Abstract

When the waist size of a Gaussian beam becomes of the order of the wavelength of light, the beam does not satisfy the paraxial condition in which it is derived. In this paper, we define the lower bound to the waist size by showing that a Gaussian beam whose waist size is larger than this bound safely satisfies the paraxial condition. A beam which is Gaussian in form but violates the paraxial condition is called a nonparaxial Gaussian beam. We clarify the range of the waist size for which the first-order correction to this beam is effective. It is shown that a distinct value of the waist size exists for which the paraxial approximation completely fails and the first-order correction never works.

© 1990 Optical Society of America

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References

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  1. G. Goubau, F. Schwering, “On the Guided Propagation of Electromagnetic Wave Beams,” IRE Trans. Antennas and Propag. AP-9, 248–256 (1961).
    [CrossRef]
  2. A. G. van Nie, “Rigorous Calculation of the Electromagnetic Field of Wave Beams,” Philips Res. Rep. 19, 378–394 (1964).
  3. W. H. Carter, “Electromagnetic Field of a Gaussian Beam with an Elliptical Cross Section,” J. Opt. Soc. Am. 62, 1195–1201 (1972).
    [CrossRef]
  4. M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to Paraxial Wave Optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [CrossRef]
  5. L. W. Davis, “Theory of Electromagnetic Beams,” Phys. Rev. A 19, 1177–1179 (1979).
    [CrossRef]
  6. G. P. Agrawal, D. N. Pattanayak, “Gaussian Beam Propagation beyond the Paraxial Approximation,” J. Opt. Soc. Am. 69, 575–578 (1979).
    [CrossRef]
  7. M. Couture, P. A. Belanger, “From Gaussian Beam to Complex-Source-Point Spherical Wave,” Phys. Rev. A 24, 355–359 (1981).
    [CrossRef]
  8. G. P. Agrawal, M. Lax, “Free-Space Wave Propagation beyond the Paraxial Approximation,” Phys. Rev. A 27, 1693–1695 (1983).
    [CrossRef]
  9. T. Takenaka, M. Yokota, O. Fukumitsu, “Propagation of Light Beams beyond the Paraxial Approximation,” J. Opt. Soc. Am. A 2, 826–829 (1985).
    [CrossRef]
  10. E. Zauderer, “Complex Argument Hermite-Gaussian and Laguerre-Gaussian Beams,” J. Opt. Soc. Am. A 3, 465–469 (1986).
    [CrossRef]
  11. H. Kogelnik, T. Li, “Laser Beams and Resonators,” Appl. Opt. 5, 1550–1567 (1966).
    [CrossRef] [PubMed]
  12. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), p. 236.
  13. R. Simon, E. C. G. Sudarshan, N. Mukunda, “Cross Polarization in Laser Beams,” Appl. Opt. 26, 1589–1593 (1987).
    [CrossRef] [PubMed]
  14. A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), p. 317.

1987 (1)

1986 (1)

1985 (1)

1983 (1)

G. P. Agrawal, M. Lax, “Free-Space Wave Propagation beyond the Paraxial Approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

1981 (1)

M. Couture, P. A. Belanger, “From Gaussian Beam to Complex-Source-Point Spherical Wave,” Phys. Rev. A 24, 355–359 (1981).
[CrossRef]

1979 (2)

1975 (1)

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to Paraxial Wave Optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

1972 (1)

1966 (1)

1964 (1)

A. G. van Nie, “Rigorous Calculation of the Electromagnetic Field of Wave Beams,” Philips Res. Rep. 19, 378–394 (1964).

1961 (1)

G. Goubau, F. Schwering, “On the Guided Propagation of Electromagnetic Wave Beams,” IRE Trans. Antennas and Propag. AP-9, 248–256 (1961).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, M. Lax, “Free-Space Wave Propagation beyond the Paraxial Approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

G. P. Agrawal, D. N. Pattanayak, “Gaussian Beam Propagation beyond the Paraxial Approximation,” J. Opt. Soc. Am. 69, 575–578 (1979).
[CrossRef]

Belanger, P. A.

M. Couture, P. A. Belanger, “From Gaussian Beam to Complex-Source-Point Spherical Wave,” Phys. Rev. A 24, 355–359 (1981).
[CrossRef]

Carter, W. H.

Couture, M.

M. Couture, P. A. Belanger, “From Gaussian Beam to Complex-Source-Point Spherical Wave,” Phys. Rev. A 24, 355–359 (1981).
[CrossRef]

Davis, L. W.

L. W. Davis, “Theory of Electromagnetic Beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

Fukumitsu, O.

Goubau, G.

G. Goubau, F. Schwering, “On the Guided Propagation of Electromagnetic Wave Beams,” IRE Trans. Antennas and Propag. AP-9, 248–256 (1961).
[CrossRef]

Kogelnik, H.

Lax, M.

G. P. Agrawal, M. Lax, “Free-Space Wave Propagation beyond the Paraxial Approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to Paraxial Wave Optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Li, T.

Louisell, W. H.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to Paraxial Wave Optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), p. 236.

McKnight, W. B.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to Paraxial Wave Optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Mukunda, N.

Pattanayak, D. N.

Schwering, F.

G. Goubau, F. Schwering, “On the Guided Propagation of Electromagnetic Wave Beams,” IRE Trans. Antennas and Propag. AP-9, 248–256 (1961).
[CrossRef]

Siegman, A. E.

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), p. 317.

Simon, R.

Sudarshan, E. C. G.

Takenaka, T.

van Nie, A. G.

A. G. van Nie, “Rigorous Calculation of the Electromagnetic Field of Wave Beams,” Philips Res. Rep. 19, 378–394 (1964).

Yokota, M.

Zauderer, E.

Appl. Opt. (2)

IRE Trans. Antennas and Propag. (1)

G. Goubau, F. Schwering, “On the Guided Propagation of Electromagnetic Wave Beams,” IRE Trans. Antennas and Propag. AP-9, 248–256 (1961).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Philips Res. Rep. (1)

A. G. van Nie, “Rigorous Calculation of the Electromagnetic Field of Wave Beams,” Philips Res. Rep. 19, 378–394 (1964).

Phys. Rev. A (4)

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to Paraxial Wave Optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

L. W. Davis, “Theory of Electromagnetic Beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

M. Couture, P. A. Belanger, “From Gaussian Beam to Complex-Source-Point Spherical Wave,” Phys. Rev. A 24, 355–359 (1981).
[CrossRef]

G. P. Agrawal, M. Lax, “Free-Space Wave Propagation beyond the Paraxial Approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

Other (2)

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), p. 236.

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), p. 317.

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Figures (16)

Fig. 1
Fig. 1

Distribution of the function F.

Fig. 2
Fig. 2

Distribution of the function G with ks0 = 1.

Fig. 3
Fig. 3

Comparison between F(r,0) and G(r,0) for ks0 = 1.

Fig. 4
Fig. 4

Comparison between F(0,z) and G(0,z) for several values of ks0.

Fig. 5
Fig. 5

Comparison between the amplitudes of the paraxial and exact fields on the beam axis for ks0 = 1,2.

Fig. 6
Fig. 6

Comparison between the amplitudes of the paraxial and exact fields on the beam axis for various values of ks0.

Fig. 7
Fig. 7

Dependence of Δp on z / k s 0 2 and ks0.

Fig. 8
Fig. 8

Dependence of δp on z / k s 0 2 and ks0.

Fig. 9
Fig. 9

Amplitudes of the paraxial, corrected, and exact fields on the beam axis for k s 0 = 1 , 2,2.

Fig. 10
Fig. 10

Phases of the paraxial, corrected, and exact fields on the beam axis for k s 0 = 1 , 2.

Fig. 11
Fig. 11

Amplitudes of the three fields in the plane z = k s 0 2 for k s 0 = 1 , 2,2.

Fig. 12
Fig. 12

Phases of the three fields in the plane z = k s 0 2 for k s 0 = 1 , 2,2.

Fig. 13
Fig. 13

Dependence of Δc on z / k s 0 2 and ks0.

Fig. 14
Fig. 14

Dependence of δc on z / k s 0 2 and ks0.

Fig. 15
Fig. 15

Comparison between Δp and Δc for ks0 = 2.

Fig. 16
Fig. 16

Comparison between δp and δc for ks0 = 2.

Equations (52)

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2 u / x 2 + 2 u / y 2 + 2 u / z 2 + k 2 u = 0 ,
u = ψ ( x , y , z ) exp ( - j k z ) .
2 ψ / z 2 2 k ψ / z ,
2 ψ / x 2 + 2 ψ / y 2 - 2 j k ψ / z = 0 ,
ψ = exp [ - j ( p + k r 2 / 2 q ) ] ,             r 2 = x 2 + y 2 ,
d p / d z = - j / q ,             d q / d z = 1.
ψ ( x , y , 0 ) = exp ( - r 2 / 2 s 0 2 ) ,
p = - j ln [ 1 + ( z / k s 0 2 ) 2 ] 1 / 2 - tan - 1 ( z / k s 0 2 ) ,
q = z + j k s 0 2 ,
Δ ( ψ / z ) = ( 2 ψ / z 2 ) Δ z .
Δ ( ψ / z ) / ( ψ / z ) 4 π ,             when Δ z = λ .
F ( r , z ) = 2 k s 0 2 ψ / z ,             G ( r , z ) = s 0 2 2 ψ / z 2 .
G ( r , z ) F ( r , z ) .
ψ / z = - ( 1 - α ) ψ / q ,
2 ψ / z 2 = ( 2 - 4 α + α 2 ) ψ / q 2 ,
α = j k r 2 / 2 q .
F ( r , z ) = f ( ρ , ζ ) ,             G ( r , z ) = g ( ρ , ζ ) ,
f ( ρ , ζ ) = [ ( 2 - ρ 2 ) 2 + 4 ζ 2 ] 1 / 2 ( 1 + ζ 2 ) - 3 / 2 × exp [ - ρ 2 / 2 ( 1 + ζ 2 ) ] ,
g ( ρ , ζ ) = ( 2 σ ) - 2 ( c 0 + c 1 ρ 2 + c 2 ρ 4 + c 3 ρ 6 + ρ 8 ) 1 / 2 × ( 1 + ζ 2 ) - 5 / 2 exp [ - ρ 2 / 2 ( 1 + ζ 2 ) ] ,
c 0 = 64 ( 1 + ζ 2 ) 2 ,             c 1 = - 128 ( 1 + ζ 2 ) ,
c 2 = 16 ( 5 + 3 ζ 2 ) ,             C 3 = - 16 ,
σ = k s 0 ,             ρ = r / s 0 ,             ζ = z / k s 0 2 .
g ( ρ , ζ ) f ( ρ , ζ ) ,
( i )             G ( 0 , 0 ) F ( 0 , 0 ) ,
( ii )             [ 0 G 2 ( r , z ) r d r ] 1 / 2 [ 0 F 2 ( r , z ) r d r ] 1 / 2 ,
( iii )             [ - G 2 ( s , z ) d z ] 1 / 2 [ - F 2 ( s , z ) d z ] 1 / 2 .
( i )             2 / σ 2 2 ,
( ii )             3 s 0 / 2 σ 2 s 0 ,
( iii )             [ 5 π exp ( - 1 ) / 16 k σ 2 ] 1 / 2 [ π exp ( - 1 ) s 0 σ ] 1 / 2 .
( i )             σ 2 1 ,
( ii )             σ 2 3 / 2 0.866 ,
( iii )             σ 2 5 / 4 0.559.
k s 0 = 1 ,             s 0 / λ = 1 / 2 π 0.1592.
k w 0 = 2 ,             w 0 / λ = 2 / 2 π 0.2251.
s 0 / λ 0 0.1592 / n ,             w 0 / λ 0 0.2251 / n .
I x = 1 + ( 4 k s 0 ) - 2 ,             I y = ( 4 k s 0 ) - 2 ,             I z = ( 2 k s 0 ) - 2 .
u e = u h + u i ,
u h = σ 2 0 1 exp ( - σ 2 b 2 / 2 ) exp [ - j k z ( 1 - b 2 ) 1 / 2 ] J 0 ( k r b ) b d b ,
u i = σ 2 1 exp ( - σ 2 b 2 / 2 ) exp [ - k z ( b 2 - 1 ) 1 / 2 ] J 0 ( k r b ) b d b ,
u p = ( 1 + ζ 2 ) - 1 / 2 exp [ - ( 1 + j ζ ) ρ 2 / 2 ( 1 + ζ 2 ) ] × exp ( - j σ 2 ζ ) exp ( j tan - 1 ζ ) ,
u p = u p exp ( j ϕ p ) ,             u e = u e exp ( j ϕ e ) ,
Δ p = ( u p - u e ) / u e ,             δ p = ϕ p - ϕ e ,
s 0 / λ = 2 / π 0.6366 ,             w 0 / λ = 2 2 / π 0.9003.
u c = u ( 0 ) + u ( 2 ) / ( k s 0 ) 2 = C u ( 0 ) ,
C = 1 + [ j ζ / σ 2 ( 1 - j ζ ) ] L 2 [ ρ 2 / 2 ( 1 - j ζ ) ] ,
C = 1 - ( A + j B ) / ( k s 0 ) 2 ,
A = ζ [ 2 ζ X + ( 1 - ζ 2 ) Y ] / ( 1 + ζ 2 ) ,
B = ζ [ 2 ζ Y - ( 1 - ζ 2 ) X ] / ( 1 + ζ 2 ) ,
X = 1 - [ ρ 2 / ( 1 + ζ 2 ) ] [ 1 - ρ 2 ( 1 - ζ 2 ) / 8 ( 1 + ζ 2 ) ] ,
Y = - [ ρ 2 ζ / ( 1 + ζ 2 ) ] [ 1 - ρ 2 / 4 ( 1 + ζ 2 ) ] .
u c = u c exp ( j ϕ c ) ,
Δ c = ( u c - u e ) / u e ,             δ c = ϕ c - ϕ e .

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