Abstract

We use a general solution to the eikonal equation to define generalized coordinates in terms of which the Maxwell equations are then cast. These coordinates are then used to obtain expressions for the electric and magnetic field vectors and the Poynting vector. An arbitrary vector function V is introduced that is subject to certain side conditions derived in this process. The k function, the arbitrary function that arises in the solution of the eikonal equation, contains only information descriptive of the geometry of a wave-front train. The vector function V contains information pertaining to the physics of the propagating energy that is distinct from the geometry of the wave-front train.

© 2000 Optical Society of America

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References

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  1. G. Joos, Theoretical Physics, translated by I. M. Freeman (Blackie, London, 1941), pp. 322ff. Most modern references start with the dipole oscillator and deduce the spherical wave fronts, such as J. B. Marion, M. A. Heald, Classical Electromagnetic Radiation, 2nd ed. (Academic, New York, 1980), pp. 232–245.
  2. O. N. Stavroudis, The Optics of Rays, Wavefronts and Caustics (Academic, New York, 1972), Chaps. VIII and IX. The method of Lagrange and Charpit is used to obtain a general solution; a detailed account can be found in A. R. Forsyth, A Treatise on Differential Equations (Dover, Mineola, N.Y., 1996), Chap. IX. See also O. N. Stavroudis, “The k function in geometrical optics and its relationship to the archetypal wave front and the caustic surface,” J. Opt. Soc. Am. A 12, 1010–1016 (1995).
    [CrossRef]
  3. O. N. Stavroudis, R. C. Fronczek, R.-S. Chang, “Geometry of the half-symmetric image,” J. Opt. Soc. Am. 68, 739–742 (1978).
    [CrossRef]
  4. O. N. Stavroudis, “The k function in geometrical optics and its relationship to the archetypal wave front and the caustic,” J. Opt. Soc. Am. 12, 1012–1016 (1995).
    [CrossRef]
  5. M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), Chap. I.
  6. The element of area D, defined in Eq. (5), vanished on the caustic surface so that the wave fronts form a cusp. Turning this around, one can regard the caustic as the cusp locus of the wave-front train.
  7. See, for example, G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers, 2nd ed. (McGraw-Hill, New York, 1961), Chap. 1.

1995 (1)

O. N. Stavroudis, “The k function in geometrical optics and its relationship to the archetypal wave front and the caustic,” J. Opt. Soc. Am. 12, 1012–1016 (1995).
[CrossRef]

1978 (1)

Born, M.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), Chap. I.

Chang, R.-S.

Fronczek, R. C.

Joos, G.

G. Joos, Theoretical Physics, translated by I. M. Freeman (Blackie, London, 1941), pp. 322ff. Most modern references start with the dipole oscillator and deduce the spherical wave fronts, such as J. B. Marion, M. A. Heald, Classical Electromagnetic Radiation, 2nd ed. (Academic, New York, 1980), pp. 232–245.

Korn, G. A.

See, for example, G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers, 2nd ed. (McGraw-Hill, New York, 1961), Chap. 1.

Korn, T. M.

See, for example, G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers, 2nd ed. (McGraw-Hill, New York, 1961), Chap. 1.

Stavroudis, O. N.

O. N. Stavroudis, “The k function in geometrical optics and its relationship to the archetypal wave front and the caustic,” J. Opt. Soc. Am. 12, 1012–1016 (1995).
[CrossRef]

O. N. Stavroudis, R. C. Fronczek, R.-S. Chang, “Geometry of the half-symmetric image,” J. Opt. Soc. Am. 68, 739–742 (1978).
[CrossRef]

O. N. Stavroudis, The Optics of Rays, Wavefronts and Caustics (Academic, New York, 1972), Chaps. VIII and IX. The method of Lagrange and Charpit is used to obtain a general solution; a detailed account can be found in A. R. Forsyth, A Treatise on Differential Equations (Dover, Mineola, N.Y., 1996), Chap. IX. See also O. N. Stavroudis, “The k function in geometrical optics and its relationship to the archetypal wave front and the caustic surface,” J. Opt. Soc. Am. A 12, 1010–1016 (1995).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), Chap. I.

J. Opt. Soc. Am. (2)

O. N. Stavroudis, R. C. Fronczek, R.-S. Chang, “Geometry of the half-symmetric image,” J. Opt. Soc. Am. 68, 739–742 (1978).
[CrossRef]

O. N. Stavroudis, “The k function in geometrical optics and its relationship to the archetypal wave front and the caustic,” J. Opt. Soc. Am. 12, 1012–1016 (1995).
[CrossRef]

Other (5)

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), Chap. I.

The element of area D, defined in Eq. (5), vanished on the caustic surface so that the wave fronts form a cusp. Turning this around, one can regard the caustic as the cusp locus of the wave-front train.

See, for example, G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers, 2nd ed. (McGraw-Hill, New York, 1961), Chap. 1.

G. Joos, Theoretical Physics, translated by I. M. Freeman (Blackie, London, 1941), pp. 322ff. Most modern references start with the dipole oscillator and deduce the spherical wave fronts, such as J. B. Marion, M. A. Heald, Classical Electromagnetic Radiation, 2nd ed. (Academic, New York, 1980), pp. 232–245.

O. N. Stavroudis, The Optics of Rays, Wavefronts and Caustics (Academic, New York, 1972), Chaps. VIII and IX. The method of Lagrange and Charpit is used to obtain a general solution; a detailed account can be found in A. R. Forsyth, A Treatise on Differential Equations (Dover, Mineola, N.Y., 1996), Chap. IX. See also O. N. Stavroudis, “The k function in geometrical optics and its relationship to the archetypal wave front and the caustic surface,” J. Opt. Soc. Am. A 12, 1010–1016 (1995).
[CrossRef]

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Equations (97)

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W(v, w, s)=[x(v, w, s),y(v, w, s),z(v, w, s)].
W(v, w, s)=qn2S-K,
S=(u, v, w),
K=(0, kv, kw),
q=(ns-k)-(vkv+wkw)=(ns-k)+SK.
Ws=1D (Wv×Ww)=1nS,
D2=(Wv×Ww)2.
×E=-μcHt,E=0,
×H=cEt,H=0,
t=d sd ts=v s=cns,
×E=-μcHt=-μHs,
×H=cEt=μEs.
fv=fWv,fw=fWw,fs=fWs.
fwWv-fvWw=(fWw)Wv-(fWv)Ww=f×(Wv×Ww)=D(f×Ws).
f=1D [(Ww×Ws)fv-(Wv×Ws)fw]+Wsfs.
V=1D [(Ww×Ws)Vv-(Wv×Ws)Vw]+WsVs,
×V=1D [(Ww×Ws)×Vv-(Wv×Ws)×Vw]+Ws×Vs,
(P)A=1D [(Ws×P)WwAv-(Ws×P)WvAw]+(WsP)As,
F=1D [(Ww×Ws)(FWv)-(Wv×Ws)(FWw)]+Ws(FWs).
Ws=Ds/D
×Ws=0.
E=1D [(Ww×Ws)Ev-(Wv×Ws)Ew]+WsEs=0.
WsEs=WsHs=0.
[Ev×Ww-Ew×Wv]Ws=0.
Ev×Ww-Ew×Wv=D(Ws×Vs).
EvWs=-(Ws×Vs)Wv,
EwWs=-(Ws×Vs)Ww.
EWvs=(Ws×Vs)Wv,
EWws=(Ws×Vs)Ww.
E×(Wvs×Wws)
=(W×Vs)WwWvs-(Ws×Vs)WvWws.
Wvs=(-v, u, 0)/nu,
Wws=(-w, 0, u)/nu,
Wvs×Wws=S /n2u=Ws/nu.
E×Ws=Q,
Q=(Ws×Vs)WwWvs-(Ws×Vs)WvWws.
WsWvs=WsWws=0.
Q=(V)Ws.
E=Ws×Q.
×E=1D [(Ww×Ws)×Ev-(Wv×Ws)×Ew]
+Ws×Es=-μHs.
-1D [(EvWs)Ww-(Ev  Ww)Ws - (EwWs)Wv
+(EwWv)Ws] + Ws×Es=-μHs.
(EvWw)-(EwWv)=0,
-1D [(EvWs)Ww-(EwWs)Wv]+Ws×Es
=-μHs.
(Ws×Q)vWw-(Ws×Q)wWv=0,
(Wvs×Q+Ws×Qv)Ww
-(Wws×Q+Ws×Qw)Wv=0,
-(Wvs×Ww)  Q - (Wv×Wws)  Q
+(Ww × Ws)  Qv - (Wv × Ws)  Qw=0.
-(Wv×Ww)sQ=-(DWs)sQ=-DsWsQ=0.
Q=0.
1D [(Ws×Vs)WvWw-(Ws×Vs)WwWv]
+ Ws×(Ws×Qs)=-μHs.
-1D [(Ws×Vs)×(Wv×Ww)]-Qs=-μHs,
Hs=μ [Qs+(Ws×Vs)×Ws],
H=μ [Q+(Ws×V)×Ws]=μ [Q+V-(WsV)Ws].
(Ws×V)Q=0.
(Ws×V)Q
=(Ww×Ws)Vs(Wvs×Ws)V-(Wv×Ws)Vs(Wws×Ws)V=0,
(Wv×Ws)Vs(Ww×Ws)Vs(Wvs×Ws)V(Wws×Ws)V=0.
(Wv×Ws)Vs=α(Wvs×Ws)V,
(Ww×Ws)Vs=α(Wws×Ws)V.
Q=α[(Wws×Ws)VWvs-(Wvs×Ws)VWws]
=α[(Ws×V)WwsWvs-(Ws×V)WvsWws]
=α(Ws×V)×(Wvs×Wws)
=(α/nu)(Ws×V)×Ws
=(α/nu)[V-(VWs)Ws].
Q=(α/nu)[V-(WsV)Ws]+(α/nu)[V-(WsV)Ws(WsV)Ds/D]=(α/nu)V-(WsV)(α/nu)s+(α/nu)×(V)-(α/nu)(Ws)(WsV)-(α/nu)(WsV)Ds/D.
D(Q)
=D[(α/nu)V]-[D(α/nu)(WsV)]s=0,
[(α/nu)V]=(1/D)[D(α/nu)(WsV)]s.
E=(α/nu)(Ws×V).
H=μ [Q+(Ws×V)×Ws]=μαnu+1(Ws×V)×Ws,
P=-c4πμαnuαnu+1(V×Ws)2Ws.
H=μαnu+1[V-(WsV)Ws]+αnu+1[V-(WsV)Ws-(WsV)(Ws)]=μαnu+1V+αnu+1V-(WsV)αnu+1s-αnu+1(WsVs)-αnu+1(WsV) DsD=0.
αnu+1V=1DDαnu+1(WsV)s.
D(V)=[D(VWs)]s,
α=nu.
E=(Ws×V),
H=2μ (Ws×V)×Ws,
P=-c2πμ (V×Ws)2Ws.
2{×V-(VWs)×Ws}=Ws×Vs,
2{×V+Ws×[(V)Ws+Vs+Ws×(×V)]}
=Ws×Vs.
Ws×Vs+2{Ws×[(V)Ws]
+Ws(×V)Ws}=0.
Ws(×V)=0.
Ws×[Vs+2(V)Ws]=0.
Vs+2(V)Ws=βWs.
β=WsVs,
Vs-(WsVs)Ws=Ws×(Vs×Ws)=-2(V)Ws.
(Ww×Ws)Vv-(Wv×Ws)Vw=Ds(VWs).
WwVu-WvVw=0.
DWs×(Vs×Ws)+(Ws×V)WwWvs
-(Ws×V)WvWws=0.

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