Abstract

Particular higher-order sources give rise to electromagnetic Gaussian beams, which are linearly polarized and have their maximum in the propagation direction. For this dipolar beam the cross-sectional shape changes in the propagation direction. Nodal surfaces exist on which the tangential component of the electric field vanishes in the standing wave that is formed by the two oppositely directed dipolar, electromagnetic Gaussian beams. These surfaces are identified as the mirror shapes for an open resonator that supports this standing wave. For standing waves that have a particular cross-sectional shape at the waist the cross section of the beam near the mirror surfaces is circular. The resonant frequencies for the fundamental transverse mode of such a resonator have been determined as a function of the geometry and the axial mode number. By a perturbation technique the resonant frequency of an open resonator with spherical mirrors has been obtained. This result is valid in only the paraxial approximation. Illustrative numerical results are included.

© 2001 Optical Society of America

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References

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  1. H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
    [CrossRef]
  2. A. L. Cullen, P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. R. Soc. London, Ser. A 366, 155–171 (1979).
    [CrossRef]
  3. G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
    [CrossRef]
  4. L. B. Felsen, “Evanescent waves,” J. Opt. Soc. Am. 66, 751–760 (1976).
    [CrossRef]
  5. C. J. R. Sheppard, S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 16, 1381–1386 (1999).
    [CrossRef]
  6. S. R. Seshadri, “Electromagnetic Gaussian beam,” J. Opt. Soc. Am. A 15, 2712–2719 (1998).
    [CrossRef]
  7. S. R. Seshadri, “Partially coherent Gaussian Schell-model electromagnetic beams,” J. Opt. Soc. Am. A 16, 1373–1380 (1999).
    [CrossRef]
  8. S. Y. Shin, L. B. Felsen, “Gaussian beam modes by multipoles with complex source points,” J. Opt. Soc. Am. 67, 699–700 (1977).
    [CrossRef]
  9. Y. Li, E. Wolf, “Radiation from anisotropic Gaussian Schell-model sources,” Opt. Lett. 7, 256–258 (1982).
    [CrossRef] [PubMed]
  10. F. Gori, G. Guattari, “A new type of optical fields,” Opt. Commun. 48, 7–12 (1983).
    [CrossRef]
  11. R. F. Harrington, Time Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961), pp. 317–321.
  12. A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1967), pp. 123–127.

1999

1998

1983

F. Gori, G. Guattari, “A new type of optical fields,” Opt. Commun. 48, 7–12 (1983).
[CrossRef]

1982

1979

A. L. Cullen, P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. R. Soc. London, Ser. A 366, 155–171 (1979).
[CrossRef]

1977

1976

1971

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[CrossRef]

1966

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Cullen, A. L.

A. L. Cullen, P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. R. Soc. London, Ser. A 366, 155–171 (1979).
[CrossRef]

Deschamps, G. A.

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[CrossRef]

Felsen, L. B.

Gori, F.

F. Gori, G. Guattari, “A new type of optical fields,” Opt. Commun. 48, 7–12 (1983).
[CrossRef]

Guattari, G.

F. Gori, G. Guattari, “A new type of optical fields,” Opt. Commun. 48, 7–12 (1983).
[CrossRef]

Harrington, R. F.

R. F. Harrington, Time Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961), pp. 317–321.

Kogelnik, H.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Li, T.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Li, Y.

Saghafi, S.

Seshadri, S. R.

Sheppard, C. J. R.

Shin, S. Y.

Wolf, E.

Yariv, A.

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1967), pp. 123–127.

Yu, P. K.

A. L. Cullen, P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. R. Soc. London, Ser. A 366, 155–171 (1979).
[CrossRef]

Electron. Lett.

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

F. Gori, G. Guattari, “A new type of optical fields,” Opt. Commun. 48, 7–12 (1983).
[CrossRef]

Opt. Lett.

Proc. IEEE

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Proc. R. Soc. London, Ser. A

A. L. Cullen, P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. R. Soc. London, Ser. A 366, 155–171 (1979).
[CrossRef]

Other

R. F. Harrington, Time Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961), pp. 317–321.

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1967), pp. 123–127.

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Figures (1)

Fig. 1
Fig. 1

Schematic diagram of the open-cavity resonator. M, perfectly conducting concave mirror. The distance between the two mirrors along the axis of symmetry is D=2d. The long-dashed curves indicate the contours of constant amplitude of the electromagnetic beam.

Equations (110)

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2x2+2y2+2z2+k2Fz(x, y, z)=0,
Ex=-Fzy,
Ey=Fzx,
Ez=0,
Hx=-1ik2Fzxz,
Hy=-1ik2Fzyz,
Hz=-1ik2z2+k2Fz.
Fz(r)=F0+(x, y, z)exp(ikz),
Et+(r)=-zˆ×Ht+(r)=zˆ×tF0+(r)exp(ikz),
2x2+2y2+2ik zF0+(x, y, z)=0.
(2+k2)Fz(r)=-0Jm0δ(x) y δ(y)δ(z),
Fz(x, y, z)=0Jm0yexp(ikr)4πr,
Fz(r)=N cos mϕρmexp(-ρ2/w02),
Fz(x, y; z=0)=Ny exp-x2w0x2-y2w0y2,
Fz(x, y, z)=-i0Jm0k exp(kb)4πb2y(1+iz/b)2×exp-k2b (x2+y2)+ikz.
F0+(x, y, z)=N1+i zbx-1/21+i zby-3/2y×exp-x2w0x2(1+iz/bx)-y2w0y2(1+iz/by),
bu=12 kw0u2,u=x, y.
Ex+(x, y, z)=Hy+(x, y, z)=-N1+i zbx-1/21+i zby-3/2×1-2y2w0y2(1+iz/by)×exp-x2w0x2(1+iz/bx)-y2w0y2(1+iz/by)exp(ikz),
Ey+(x, y, z)=-Hx+(x, y, z)=-N1+i zbx-3/21+i zby-3/22xyw0x2×exp-x2w0x2(1+iz/bx)-y2w0y2(1+iz/by)exp(ikz),
Ez+(x, y, z)=Hz+(x, y, z)=0.
S+(x, y, z)=zˆSz+(x, y, z)=zˆ c2Re[Ex+(x, y, z)Hy+*(x, y, z)-Ey+(x, y, z)Hx+*(x, y, z)]=zˆ N2cw0xw0y32wfx(z)wfy3(z)1-4y2wfy2(z)+4y4w0y2wfy2(z)+4x2y2w0x2wfx2(z)exp-2x2wfx2(z)-2y2wfy2(z),
wfu2(z)=w0u21+z2bu2,u=x, y.
P+=--dxdySz+(x, y, z)=N2cπw0y2D04,
D0=w0y4w0x+3w0x4w0y.
N2=4cπw0y2D0.
(u)av=--dxdyuSz+(x, y, z)=0,u=x, y.
(x2)av=--dxdyx2Sz+(x, y, z)=3w0x216D0w0yw0x+w0xw0y+3z24k2w0x2D0w0yw0x+w0xw0y,
(y2)av=w0y216D03w0yw0x+7w0xw0y+3z24k2w0y2D0w0yw0x+5w0xw0y,
(xy)av=0.
(σϕ2)av=--dxdy(x cos ϕ0+y sin ϕ0)2Sz+(x, y, z)=(x2)avcos2 ϕ0+(y2)avsin2 ϕ0.
(x2)av=38 w02+z2k232w02,
(y2)av=58 w02+z2k292w02.
Ex+(x, y, z)=Hy+(x, y, z)=-N(w0x)1/2(w0y)3/2[wfx(z)]1/2[wfy(z)]3/2×exp-x2wfx2(z)-y2wfy2(z)×1-2y2w0ywfy(z)exp[-iΦy(z)]×exp[iqx1(z)],
Ey+(x, y, z)=-Hx+(x, y, z)=-N(w0x)3/2(w0y)3/2[wfx(z)]3/2[wfy(z)]3/2×exp-x2wfx2(z)-y2wfy2(z)×2xyw0x2exp[iqy(z)],
qx1(z)=kz-12 Φx(z)-32 Φy(z)+kx22Rx(z)+ky22Ry(z),
qy(z)=qx1(z)-Φx(z),
Φu(z)=tan-1zbu,u=x, y,
Ru(z)=z+bu2z,u=x, y.
Eeven=Eeven*=12 (E++E+*),
Heven=-Heven*=12 (H+-H+*),
Eodd=-Eodd*=12 (E+-E+*),
Hodd=Hodd*=12 (H++H+*).
Ex,odd=-iN(w0x)1/2(w0y)3/2[wfx(z)]1/2[wfy(z)]3/2exp-x2wfx2(z)-y2wfy2(z)×sin qx1(z)-2y2w0ywfy(z)sin qx2(z),
Ey,odd=-iN(w0x)3/2(w0y)3/2[wfx(z)]3/2[wfy(z)]3/2exp-x2wfx2(z)-y2wfy2(z)×2xyw0x2sin qy(z),
Ez,odd=0,
qx2(z)=qx1(z)-Φy(z).
Hx,odd=N(w0x)3/2(w0y)3/2[wfx(z)]3/2[wfy(z)]3/2exp-x2wfx2(z)-y2wfy2(z)×2xyw0x2cos qy(z),
Hy,odd=-N(w0x)1/2(w0y)3/2[wfx(z)]1/2[wfy(z)]3/2exp-x2wfx2(z)-y2wfy2(z)×cos qx1(z)-2y2w0ywfy(z)cos qx2(z),
Hz,odd=0.
Ex,odd=-iN(w0x)1/2(w0y)3/2[wfx(z)]1/2[wfy(z)]3/2sin qx1(z).
kd-12 Φx(d)-32 Φy(d)=pπ,
p=-2, -1, 0, 1, 2, .
ff0=1+12πptan-1D2r2πp(f/f0)w02+3 tan-1D22πp(f/f0)w02r,
D=2d,
w02=w0xw0y,
r=w0y/w0x,
f0=pcD.
ϕ(x, y, z)=z-d+x22Rxd+y22Ryd=0,
Rud=Ru(d),u=x, y.
Ex(x, y, z)=0,
Ey(x, y , z)=0.
sin qx1(z)=2y2w0ywfy(z)sin qx2(z),
sin qy(z)=0.
Φu(z)=Φu(d)+Φu1,
Φu1=-1kwfu2(d)x2Rxd+y2Ryd.
ku22Ru(z)=ku22Rud+Fu1,u=x, y,
Fu1=ku24Rud21-bu2d2x2Rxd+y2Ryd,u=x, y.
qx1(z)=kd-12 Φx(d)-32 Φy(d)+kz-d+x22Rxd+y22Ryd-12 Φx1-32 Φy1+Fx1+Fy1.
sin qx1(z)=(-1)pk(z-d)+kx22Rxd+ky22Ryd-12 Φx1-32 Φy1+Fx1+Fy1.
qx2(z)=pπ-Φy(d)+kz-d+x22Rxd+y22Ryd-12 Φx1-52 Φy1+Fx1+Fy1.
sin qx2(z)=(-1)pw0ywfy(d)k(z-d)+kx22Rxd+ky22Ryd-12 Φx1-52 Φy1+Fx1+Fy1-dby.
qy(z)=pπ-Φx(d)+kz-d+x22Rxd+y22Ryd-32 Φx1-32 Φy1+Fx1+Fy1.
k(z-d)+kx22Rxd+ky22Ryd-32 Φx1-32 Φy1+Fx1
+Fy1-dbx=0.
sin qx2(z)=(-1)pw0ywfy(d)dbx-dby+Φx1-Φy1.
k(z-d)+kx22Rxd+ky22Ryd-12 Φx1-32 Φy1+Fx1+Fy1
=2y2wfy2(d)dbx-dby+Φx1-Φy1-1bx-1byd2by Φy1.
z=zM=d-x2Sx-y2Sy-x4Tx-2x2y2Txy-y4Ty,
1Sx=12Rxd1+1k2wfx2(d)+3k2wfy2(d),
1Sy=12Ryd1+1k2wfx2(d)+3k2wfy2(d)-2by1bx-1by,
1Tx=12Rxd31-Rxd2d,
1Txy=14Rxd2Ryd1-Rxd2d+14RxdRyd21-Ryd2d+1k2wfx2(d)wfy2(d)Rxd-1k2wfy4(d)Rxd-1bx-1byd2k2bywfy4(d)Rxd,
1Ty=12Ryd31-Ryd2d+2k2wfx2(d)wfy2(d)Ryd-2k2wfy4(d)Ryd-1bx-1by2d2k2bywfy4(d)Ryd.
zs=d-x22R-y22R-x48R3-2x2y28R3-y48R3.
Δz=zM-zS=x212R-1Sx+y212R-1Sy+x418R3-1Tx+2x2y218R3-1Txy+y418R3-1Ty.
dff=dWW,
W=V(H.H*+E.E*)dV,
dW=ΔV(H.H*-E.E*)dV.
H.H*+E.E*=14 [(Ey++Ey+*)2+(Ex++Ex+*)2-(Ex+-Ex+*)2-(Ey+-Ey+*)2]=Ex+Ex+*+Ey+Ey+*=2c Sz+(x, y, z).
W=-dx-dy-dddz 2c Sz+(x, y, z)=4dc=2Dc.
Ex,oddEx,odd*=N2w0xw0y3wfx(z)wfy3(z)exp-2x2wfx2(z)-2y2wfy2(z)×sin qx1(z)-2y2w0ywfy(z)sin qx2(z)2.
sin qx1(z)=0.
sin qx2(z)=-(-1)psin Φy(d).
Ex,oddEx,odd*=N2w0xw0y3wfx(z)wfy3(z)exp-2x2wfx2(z)-2y2wfy2(z)×4y4w0y2wfy2(z)sin2 Φy(d),
sin2 Φu(d)=w0u2d2wfu2(d)bu2,u=x,y.
Ey,oddEy,odd*=N2w0x3w0y3wfx3(z)wfy3(z)exp-2x2wfx2(z)-2y2wfy2(z)×4x2y2w0x4sin2 qy(z).
Ey,oddEy,odd*=N2w0xw0y3wfx(z)wfy3(z)exp-2x2wfx2(z)-2y2wfy2(z)×4x2y2w0x2wfx2(z)sin2 Φx(d).
H.H*-E.E*=H.H*+E.E*-2E.E*=N2w0xw0y3wfx(d)wfy3(d)1-4y2wfy2(d)+4y4wfy4(d)×1-d2by2+4x2y2wfx4(d)1-d2bx2×exp-2x2wfx2(d)-2y2wfy2(d).
dW=2-dxdy(zM-zS)(H.H*-E.E*).
dff=2w0xw0yDD0 (Ax+Ay+Bx+By+Bxy),
αx=1-d2bx2wfy2(d),
αy=1-d2by2wfx2(d),
Ax=12R-1Sx316wfy2(d) (αx+αy),
Ay=12R-1Sy-12+316wfx2(d) (αx+5αy),
Bx=18R3-1Tx3wfx2(d)64wfy2(d) (5αx+3αy),
By=18R3-1Ty-3wfy2(d)4+15wfy2(d)64wfx2(d) (αx+7αy),
Bxy=18R3-1Txy-wfx2(d)4+332 (αx+5αy).
w04=3D4(5r-2-r2)4π2p2(f/f0)2(3r-2-4-3r2).
w02=4(x2)av(3r-1+r)(5r-2-r2)(r-1+r)(15r-3+3r-1-7r-3r3).
D2(x2)av2=643 π2p2ff02(3r-1+r)2×(3r-2-4-3r2)(5r-2-r2)(r-1+r)2(15r-3+3r-1-7r-3r3)2.

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