Abstract

A detailed analysis of the noise properties of wideband optical FM systems is presented. The equations, which are extensions of the results derived by Middleton [Q. Appl. Math. 7, 129 (1949)] and Rice [Bell Syst. Tech. J. 27, 109 (1948)] over 50 years ago, can be used to determine the final noise spectral density for all degrees of limiting, with and without sinusoidal frequency modulation. These results are perfectly general and can be used for any FM system (radio, microwave, or optical) having the particular type of limiter described herein.

© 2001 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. O. Rice, “Noise in FM receivers,” in Selected Papers on Frequency Modulation, J. Klapper, ed. (Dover, New York, 1970).
  2. G. L. Abbas, V. W. S. Chan, T. K. Yee, “A dual-detector heterodyne receiver for local oscillator noise suppression,” J. Lightwave Technol. LT-3, 1110–1122 (1985).
    [CrossRef]
  3. W. Sorin, K. W. Chang, G. A. Conrad, P. R. Hernday, “Frequency domain analysis of an optical FM discriminator,” J. Lightwave Technol. 10, 787–793 (1992).
    [CrossRef]
  4. R. V. Schmidt, “Integrated optics switches and modulators,” in Integrated Optics—Physics and Applications, NATO Advanced Science Institute Series B, S. Martellucci, A. N. Chester, eds. (Plenum, New York, 1981), Vol. 91, pp. 194.
  5. C. H. Henry, R. F. Kazarinov, “Quantum noise in photonics,” Rev. Mod. Phys. 68, 801–853 (1996).
    [CrossRef]
  6. N. A. Olsson, “Lightwave systems with optical amplifiers,” J. Lightwave Technol. 7, 1071–1082 (1989).
    [CrossRef]
  7. R. Loudon, T. J. Shepherd, “Properties of the optical quantum amplifier,” Opt. Acta. 31, 1243–1269 (1984).
    [CrossRef]
  8. H. Kogelnik, A. Yariv, “Considerations of noise and schemes for its reduction in laser amplifiers,” Proc. IEEE. 52, 165–172 (1964).
    [CrossRef]
  9. S. O. Rice, “Statistical properties of a sine-wave and random noise,” Bell Syst. Techn. J. 27, 109–157 (1948).
    [CrossRef]
  10. D. Middleton, “The spectrum of frequency-modulated waves after reception in random noise—I,” Q. Appl. Math. 7, 129–174 (1949).
  11. D. Middleton, “The spectrum of frequency-modulated waves after reception in random noise—II,” Q. Appl. Math. 8, 59–80 (1950).
  12. E. Goobar, R. Schatz, “Broadband measurements of frequency noise spectrum in two section DBR laser,” Electron. Lett. 27, 289–291 (1991).
    [CrossRef]
  13. Y. Yamamoto, “AM and FM quantum noise in semiconductor lasers—Part I: Theoretical analysis,” IEEE J. Quantum Electron. QE-19, 34–46 (1983).
    [CrossRef]
  14. D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).

1996

C. H. Henry, R. F. Kazarinov, “Quantum noise in photonics,” Rev. Mod. Phys. 68, 801–853 (1996).
[CrossRef]

1992

W. Sorin, K. W. Chang, G. A. Conrad, P. R. Hernday, “Frequency domain analysis of an optical FM discriminator,” J. Lightwave Technol. 10, 787–793 (1992).
[CrossRef]

1991

E. Goobar, R. Schatz, “Broadband measurements of frequency noise spectrum in two section DBR laser,” Electron. Lett. 27, 289–291 (1991).
[CrossRef]

1989

N. A. Olsson, “Lightwave systems with optical amplifiers,” J. Lightwave Technol. 7, 1071–1082 (1989).
[CrossRef]

1985

G. L. Abbas, V. W. S. Chan, T. K. Yee, “A dual-detector heterodyne receiver for local oscillator noise suppression,” J. Lightwave Technol. LT-3, 1110–1122 (1985).
[CrossRef]

1984

R. Loudon, T. J. Shepherd, “Properties of the optical quantum amplifier,” Opt. Acta. 31, 1243–1269 (1984).
[CrossRef]

1983

Y. Yamamoto, “AM and FM quantum noise in semiconductor lasers—Part I: Theoretical analysis,” IEEE J. Quantum Electron. QE-19, 34–46 (1983).
[CrossRef]

1964

H. Kogelnik, A. Yariv, “Considerations of noise and schemes for its reduction in laser amplifiers,” Proc. IEEE. 52, 165–172 (1964).
[CrossRef]

1950

D. Middleton, “The spectrum of frequency-modulated waves after reception in random noise—II,” Q. Appl. Math. 8, 59–80 (1950).

1949

D. Middleton, “The spectrum of frequency-modulated waves after reception in random noise—I,” Q. Appl. Math. 7, 129–174 (1949).

1948

S. O. Rice, “Statistical properties of a sine-wave and random noise,” Bell Syst. Techn. J. 27, 109–157 (1948).
[CrossRef]

Abbas, G. L.

G. L. Abbas, V. W. S. Chan, T. K. Yee, “A dual-detector heterodyne receiver for local oscillator noise suppression,” J. Lightwave Technol. LT-3, 1110–1122 (1985).
[CrossRef]

Chan, V. W. S.

G. L. Abbas, V. W. S. Chan, T. K. Yee, “A dual-detector heterodyne receiver for local oscillator noise suppression,” J. Lightwave Technol. LT-3, 1110–1122 (1985).
[CrossRef]

Chang, K. W.

W. Sorin, K. W. Chang, G. A. Conrad, P. R. Hernday, “Frequency domain analysis of an optical FM discriminator,” J. Lightwave Technol. 10, 787–793 (1992).
[CrossRef]

Conrad, G. A.

W. Sorin, K. W. Chang, G. A. Conrad, P. R. Hernday, “Frequency domain analysis of an optical FM discriminator,” J. Lightwave Technol. 10, 787–793 (1992).
[CrossRef]

Goobar, E.

E. Goobar, R. Schatz, “Broadband measurements of frequency noise spectrum in two section DBR laser,” Electron. Lett. 27, 289–291 (1991).
[CrossRef]

Henry, C. H.

C. H. Henry, R. F. Kazarinov, “Quantum noise in photonics,” Rev. Mod. Phys. 68, 801–853 (1996).
[CrossRef]

Hernday, P. R.

W. Sorin, K. W. Chang, G. A. Conrad, P. R. Hernday, “Frequency domain analysis of an optical FM discriminator,” J. Lightwave Technol. 10, 787–793 (1992).
[CrossRef]

Kazarinov, R. F.

C. H. Henry, R. F. Kazarinov, “Quantum noise in photonics,” Rev. Mod. Phys. 68, 801–853 (1996).
[CrossRef]

Kogelnik, H.

H. Kogelnik, A. Yariv, “Considerations of noise and schemes for its reduction in laser amplifiers,” Proc. IEEE. 52, 165–172 (1964).
[CrossRef]

Loudon, R.

R. Loudon, T. J. Shepherd, “Properties of the optical quantum amplifier,” Opt. Acta. 31, 1243–1269 (1984).
[CrossRef]

Middleton, D.

D. Middleton, “The spectrum of frequency-modulated waves after reception in random noise—II,” Q. Appl. Math. 8, 59–80 (1950).

D. Middleton, “The spectrum of frequency-modulated waves after reception in random noise—I,” Q. Appl. Math. 7, 129–174 (1949).

D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).

Olsson, N. A.

N. A. Olsson, “Lightwave systems with optical amplifiers,” J. Lightwave Technol. 7, 1071–1082 (1989).
[CrossRef]

Rice, S. O.

S. O. Rice, “Statistical properties of a sine-wave and random noise,” Bell Syst. Techn. J. 27, 109–157 (1948).
[CrossRef]

S. O. Rice, “Noise in FM receivers,” in Selected Papers on Frequency Modulation, J. Klapper, ed. (Dover, New York, 1970).

Schatz, R.

E. Goobar, R. Schatz, “Broadband measurements of frequency noise spectrum in two section DBR laser,” Electron. Lett. 27, 289–291 (1991).
[CrossRef]

Schmidt, R. V.

R. V. Schmidt, “Integrated optics switches and modulators,” in Integrated Optics—Physics and Applications, NATO Advanced Science Institute Series B, S. Martellucci, A. N. Chester, eds. (Plenum, New York, 1981), Vol. 91, pp. 194.

Shepherd, T. J.

R. Loudon, T. J. Shepherd, “Properties of the optical quantum amplifier,” Opt. Acta. 31, 1243–1269 (1984).
[CrossRef]

Sorin, W.

W. Sorin, K. W. Chang, G. A. Conrad, P. R. Hernday, “Frequency domain analysis of an optical FM discriminator,” J. Lightwave Technol. 10, 787–793 (1992).
[CrossRef]

Yamamoto, Y.

Y. Yamamoto, “AM and FM quantum noise in semiconductor lasers—Part I: Theoretical analysis,” IEEE J. Quantum Electron. QE-19, 34–46 (1983).
[CrossRef]

Yariv, A.

H. Kogelnik, A. Yariv, “Considerations of noise and schemes for its reduction in laser amplifiers,” Proc. IEEE. 52, 165–172 (1964).
[CrossRef]

Yee, T. K.

G. L. Abbas, V. W. S. Chan, T. K. Yee, “A dual-detector heterodyne receiver for local oscillator noise suppression,” J. Lightwave Technol. LT-3, 1110–1122 (1985).
[CrossRef]

Bell Syst. Techn. J.

S. O. Rice, “Statistical properties of a sine-wave and random noise,” Bell Syst. Techn. J. 27, 109–157 (1948).
[CrossRef]

Electron. Lett.

E. Goobar, R. Schatz, “Broadband measurements of frequency noise spectrum in two section DBR laser,” Electron. Lett. 27, 289–291 (1991).
[CrossRef]

IEEE J. Quantum Electron.

Y. Yamamoto, “AM and FM quantum noise in semiconductor lasers—Part I: Theoretical analysis,” IEEE J. Quantum Electron. QE-19, 34–46 (1983).
[CrossRef]

J. Lightwave Technol.

N. A. Olsson, “Lightwave systems with optical amplifiers,” J. Lightwave Technol. 7, 1071–1082 (1989).
[CrossRef]

G. L. Abbas, V. W. S. Chan, T. K. Yee, “A dual-detector heterodyne receiver for local oscillator noise suppression,” J. Lightwave Technol. LT-3, 1110–1122 (1985).
[CrossRef]

W. Sorin, K. W. Chang, G. A. Conrad, P. R. Hernday, “Frequency domain analysis of an optical FM discriminator,” J. Lightwave Technol. 10, 787–793 (1992).
[CrossRef]

Opt. Acta.

R. Loudon, T. J. Shepherd, “Properties of the optical quantum amplifier,” Opt. Acta. 31, 1243–1269 (1984).
[CrossRef]

Proc. IEEE.

H. Kogelnik, A. Yariv, “Considerations of noise and schemes for its reduction in laser amplifiers,” Proc. IEEE. 52, 165–172 (1964).
[CrossRef]

Q. Appl. Math.

D. Middleton, “The spectrum of frequency-modulated waves after reception in random noise—I,” Q. Appl. Math. 7, 129–174 (1949).

D. Middleton, “The spectrum of frequency-modulated waves after reception in random noise—II,” Q. Appl. Math. 8, 59–80 (1950).

Rev. Mod. Phys.

C. H. Henry, R. F. Kazarinov, “Quantum noise in photonics,” Rev. Mod. Phys. 68, 801–853 (1996).
[CrossRef]

Other

S. O. Rice, “Noise in FM receivers,” in Selected Papers on Frequency Modulation, J. Klapper, ed. (Dover, New York, 1970).

R. V. Schmidt, “Integrated optics switches and modulators,” in Integrated Optics—Physics and Applications, NATO Advanced Science Institute Series B, S. Martellucci, A. N. Chester, eds. (Plenum, New York, 1981), Vol. 91, pp. 194.

D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (21)

Fig. 1
Fig. 1

Argand plot of the signal-plus-noise vector.

Fig. 2
Fig. 2

Relation between phase and frequency noise.

Fig. 3
Fig. 3

Model for ideal limiter.

Fig. 4
Fig. 4

Experimental technique for measuring the electrical suppression of an optical limiter. The light is converted into a photocurrent, and the noise power of this photocurrent is measured by a spectrum analyzer or power meter.

Fig. 5
Fig. 5

All-optical FM receiver employing a delay-line discriminator and a dual-balanced detector.

Fig. 6
Fig. 6

Gain profile (in dB) for an erbium-doped fiber amplifier.

Fig. 7
Fig. 7

Noise suppression by limiter. Note that only the in-phase component is suppressed. The quadrature component is unaffected.

Fig. 8
Fig. 8

Spectral density for (a) the process n Q (t) and (b) the process n Q (t) - n Q (t - τ), assuming that the optical filter can be characterized by a rectangular passband.

Fig. 9
Fig. 9

Spectral density of the first term of Eq. (28).

Fig. 10
Fig. 10

Noise spectral density for the spontaneous-spontaneous term. The frequency is expressed in units of B, the noise density in units of 2BS Q S I.

Fig. 11
Fig. 11

Noise spectral density for the signal × noise terms (in units of σ4 Bτ 2) for a constant frequency offset equal to the band edge.

Fig. 12
Fig. 12

FM discriminator.

Fig. 13
Fig. 13

Noise spectral density (in units of σ4 Bτ 2) for sinusoidal frequency modulation having a deviation of B and a modulation rate of 0.1B.

Fig. 14
Fig. 14

Graphical depiction of the relation between the various components.

Fig. 15
Fig. 15

Noise spectral density (in units of Bσ 4τ2) for an FM receiver with no limiting and no frequency modulation, for the cases of (a) CNR = 0, (b) CNR = 1, and (c) CNR = 10.

Fig. 16
Fig. 16

Noise spectral density for the case of no modulation and perfect limiting, for the cases of (a) CNR = 0, (b) CNR = 1, and (c) CNR = 10.

Fig. 17
Fig. 17

Noise spectral density as a function of the various degrees of limiting (suppression).

Fig. 18
Fig. 18

Noise spectral density over the baseband bandwidth (0.1B).

Fig. 19
Fig. 19

Noise spectral density for the case of perfect limiting (180-dB suppression) and a CNR of 13 dB. Note the complete elimination of noise at f equal zero.

Fig. 20
Fig. 20

Spectral density of laser frequency noise for (a) 2 mW, (b) 4.75 mW, and (c) 6.5 mW, for a two-electrode distributed Bragg reflector laser (from Ref. 12).

Fig. 21
Fig. 21

Noise spectral density for a CNR of 10, a frequency deviation of B, a modulation frequency of 0.1B, and perfect limiting, for two different analytic approaches.

Tables (1)

Tables Icon

Table 1 Value of the Noise Spectral Density at Zero Frequency As a Function of the CNR (for the Case of No Modulation)

Equations (87)

Equations on this page are rendered with MathJax. Learn more.

SNR=3 CNRΔF/Δfs3.
FM advantageΔF/Δfs2.
Pout=1+AαPrefPin1+AαPin.
Suppression=dPout/PoutdPin/Pin=PinPoutdPindPout=11+AαPin.
0t2=σ22+2p+22σ2+t2 t2,
SuppressiondB=20 log1022+2p+2.
I1-I2=14at-at-τ2-at+at-τ2=-atat-τ,
at=a0 cosω0t+ϕt.
I1-I2=a02cosω0t+ϕtsinω0t+ϕt-τ.
I1-I2=12a02 sinϕt-ϕt-τ+a02sin2ω0t+ϕt+ϕt-τ.
I1-I2=12a02ϕt-ϕt-τ12a02dϕdt τ=12a02ωtτ,
P0=nsphνG-1,
at=carrier+noise=a0 cosω0t+nt,
at=carrier+noise=a0 cosω0t+nItcosω0t+nQtsinω0t,
I1-I2=12a0nQt-nQt-τ+12nQtnIt-τ-nQt-τnIt.
at=a0+nItcosω0t+nQtsinω0t,
Sθf=- θtθt-texp-j2πftdt,
θtθt-t=- Sθfexpj2πftdt,
θt=a0nQt-nQt-τ.
θtθt-t=a02nQt-nQt-τnQt-t-nQt-t-τ=a02nQtnQt-t+nQt-τnQt-τ-t-nQt-τnQt-t-nQtnQt-τ-t.
θtθt-t=a022nQtnQt-t-nQtnQt-t-τ-nQtnQt-t+τ.
Fht-t0=expj2πft0Fht,
Sf=a02SQ1-cos2πfτ |f|<BSf=0 otherwise.
βt=nQtnIt-τ-nQt-τnIt.
βtβt-t=nQtnIt-τ-nQt-τnItnQt-tnIt-t-τ-nQt-t-τnIt-t.
βtβt-t=nQtnQt-tnIt-τnIt-t-τ+nQt-τnQt-t-τnItnIt-t-nQt-τnQt-tnItnIt-t-τ-nQtnQt-t-τ×nIt-τnIt-t.
RXt=nXtnXt-t,
βtβt-t=2RQtRIt-RQt+τRIt-τ-RQt-τRIt+τ.
FR1tR2t=- S1fS2f-fdf,
FRt-τRt+τ=expj2πfτ  exp-j4πfτ×SfSf-fdf=SQSI sin2π2B-fτ2πτ.
Sf=SQSI2B-f1-sin2π2B-fτ2π2B-fτfor 0<f<2B,Sf=0  otherwise,
Sf=a02SQ1-cos2πfτ+SQSI2B-f×1-sin2π2B-fτ2π2B-fτ,
I1-I2=12a02Δωτ cos2πft.
Ps=18a04Δωτ2.
SNR=IsΔωτ24nspeΔf1-cos 2πfτ.
SNRIsΔωτ22nspeΔf2πfτ2=Is2nspeΔfΔFf2,
Pnoise=2nspe 0Δf1-cos 2πfτdf=2nspeΔf1-sin 2πΔfτ2πΔfτ13nspeΔf2πΔfτ2,
SNR=3Is2nspeΔfΔFΔf2.
SNR=3 CNRΔFΔf3,
SQ=SI=nspe=σ22B,  p=CNR=Is2nspeB=a022σ2.
Sf=σ4Bτ2π22pfB2+162-fB3.
at=a0 cosω0t+ϕt+nItcosω0t+nQtsinω0t=a0 cos ϕt+nItcosω0t+a0 sin ϕt+nQtsinω0t.
I1-I2=12a02 sinϕt-τ-ϕt+a0cos ϕt-τnQt-cos ϕtnQt-τ+a0sin ϕt-τnIt-sin ϕtnIt-τ+nQtnIt-τ-nItnQt-τ.
θt=a0cos ϕt-τnQt-cos ϕtnQt-τ,
θtθt-t=a0242RmtRQt-Rmt-τRQt+τ-Rmt+τRQt-τ,
RQt=nQtnQt-t, Rmt=cos ϕtcos ϕt-t.
Rsxn=2θtθt-t=a02τ222R˙mtR˙Qt-R¨mtRQt-RmtR¨Qt,
Sf=-exp-j2πftRtsxndt.
Rmt=cosωdtcosωdt-ωdt=12cosωdt+cos2ωdt-ωdt=12cosωdt,
Sf=a02S01-cos4πfdτcos2πfτ   for f<B-fd, Sf=12 a02S01-cos2πτ2fd-f  for B-fd<f<B-fd,
Sf=σ4Bτ2π28pfdB2+2pfB2  for f<B-fd, Sf=σ4Bτ2π2p2fd-fB2  for B-fd<f<B+fd,
ϕt=Δϕ cos ωmt,
Δϕ=Δω/ωm.
Rmt=cos ϕtcos ϕt-t=12J0Δϕ2 sinωmt/2.
Rtotal=Rsxn+Rnxn=a02τ222R˙mtR˙Qt-R¨mtRQt-RmtR¨Qt+τ22R˙2-RR¨.
at=a0 cos ϕt+nItcos ω0t+a0 sin ϕt+nQtsin ω0t.
at=tcosω0t+ψt,
2=a02+2a0nI cos ϕ+2a0nQ sin ϕ+nI2+nQ2.
tan ψ=a0 sin ϕ+nQa0 cos ϕ+nI.
atat-τ=0t0t-τcosω0t+ψt×cosω0t-τ+ψt-τ=120t0t-τ×sinψt-ψt-τ,
sinψt-ψt-τ=sin ψtcos ψt-τ-cos ψtsin ψt-τ.
0t2=σ22+2p+22σ2+t2 t2.
atat-τ=I1-I2=σ222+2p+2a02 sinϕt-ϕt-τ+a0cos ϕtnQt-τ-cos×ϕt-τnQt+a0sin ϕtnIt-τ-sin ϕt-τnIt+nQtnIt-τ-nItnQt-τ2σ2+t2×2σ2+t-τ2-1/2.
I1-I2=σ2τ22+2p+22σ2+t2a02ϕ˙+a0cos ϕn˙Q+cos ϕϕ˙nI-sin ϕn˙I+sin ϕϕ˙nQ+n˙QnI-n˙InQ.
Rt= dnI1  dnI2  dnQ1 dn˙Q2×W2nI1, nI2, nQ1, nQ2n˙Q2θ1θ2,
F2z1, z2, z8; t=exp-b02z12+z22+z32+z42-b22z52+z62+z72+z82-ϕ0tz1z2+z3z4-ϕ1t×z1z6-z2z5+z3z8-z4z7+ϕ2tz5z6+z7z8,
H1=-dx -dy x2+x2+y2exp-ixz-iyξ.
H1=2πizz2+ξ2 K1z2+ξ2 ,
Hk,2m+n,qnp=2n+2/22+2p+2b0n-1/2m!m+k!1/20 x2m+n+k+1×exp-x2Jq2pxK12xdx.
Rt=k=0m=0 rot2m+k-ϕ2tk2Hk,2m,k+10  2 cosk+1η2-η1+Hk,2m,|k-1|0  2 cosk-1η2-η1+ϕ1t2k2Hk,2m+1,k1  2 cos kη2-η1-12Hk,2m+1,k+21  2 cosk+2η2-η1-12Hk,2m+1,|k-2|1  2 cosk-2×η2-η1+2b0p1/2ϕ1tη˙1+η˙221-δk0Hk,2m,k+10Hk,2m+1,k+21-Hk,2m+1,k1sink+1η2-η1-Hk,2m,|k-1|0Hk,2m+1,k1-Hk,2m+1,|k-2|1sin|k-1|η2-η1+2b0pη˙1η˙2 cos kη2-η1δk0Hk,2m,k+10  2+12Hk,2m,k+10-Hk,2m,|k-1|02.
Pnoise=2eI1Δf+2eI2Δf=2eI1+I2Δf=2eIsGΔf.
Pnoise=Is2τ2SffΔf.
SNR=12Is2Δωτ22IsnspeΔf1-cos2πfτ+2eIsΔfG+NAkTG2RA+Is2τ2SffΔf,
Rt=14π2-dz1-dz4ϕ2tz1z2+z3z4+ϕ1t2z1z4-z2z32-iϕ1tz1z4-z2z3×α˙1z3+α˙2z4-β˙1z1-β˙2z2-β˙1β˙2z1z2+α˙1α˙2z3z4-α˙2β˙1z1z4-α˙1β˙2z2z4K1z12+z32z12+z32K1z22+z42z22+z42×exp-12b0z12+z22+z32+z42-ϕ0tz1z2+z3z4+iα1z1+α2z2+β1z3+β2z4.
-dz1 -dzn expiξ · z-12z · A · z=2πndet A1/2 exp-12ξ · A-1 · ξ.
K1z12+z32z12+z32=120du exp-122u+uz12+z32.
Rt=limγ014π21T0Tdt 0du 0dv×exp-221u+1v-ϕ2ϕ0+ϕ122γ2+ϕ1γα˙1β1+α˙2β2-β˙1α1-β˙2α2+β˙1β˙22α1α2+α˙1α˙22β1β2-α˙2β˙12α1β2-α˙1β˙22α2β1-dz1 -dz4×exp-12b0+uz12+z32-12b0+vz22+z42-ϕ0tz1z2+z3z4-γz1z4-z2z3+iα1z1+α2z2+β1z3+β2z4.
Rt=1T0Tdt 0du 0dv exp-221u+1v×14D3c0+ϕ1tcϕ1+ϕ2tcϕ2+ϕ1t2cϕ12×exp-a022b0+u+v-2ϕ0tcosθ1-θ22bo+ubo+v-ϕ0t2,
D=bo+ubo+v-ϕ0t2, c0=a02θ˙1θ˙2a02D+2ϕ02+ϕ0D-2p2+u+v×cosθ1-θ2-a02ϕ02 sin2θ1-θ2, cϕ1=a02 sinθ1-θ2θ˙1+θ˙2×D+a02ϕ0 cosθ1-θ2-a02b0-a02vθ˙1+uθ˙2, cϕ2=-2ϕ0+a02 cosθ1-θ2D+a02ϕ0-2ϕ0 cosθ1-θ2+2b0+u+v, cϕ12=2D+a02a02 sin2θ1-θ2+2ϕ0 cosθ1-θ2-2b0-u-v.
θ1=θ2=θ˙1=θ˙2=0,  c0=cϕ1=0.
θt=ωdt, cosθ1-θ2=cosωdt-ωdt-t=cos ωdt, θ˙1+θ˙2=2ωd, sinθ1-θ2=-sin ωdt, 1T0Tdt=1.
θt=Δϕ cosωmt, cosθ1-θ2=cos2Δϕ sinωmt/2sinωmt+ωmt/2, θ˙1+θ˙2=-2Δϕ cosωmt/2sinωmt+ωmt/2, sinθ1-θ2=sin2Δϕ sinωmt/2sinωmt-ωmt/2.
12π02πdψ expa cosz sin ψ1cosz sin ψsinz sin ψsin ψsin2z sin ψcos 2ψ,
Int1a, z=12π02πdψ expa cosz sin ψ,
Int1a=12π02πdψ expa cosz sin ψcosz sin ψ, Int1z=12π02πdψ expa cosz sin ψsinz sin ψ×sin ψ.
Int2a, z=12π02πdψ expa cosz sin ψcos 2ψ.
Int1a, z=12π02πdψ expa cosz sin ψ=I0a+2 k=1 IkaJ0kz, Int2a, z=12π02πdψ expa cosz sin ψcos 2ψ=2 k=1 IkaJ2kz,

Metrics