Abstract

The motion of a charged particle in a space- and time-dependent potential and the motion of an optical pulse in an inhomogeneous anisotropic medium coincide when the dispersion surfaces are the same. For the trajectories in space to coincide, it is sufficient that the wave vectors be proportional. It follows from the general expression of the average stress-energy density (L¯/K)K where L¯ denotes the average lagrangian density and K denotes the 4 wave vector, that radiation forces are proportional to wave vectors for charged particles as well as for optical pulses. Because of these relations, many results in mechanics are applicable to optics. In particular, the constancy of the horizontal component of the velocity of a bullet on earth has, as a counterpart in optics, the constancy of the axial component of the group velocity of optical pulses propagating in thick tapered dielectric slabs. It follows from this observation that thick tapered dielectric slabs are not suited for long-distance communication, because of the large pulse spreading that they introduce. Slabs with moderate thickness, however, may exhibit low pulse spreading.

© 1975 Optical Society of America

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  1. V. A. Fock, Bull. Acad. Sci. l’URSS, Ser. Phys. 14, 70 (1950), translated in V. A. Fock, Electromagnetic Diffraction and Propagation Problems (Pergamon, Oxford, 1965), Ch. 14.
  2. G. B. Whitham, J. Fluid Mech. 44, 373 (1970).
    [CrossRef]
  3. O. Costa de Beauregard, C. R. Acad. Sci. (Paris) B 274, 164 (1972); Nuovo Cimento, Ser. 10 63B, 611 (1969).
  4. P. Penfield and H. A. Haus, Electrodynamics of Moving Media (M.I.T. Press, Cambridge, 1967), p. 211.
  5. S. R. De Groot and L. G. Suttorp, Foundations of Electrodynamics (North–Holland, Amsterdam, 1972).
  6. H. A. Haus, Physica 43, 77 (1969).
    [CrossRef]
  7. J. A. Arnaud, (a)Electr. Lett. 8, 541 (1972); (b)Opt. Commun. 7, 313 (1973); (c)Am. J. Phys. 42, 71 (1974).
  8. J. P. Gordon, Phys. Rev. A 8, 14 (1973).
    [CrossRef]
  9. W. B. Joyce, Phys. Rev. D 9, 3234 (1974).
    [CrossRef]
  10. R. Lucas, in Louis de Broglie, sa Conception du Monde Physique (Gauthier-Villars, Paris, 1973).
  11. A. Kastler, C. R. Acad. Sci. (Paris) B 278, 1013 (1974).
  12. D. Marcuse, Engineering Quantum Electrodynamics (Harcourt, Brace and World, New York, 1970), p. 80.
  13. D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972), p. 100.
  14. R. V. Jones and J. C. S. Richard, Proc. R. Soc. (Lond.) A 221, 480 (1954).
    [CrossRef]
  15. A. Ashkin, Appl. Phys. Lett. 19, 283 (1971); A. Ashkin and J. M. Diedzic, Phys. Rev. Lett. 30, 139 (1973).
    [CrossRef]
  16. T. H. Boyer, Phys. Rev. D 8, 1669 (1973).
  17. J. A. Arnaud, J. Opt. Soc. Am. 62, 290 (1972) and J. A. Arnaud, in Progress in Optics, Vol. 11, edited by E. Wolf (North–Holland, Amsterdam, 1973), p. 247.
    [CrossRef]
  18. K. Husini, Prog. Theor. Phys. 9, 381 (1953).
    [CrossRef]
  19. J. A. Arnaud, Bell Syst. Tech. J. 49, 2311 (1970).
    [CrossRef]
  20. J. M. Kelso, Radio Ray Propagation in the Ionosphere (McGraw-Hill, New York, 1964), p. 220.
  21. J. A. Arnaud, Bell Syst. Tech. J. 53, 1599 (1974).
    [CrossRef]
  22. E. A. J. Marcatili, Bell Syst. Tech. J. 53, 645 (1974).
    [CrossRef]

1974 (4)

W. B. Joyce, Phys. Rev. D 9, 3234 (1974).
[CrossRef]

A. Kastler, C. R. Acad. Sci. (Paris) B 278, 1013 (1974).

J. A. Arnaud, Bell Syst. Tech. J. 53, 1599 (1974).
[CrossRef]

E. A. J. Marcatili, Bell Syst. Tech. J. 53, 645 (1974).
[CrossRef]

1973 (2)

T. H. Boyer, Phys. Rev. D 8, 1669 (1973).

J. P. Gordon, Phys. Rev. A 8, 14 (1973).
[CrossRef]

1972 (2)

1971 (1)

A. Ashkin, Appl. Phys. Lett. 19, 283 (1971); A. Ashkin and J. M. Diedzic, Phys. Rev. Lett. 30, 139 (1973).
[CrossRef]

1970 (2)

G. B. Whitham, J. Fluid Mech. 44, 373 (1970).
[CrossRef]

J. A. Arnaud, Bell Syst. Tech. J. 49, 2311 (1970).
[CrossRef]

1969 (1)

H. A. Haus, Physica 43, 77 (1969).
[CrossRef]

1954 (1)

R. V. Jones and J. C. S. Richard, Proc. R. Soc. (Lond.) A 221, 480 (1954).
[CrossRef]

1953 (1)

K. Husini, Prog. Theor. Phys. 9, 381 (1953).
[CrossRef]

1950 (1)

V. A. Fock, Bull. Acad. Sci. l’URSS, Ser. Phys. 14, 70 (1950), translated in V. A. Fock, Electromagnetic Diffraction and Propagation Problems (Pergamon, Oxford, 1965), Ch. 14.

Arnaud, J. A.

J. A. Arnaud, Bell Syst. Tech. J. 53, 1599 (1974).
[CrossRef]

J. A. Arnaud, J. Opt. Soc. Am. 62, 290 (1972) and J. A. Arnaud, in Progress in Optics, Vol. 11, edited by E. Wolf (North–Holland, Amsterdam, 1973), p. 247.
[CrossRef]

J. A. Arnaud, Bell Syst. Tech. J. 49, 2311 (1970).
[CrossRef]

J. A. Arnaud, (a)Electr. Lett. 8, 541 (1972); (b)Opt. Commun. 7, 313 (1973); (c)Am. J. Phys. 42, 71 (1974).

Ashkin, A.

A. Ashkin, Appl. Phys. Lett. 19, 283 (1971); A. Ashkin and J. M. Diedzic, Phys. Rev. Lett. 30, 139 (1973).
[CrossRef]

Boyer, T. H.

T. H. Boyer, Phys. Rev. D 8, 1669 (1973).

Costa de Beauregard, O.

O. Costa de Beauregard, C. R. Acad. Sci. (Paris) B 274, 164 (1972); Nuovo Cimento, Ser. 10 63B, 611 (1969).

De Groot, S. R.

S. R. De Groot and L. G. Suttorp, Foundations of Electrodynamics (North–Holland, Amsterdam, 1972).

Fock, V. A.

V. A. Fock, Bull. Acad. Sci. l’URSS, Ser. Phys. 14, 70 (1950), translated in V. A. Fock, Electromagnetic Diffraction and Propagation Problems (Pergamon, Oxford, 1965), Ch. 14.

Gordon, J. P.

J. P. Gordon, Phys. Rev. A 8, 14 (1973).
[CrossRef]

Haus, H. A.

H. A. Haus, Physica 43, 77 (1969).
[CrossRef]

P. Penfield and H. A. Haus, Electrodynamics of Moving Media (M.I.T. Press, Cambridge, 1967), p. 211.

Husini, K.

K. Husini, Prog. Theor. Phys. 9, 381 (1953).
[CrossRef]

Jones, R. V.

R. V. Jones and J. C. S. Richard, Proc. R. Soc. (Lond.) A 221, 480 (1954).
[CrossRef]

Joyce, W. B.

W. B. Joyce, Phys. Rev. D 9, 3234 (1974).
[CrossRef]

Kastler, A.

A. Kastler, C. R. Acad. Sci. (Paris) B 278, 1013 (1974).

Kelso, J. M.

J. M. Kelso, Radio Ray Propagation in the Ionosphere (McGraw-Hill, New York, 1964), p. 220.

Lucas, R.

R. Lucas, in Louis de Broglie, sa Conception du Monde Physique (Gauthier-Villars, Paris, 1973).

Marcatili, E. A. J.

E. A. J. Marcatili, Bell Syst. Tech. J. 53, 645 (1974).
[CrossRef]

Marcuse, D.

D. Marcuse, Engineering Quantum Electrodynamics (Harcourt, Brace and World, New York, 1970), p. 80.

D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972), p. 100.

Penfield, P.

P. Penfield and H. A. Haus, Electrodynamics of Moving Media (M.I.T. Press, Cambridge, 1967), p. 211.

Richard, J. C. S.

R. V. Jones and J. C. S. Richard, Proc. R. Soc. (Lond.) A 221, 480 (1954).
[CrossRef]

Suttorp, L. G.

S. R. De Groot and L. G. Suttorp, Foundations of Electrodynamics (North–Holland, Amsterdam, 1972).

Whitham, G. B.

G. B. Whitham, J. Fluid Mech. 44, 373 (1970).
[CrossRef]

Appl. Phys. Lett. (1)

A. Ashkin, Appl. Phys. Lett. 19, 283 (1971); A. Ashkin and J. M. Diedzic, Phys. Rev. Lett. 30, 139 (1973).
[CrossRef]

Bell Syst. Tech. J. (3)

J. A. Arnaud, Bell Syst. Tech. J. 49, 2311 (1970).
[CrossRef]

J. A. Arnaud, Bell Syst. Tech. J. 53, 1599 (1974).
[CrossRef]

E. A. J. Marcatili, Bell Syst. Tech. J. 53, 645 (1974).
[CrossRef]

Bull. Acad. Sci. l’URSS, Ser. Phys. (1)

V. A. Fock, Bull. Acad. Sci. l’URSS, Ser. Phys. 14, 70 (1950), translated in V. A. Fock, Electromagnetic Diffraction and Propagation Problems (Pergamon, Oxford, 1965), Ch. 14.

C. R. Acad. Sci. (Paris) B (2)

O. Costa de Beauregard, C. R. Acad. Sci. (Paris) B 274, 164 (1972); Nuovo Cimento, Ser. 10 63B, 611 (1969).

A. Kastler, C. R. Acad. Sci. (Paris) B 278, 1013 (1974).

J. Fluid Mech. (1)

G. B. Whitham, J. Fluid Mech. 44, 373 (1970).
[CrossRef]

J. Opt. Soc. Am. (1)

Phys. Rev. A (1)

J. P. Gordon, Phys. Rev. A 8, 14 (1973).
[CrossRef]

Phys. Rev. D (2)

W. B. Joyce, Phys. Rev. D 9, 3234 (1974).
[CrossRef]

T. H. Boyer, Phys. Rev. D 8, 1669 (1973).

Physica (1)

H. A. Haus, Physica 43, 77 (1969).
[CrossRef]

Proc. R. Soc. (Lond.) A (1)

R. V. Jones and J. C. S. Richard, Proc. R. Soc. (Lond.) A 221, 480 (1954).
[CrossRef]

Prog. Theor. Phys. (1)

K. Husini, Prog. Theor. Phys. 9, 381 (1953).
[CrossRef]

Other (7)

J. M. Kelso, Radio Ray Propagation in the Ionosphere (McGraw-Hill, New York, 1964), p. 220.

D. Marcuse, Engineering Quantum Electrodynamics (Harcourt, Brace and World, New York, 1970), p. 80.

D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972), p. 100.

J. A. Arnaud, (a)Electr. Lett. 8, 541 (1972); (b)Opt. Commun. 7, 313 (1973); (c)Am. J. Phys. 42, 71 (1974).

R. Lucas, in Louis de Broglie, sa Conception du Monde Physique (Gauthier-Villars, Paris, 1973).

P. Penfield and H. A. Haus, Electrodynamics of Moving Media (M.I.T. Press, Cambridge, 1967), p. 211.

S. R. De Groot and L. G. Suttorp, Foundations of Electrodynamics (North–Holland, Amsterdam, 1972).

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Figures (11)

FIG. 1
FIG. 1

Illustration of the Descartes mechanical analogy for the refraction of light rays. (a) A ball traverses a sheet that reduces its momentum mu by a factor independent of the angle of incidence. The tangential component of m u is invariant because the sheet does not exert a force on the ball in that direction. (b) A light ray is refracted away from the normal by going to a less-dense medium. The tangential component of a vector k , whose magnitude is independent of the incidence angle, is invariant. sin(i)/sin(i) is a constant in both cases.

FIG. 2
FIG. 2

(a) Motion of a steel ball in a gutter. The ball is assumed to have zero velocity (zero energy, by definition) when located on the top. The ball falls to the bottom of the gutter where it is deflected by a plate. Its subsequent motion is defined by the gravitational potential V(x), which corresponds to the profile of the gutter if the gutter is sufficiently shallow. The axial velocity is a constant of motion. (b) Waveguide whose wall-to-wall spacing 2d varies slowly with x. The ray trajectories (in space only) are the same as in (a) when the local wave vector squared k2(x) is proportional to V(x).

FIG. 3
FIG. 3

(a) A modern version of the Descartes mechanical analogy, the sheet being replaced by a pair of electrodes. m* = meV denotes the moving mass. The ratio of the sines of the angles to the normal is a constant, as a result of isotropy. (b) The charged particle traverses a double sheet of current, creating a discontinuity in potential vector a . The tangential component of the canonical momentum J k m * u + e a is invariant. The momentum transferred to the sheet is opposite to the change in m * u , as can be seen by considering a steady flow of charge (current) from the emitter to the collector.

FIG. 4
FIG. 4

Displacement of a slab traversed by a particle. Losses and reflection are neglected. (a) An electron traverses a potential box. Because u decreases as the electron enters the box for the voltage shown, the box displacement is forward. (b) In the optical case the slab displacement is always forward (u < c). In such arrangements, only the mass-carrying momentum is relevant.

FIG. 5
FIG. 5

Experiments in dynamics. (a) A light beam emitting a power P exerts on an absorber immersed in a fluid a force ( P / ω ) k (Jones and Richards experiment). Perforations allow the absorber to move vertically in the fluid. The liquid is forced upward through the perforations and makes the surface of the liquid bulge out, in the steady state. (b) Equivalent model, in which the absorber can slide between two wires. These wires, representing the liquid with index n = 1/cosα, make an angle α with the vertical. (c) Proposed experiment to verify that the force on an absorber need not have the direction of the beam if the medium lacks isotropy. The disk is a lossy dielectric that picks up the optical power from the film through tunneling.

FIG. 6
FIG. 6

Dispersion surfaces for optical waves in isotropic nondispersive media (a) and massive particle (mass m) in free space. The parabolic approximation used in beam optics consists of replacing the circle ( k x 2 + k z 2 k 2 = 0 ) at some ω = ω0 by a parabola. This parabola is shown as a dotted line in (a). The nonrelativistic approximation in mechanics consists of replacing the hyperboloid in (b) by a paraboloid in the neighborhood of ω = m. The laws of diffraction in free space and the law of spreading of pulses in dispersive media have the same general form. They depend on the curvatures of the dispersion surface in the kx, kz planes and k, ω planes, respectively.

FIG. 7
FIG. 7

Deflection of an electron or optical wave by wave-optics gratings that consist of metal tubes (a) and electro-optic crystals (b) respectively. In both the mechanical and optical systems, the wave packet is deflected upward, even though no classical force is exerted on the particle when the voltages are turned on and off while the wave packet is still traveling in the tubes or crystals (second Aharonov and Bohm effect).

FIG. 8
FIG. 8

Optical fibers with small pulse spreading. (a) Tapered dielectric slab. Pulse spreading is very small for certain profiles (see Fig. 11). (b) A step in the thickness introduces coupling between modes with different mode numbers in the y direction. For properly selected dimensions, only one mode is free of radiation loss (Ref. 22). (c) Coupling between a rod carrying trapped modes and a slab carrying radiation modes. (d) The configuration in (c) can be analyzed by replacing the rod by a distribution of electric and magnetic currents (only one line of current is shown) and evaluating the coupling to the surface waves.

FIG. 9
FIG. 9

(a) A ray, at some z, is represented by a point in the phase space kx, x. The area enclosed by three neighboring rays, (α, β, γ) is invariant as z varies (Lagrange ray invariant). It follows that the density f of rays, for any continuous distribution, is invariant, df/dz = 0 (Liouville theorem). (b) The propagation of gaussian beams in uniform square-law media is represented by an ellipse that rotates in phase space with a constant rate Ω. If the beam is injected off axis, the center of the ellipse is off-set.

FIG. 10
FIG. 10

For thick dielectric slabs, or metallic boundaries, and nonrelativistic particles, the theorem of Breit and Tuive, originally derived for cold plasmas, applies. The time of flight of a pulse can be obtained by extending the ray direction along a straight line from the origin, and assuming the medium homogeneous up to the reflection point. This is because the local group velocity is proportional the local k vector. Thus, uzkz, is a constant of motion, that is, the pulse of light moves at a constant speed along the z axis.

FIG. 11
FIG. 11

(a) Quadratically tapered dielectric slab. θ0 denotes the angle of a ray (H0 mode in the direction perpendicular to the slab). (b) Ratio of the local magnitudes of the phase (υ) to group (u) velocities as a function of the slab thickness [ωd/c = (n2 − 1)−1/2ϕ/cosϕ] for n = 10, 1.4, and 1.01. The ratio υ/u is stationary at ϕopt. (c) Pulse delay as a function of the angle θ0 that the ray makes at the origin with the z axis. (The number of modes is approximately proportional to θ 0 2.) For 15 modes, λ0 = 1 µm, n =1.01, pulse spreading is less than 0.05 ns/km.

Equations (20)

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u n ,
d 2 ( x ) d t 2 = V ( x ) .
d 2 ( x ) d t 2 = [ 1 2 n 2 ( x ) ] .
( d x 1 d τ ) 2 + ( d x 2 d τ ) 2 + ( d x 3 d τ ) 2 ( d t d τ ) 2 + 1 = 0 .
J k = m ( d x / d τ ) .
J ω = m ( d t / d τ ) .
| J k ' | | J k | = | d x / d τ | | d x / d τ | = [ | ( d t / d τ ) 2 1 | ( d t / d τ ) 2 1 ] 1 / 2 = [ ( J ω e V ) 2 / m 2 1 ( J ω ) 2 / m 2 1 ] 1 / 2 .
J k = m d x d τ + e a .
f = ( P / ω ) k .
f = ( P / c ) ( n 1 ) ,
k x 2 + k z 2 = k 2 ( x ) .
[ 2 x 2 + 2 z 2 + k 2 ( x ) ] ψ = 0 .
k x 2 + k z 2 e ¯ 2 V 2 ( x ) + 2 m ¯ e ¯ V ( x ) = 0 .
[ 2 x 2 + 2 z 2 + e ¯ 2 V 2 ( x ) 2 m ¯ e ¯ V ( x ) ] ψ = 0 ,
k 2 ( x ) = e ¯ 2 V 2 ( x ) 2 m ¯ e ¯ V ( x ) .
k z k ( 1 / 2 k ) k x 2 .
i ψ z = { k ( x , z ) + [ 1 / 2 k 0 ( z ) ] 2 x 2 } ψ ,
i ψ t = [ e ¯ U ( x , z ) + ( 1 / 2 m ¯ ) 2 x 2 ] ψ .
k z ( e ¯ 2 V 2 2 m ¯ e ¯ V ) 1 / 2 1 2 ( e ¯ V 2 2 m ¯ e ¯ V ) 1 / 2 k x 2 .
i ψ z = { [ 2 m ¯ e ¯ V ( x , z ) ] 1 / 2 + 1 2 [ 2 m ¯ e ¯ V 0 ( z ) ] 1 / 2 2 x 2 } ψ ,