvable spots at plane are large, P4 thus the depth of field is also large. Since P3 is a focal distance from L3, the deflected beams to the right of L3 are parallel to one another.

Equations (34)

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W 2 ( z ) = W 0 2 + θ 0 2 z 2 ,
θ 0 = ( 4 / π ) ( λ / W 0 ) = λ e / W 0 .
W 2 = λ e R / W 1 .
W 2 2 = W 0 2 + ( λ e 2 / W 0 2 ) z 2 2
W 1 2 = W 0 2 + ( λ e 2 / W 0 2 ) z 1 2
R = z 1 + z 2
z 1 ( W 2 2 / W 0 2 ) + z 2 ( W 1 2 / W 0 2 )
z 1 z 2 = W 0 2 / θ 0 2 .
W 2 = W 0 W 1 / θ 0 z 1
R = W 1 2 / θ 0 2 z 1 .
θ 1 = λ e / W 1 .
R ( z ) = z [ 1 + ( W 0 2 / λ e z ) 2 ] .
R ( z 2 ) = z 2 + ( W 0 4 / λ e 2 z 2 ) = z 2 + z 1 = R ( z 1 ) = R ,
F 1 = R ( z 1 ) / W 1 = R / W 1 = W 2 / λ e .
N R = ϕ ( z 1 + z 2 ) / W ( z 2 ) .
N R = ϕ W 1 / λ e = ϕ / θ 1 ,
X = i ϕ i ( d i + d 2 ) ,
ϕ = i ϕ i .
z 1 = Σ i ϕ i d i / Σ i ϕ i ,
N R = ( X / W 2 ) = [ ϕ ( z 1 + z 2 ) ] / W 2 ,
z 1 = z i z f z d ϕ / z i z f d ϕ = z i z f z d z / z i z f d z = 1 2 ( z i + z f ) .
( N R ) max / ( N R ) z 2 = 0 = ( ϕ W 1 / λ e ) ( W 0 / ϕ z 1 ) = [ 1 + ( z 2 / z 1 ) ] 1 2 .
z 1 / z 2 = W 1 2 / W 2 2 = ( W 1 2 / λ e R ) 2 ,
z 1 z 2 = W 0 2 / θ 0 2 ,
W 1 W 2 = λ e R ,
W 1 2 / z 1 = W 2 2 / z 2 = θ 0 2 R ,
W 1 2 z 2 = W 2 2 z 1 = W 0 2 R .
W 2 = R λ e / W 1 = R θ 1 = F 1 λ e .
( 1 / R ) + ( 1 / R ) = ( 1 / f ) and W 2 / W 1 = R / R ,
N R = ϕ ( R + Δ R ) / [ W 0 2 + θ 0 2 ( z 2 + Δ R ) 2 ] 1 2 .
N R = N R max [ 1 - 1 2 z 1 z 2 ( Δ R R ) 2 + z 1 z 2 ( Δ R R ) 3 + ] .
z 1 / z 2 = W 1 2 / W 2 2 = W 0 2 / λ z 2 = λ z 1 / W 0 2
( N R max - N R ) / N R max = 1 2 ( 1 / F 1 2 ) ( Δ R / W 2 ) 2
= 1 2 ( 1 / F 1 4 ) ( Δ R / λ ) 2 ,

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