Abstract

The existence of unguided TM and TE Gaussian beams is derived in a simple way from the wave equation for the vector potential, and the field configurations are discussed. We point out the possibility of a new type of free-electron laser.

© 1981 Optical Society of America

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References

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  1. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177 (1979).
    [CrossRef]
  2. G. Goubau, F. Schwering, “On the guided propagation of electromagnetic wave beams,” IRE Trans. Antennas Propag. AP-9, 248 (1961).
    [CrossRef]
  3. H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312 (1966).
    [CrossRef]
  4. P. Lorrain, D. R. Corson, Electromagnetic Fields and Waves, 2nd ed. (Freeman, San Francisco, 1970).
  5. In the paraxial approximation, the equation for the phase fronts of Az is z = const. − r2/2R.
  6. Simple considerations about the nature of field lines make it obvious that any wave of finite lateral extent has longitudinal field components. In the case of the so-called TEM00 mode, it is seen that the longitudinal electric field vanishes along the z axis.
  7. K. Shimoda, “Proposal for an electron accelerator using an optical maser,” Appl. Opt. 1, 33 (1962).
    [CrossRef]

1979 (1)

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177 (1979).
[CrossRef]

1966 (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312 (1966).
[CrossRef]

1962 (1)

1961 (1)

G. Goubau, F. Schwering, “On the guided propagation of electromagnetic wave beams,” IRE Trans. Antennas Propag. AP-9, 248 (1961).
[CrossRef]

Corson, D. R.

P. Lorrain, D. R. Corson, Electromagnetic Fields and Waves, 2nd ed. (Freeman, San Francisco, 1970).

Davis, L. W.

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177 (1979).
[CrossRef]

Goubau, G.

G. Goubau, F. Schwering, “On the guided propagation of electromagnetic wave beams,” IRE Trans. Antennas Propag. AP-9, 248 (1961).
[CrossRef]

Kogelnik, H.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312 (1966).
[CrossRef]

Li, T.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312 (1966).
[CrossRef]

Lorrain, P.

P. Lorrain, D. R. Corson, Electromagnetic Fields and Waves, 2nd ed. (Freeman, San Francisco, 1970).

Schwering, F.

G. Goubau, F. Schwering, “On the guided propagation of electromagnetic wave beams,” IRE Trans. Antennas Propag. AP-9, 248 (1961).
[CrossRef]

Shimoda, K.

Appl. Opt. (1)

IRE Trans. Antennas Propag. (1)

G. Goubau, F. Schwering, “On the guided propagation of electromagnetic wave beams,” IRE Trans. Antennas Propag. AP-9, 248 (1961).
[CrossRef]

Phys. Rev. A (1)

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177 (1979).
[CrossRef]

Proc. IEEE (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312 (1966).
[CrossRef]

Other (3)

P. Lorrain, D. R. Corson, Electromagnetic Fields and Waves, 2nd ed. (Freeman, San Francisco, 1970).

In the paraxial approximation, the equation for the phase fronts of Az is z = const. − r2/2R.

Simple considerations about the nature of field lines make it obvious that any wave of finite lateral extent has longitudinal field components. In the case of the so-called TEM00 mode, it is seen that the longitudinal electric field vanishes along the z axis.

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Figures (1)

Fig. 1
Fig. 1

Field distributions of unguided electromagnetic beams in a longitudinal section passing through z axis. (a) The TEM00 mode and (b) the fundamental TM mode. The ratio λ/w0 normally would be much smaller than depicted here. Note that along the axis (r = 0), the longitudinal electric field of the TEM00 mode vanishes, whereas that of the lowest-order TM mode has maximum amplitude.

Equations (16)

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2 A + k 2 A = 0 .
A z = ψ e i k z ,
ψ = i Q e i Q ρ 2 , ρ 2 = x 2 + y 2 w 0 2 ,
i Q = 1 1 i z / l ,
H ϕ = 2 i Q ρ w 0 A z
i Q = w 0 w e i Φ ,
E = i k × H .
E r = H ϕ ,
E ϕ = 0 ,
E z = 2 Q l ( 1 i Q ρ 2 ) A z ,
A z z = i k A z .
r 2 w 2 e r 2 / w 2 cos ( ω t k z k r 2 2 R + 2 Φ ) = K ,
r 2 w 2 e r 2 / w 2 cos ω t cos ( k z + k r 2 z R 2 Φ ) = K .
A z = g ( r w ) cos n θ w 0 w exp [ i ( Φ K z Q ρ 2 ) ] ,
g ( r w ) = ( 2 r w ) n L p n ( 2 r 2 w 2 ) ,
Φ ( p , n ; z ) = ( 2 p + n + 1 ) arctan ( z / l ) .

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