Abstract

A procedure is described for calculating the power coupled between partially coherent waveguide fields that are in different states of coherence. The method becomes important when it is necessary to calculate the power transferred from a distributed source S to a distributed load L through a length of multimode metallic, or dielectric, waveguide. It is shown that if the correlations between the transverse components of the electric and magnetic fields of S and L are described by coherence matrices M and M, respectively, then the normalized average power coupled between them is η=Tr[MM]/Tr[M]Tr[M], where Tr denotes the trace. When the modal impedances are equal, this expression for the coupled power reduces to an equation derived in a previous paper [J. Opt. Soc. Am. A 18, 3061 (2001)], by use of thermodynamic arguments, for the power coupled between partially coherent free-space beams.

© 2002 Optical Society of America

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References

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  1. M. J. Griffin, “Bolometers for far-infrared and submillimetre astronomy,” Nucl. Instrum. Methods Phys. Res. A 444, 397–403 (2000).
    [CrossRef]
  2. P. L. Richards, “Bolometers for infrared and millimeter waves,” J. Appl. Phys. 76, 1–24 (1994).
    [CrossRef]
  3. J. A. Murphy, “Radiation patterns of few-moded horns and condensing lightpipes,” Infrared Phys. 31, 291–299 (1991).
    [CrossRef]
  4. R. Padman, J. A. Murphy, “Radiation patterns of scalar lightpipes,” Infrared Phys. 31, 441–446 (1991).
    [CrossRef]
  5. R. E. Collin, Field Theory of Guided Waves (IEEE Press, New York, 1991), Chap. 5.
  6. P. F. Goldsmith, Quasioptical Systems (IEEE Press, New York, 1998), Chap. 4.
  7. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Chap. 3.
  8. S. Withington, G. Yassin, J. A. Murphy, “Dyadic analysis of partially coherent submillimetre-wave antenna systems,” IEEE Trans. Antennas Propag. 49, 1226–1234 (2001).
    [CrossRef]
  9. A. Wexler, “Solution of waveguide discontinuities by modal analysis,” IEEE Trans. Microwave Theory Tech. 15, 508–517 (1967).
    [CrossRef]
  10. S. Withington, G. Yassin, “Power coupled between partially coherent vector fields in different states of coherence,” J. Opt. Soc. Am. A 18, 3061–3071 (2001).
    [CrossRef]
  11. T. S. Maclean, Principles on Antennas: Wire Aperture (Cambridge U. Press, Cambridge, UK, 1986), Sec. 14.3.
  12. E. Wolf, “New theory of partial coherence in the space–frequency domain. Part I: spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
    [CrossRef]
  13. T. Rozzi, M. Mongiardo, Open Electromagnetic Waveguides, Vol. 43 of Electromagnetic Wave Series 4 (Institution of Electrical Engineers, London, 1977).

2001 (2)

S. Withington, G. Yassin, J. A. Murphy, “Dyadic analysis of partially coherent submillimetre-wave antenna systems,” IEEE Trans. Antennas Propag. 49, 1226–1234 (2001).
[CrossRef]

S. Withington, G. Yassin, “Power coupled between partially coherent vector fields in different states of coherence,” J. Opt. Soc. Am. A 18, 3061–3071 (2001).
[CrossRef]

2000 (1)

M. J. Griffin, “Bolometers for far-infrared and submillimetre astronomy,” Nucl. Instrum. Methods Phys. Res. A 444, 397–403 (2000).
[CrossRef]

1994 (1)

P. L. Richards, “Bolometers for infrared and millimeter waves,” J. Appl. Phys. 76, 1–24 (1994).
[CrossRef]

1991 (2)

J. A. Murphy, “Radiation patterns of few-moded horns and condensing lightpipes,” Infrared Phys. 31, 291–299 (1991).
[CrossRef]

R. Padman, J. A. Murphy, “Radiation patterns of scalar lightpipes,” Infrared Phys. 31, 441–446 (1991).
[CrossRef]

1982 (1)

1967 (1)

A. Wexler, “Solution of waveguide discontinuities by modal analysis,” IEEE Trans. Microwave Theory Tech. 15, 508–517 (1967).
[CrossRef]

Collin, R. E.

R. E. Collin, Field Theory of Guided Waves (IEEE Press, New York, 1991), Chap. 5.

Goldsmith, P. F.

P. F. Goldsmith, Quasioptical Systems (IEEE Press, New York, 1998), Chap. 4.

Griffin, M. J.

M. J. Griffin, “Bolometers for far-infrared and submillimetre astronomy,” Nucl. Instrum. Methods Phys. Res. A 444, 397–403 (2000).
[CrossRef]

Maclean, T. S.

T. S. Maclean, Principles on Antennas: Wire Aperture (Cambridge U. Press, Cambridge, UK, 1986), Sec. 14.3.

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Chap. 3.

Mongiardo, M.

T. Rozzi, M. Mongiardo, Open Electromagnetic Waveguides, Vol. 43 of Electromagnetic Wave Series 4 (Institution of Electrical Engineers, London, 1977).

Murphy, J. A.

S. Withington, G. Yassin, J. A. Murphy, “Dyadic analysis of partially coherent submillimetre-wave antenna systems,” IEEE Trans. Antennas Propag. 49, 1226–1234 (2001).
[CrossRef]

R. Padman, J. A. Murphy, “Radiation patterns of scalar lightpipes,” Infrared Phys. 31, 441–446 (1991).
[CrossRef]

J. A. Murphy, “Radiation patterns of few-moded horns and condensing lightpipes,” Infrared Phys. 31, 291–299 (1991).
[CrossRef]

Padman, R.

R. Padman, J. A. Murphy, “Radiation patterns of scalar lightpipes,” Infrared Phys. 31, 441–446 (1991).
[CrossRef]

Richards, P. L.

P. L. Richards, “Bolometers for infrared and millimeter waves,” J. Appl. Phys. 76, 1–24 (1994).
[CrossRef]

Rozzi, T.

T. Rozzi, M. Mongiardo, Open Electromagnetic Waveguides, Vol. 43 of Electromagnetic Wave Series 4 (Institution of Electrical Engineers, London, 1977).

Wexler, A.

A. Wexler, “Solution of waveguide discontinuities by modal analysis,” IEEE Trans. Microwave Theory Tech. 15, 508–517 (1967).
[CrossRef]

Withington, S.

S. Withington, G. Yassin, J. A. Murphy, “Dyadic analysis of partially coherent submillimetre-wave antenna systems,” IEEE Trans. Antennas Propag. 49, 1226–1234 (2001).
[CrossRef]

S. Withington, G. Yassin, “Power coupled between partially coherent vector fields in different states of coherence,” J. Opt. Soc. Am. A 18, 3061–3071 (2001).
[CrossRef]

Wolf, E.

Yassin, G.

S. Withington, G. Yassin, “Power coupled between partially coherent vector fields in different states of coherence,” J. Opt. Soc. Am. A 18, 3061–3071 (2001).
[CrossRef]

S. Withington, G. Yassin, J. A. Murphy, “Dyadic analysis of partially coherent submillimetre-wave antenna systems,” IEEE Trans. Antennas Propag. 49, 1226–1234 (2001).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

S. Withington, G. Yassin, J. A. Murphy, “Dyadic analysis of partially coherent submillimetre-wave antenna systems,” IEEE Trans. Antennas Propag. 49, 1226–1234 (2001).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

A. Wexler, “Solution of waveguide discontinuities by modal analysis,” IEEE Trans. Microwave Theory Tech. 15, 508–517 (1967).
[CrossRef]

Infrared Phys. (2)

J. A. Murphy, “Radiation patterns of few-moded horns and condensing lightpipes,” Infrared Phys. 31, 291–299 (1991).
[CrossRef]

R. Padman, J. A. Murphy, “Radiation patterns of scalar lightpipes,” Infrared Phys. 31, 441–446 (1991).
[CrossRef]

J. Appl. Phys. (1)

P. L. Richards, “Bolometers for infrared and millimeter waves,” J. Appl. Phys. 76, 1–24 (1994).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Nucl. Instrum. Methods Phys. Res. A (1)

M. J. Griffin, “Bolometers for far-infrared and submillimetre astronomy,” Nucl. Instrum. Methods Phys. Res. A 444, 397–403 (2000).
[CrossRef]

Other (5)

R. E. Collin, Field Theory of Guided Waves (IEEE Press, New York, 1991), Chap. 5.

P. F. Goldsmith, Quasioptical Systems (IEEE Press, New York, 1998), Chap. 4.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Chap. 3.

T. S. Maclean, Principles on Antennas: Wire Aperture (Cambridge U. Press, Cambridge, UK, 1986), Sec. 14.3.

T. Rozzi, M. Mongiardo, Open Electromagnetic Waveguides, Vol. 43 of Electromagnetic Wave Series 4 (Institution of Electrical Engineers, London, 1977).

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Figures (1)

Fig. 1
Fig. 1

Diagram shows two distributed sources of radiation S and L connected by a length of a uniform metallic waveguide (solid bold lines) having an arbitrary cross section. We wish to calculate the power coupled from S to L, when L is regarded as a load. The dotted–dashed line, labeled B, is a closed surface of integration that, within the waveguide, has elemental surface areas of zˆdS.

Equations (58)

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E¯t(r¯t, z)exp(-jωt)=nE¯nt(r¯t, z)exp(-jωt)=nan1Cn e¯n(r¯t)exp(jβnz)×exp(-jωt)=nanψ¯n(r¯t, z)exp(-jωt),
H¯t(r¯t, z)exp(-jωt)=nH¯nt(r¯t, z)exp(-jωt)=nbn1Cn h¯n(r¯t)exp(jβnz)×exp(-jωt)=nbnϕ¯n(r¯t, z)exp(-jωt),
E¯nt(r¯t, z)=-Znzˆ×H¯nt(r¯t, z)forTEmodes,
H¯nt(r¯t, z)=Ynzˆ×E¯nt(r¯t, z)forTMmodes,
Zn=Z 1[1-(fcn/f )2]1/2 forTEmodes,
Yn=1Z1[1-(fcn/f )2]1/2 forTMmodes,
an=bnZn=bn/Yn,
ψ¯n(r¯t, z)=-zˆ×ϕ¯n(r¯t, z),
ϕ¯n(r¯t, z)=zˆ×ψ¯n(r¯t, z).
S[E¯nt(r¯t, z)×H¯mt*(r¯t, z)]zˆdS=0,mn,
Cn=S[e¯n(r¯t)×h¯n(r¯t)]zˆdS,
S[ψ¯n(r¯t, z)×ϕ¯m*(r¯t, z)]zˆdS=δnm,
Sψ¯n(r¯t, z)ψ¯m*(r¯t,z)dS=Sϕ¯n(r¯t, z)ϕ¯m*(r¯t, z)dS=δnm.
Pf=12ReS[E¯t(r¯t, z)×H¯t*(r¯t, z)]zˆdS,
Pf=12nanbn*=12n|an|2Zn.
Pf=12SE¯t(r¯t, z)E¯t*(r¯t, z)dS.
Pf=12n|an|2,
Pf=121ZSE¯t(r¯t, z)E¯t*(r¯t, z)dS.
E¯¯(r¯1, r¯2)=E¯t(r¯2)E¯t*(r¯1),
H¯¯(r¯1, r¯2)=H¯t(r¯2)H¯t*(r¯1),
M¯¯(r¯1, r¯2)=E¯t(r¯2)H¯t*(r¯1),
N¯¯(r¯1, r¯2)=H¯t(r¯2)E¯t*(r¯1).
E¯¯(r¯1, r¯2)=m,nEnmψ¯n(r¯t2, z)ψ¯m*(r¯t1, z),
H¯¯(r¯1, r¯2)=m,nHnmϕ¯n(r¯t2, z)ϕ¯m*(r¯t1, z),
M¯¯(r¯1, r¯2)=m,nMnmψ¯n(r¯t2, z)ϕ¯m*(r¯t1, z),
N¯¯(r¯1, r¯2)=m,nNnmϕ¯n(r¯t2, z)ψ¯m*(r¯t1, z),
Enm=anam*,Hnm=bnbm*,
Mnm=anbm*,Nnm=bnam*,
Enm=ψ¯n*(r¯t2, z)E¯¯(r¯1,r¯2)ψ¯m(r¯t1, z)d2r¯t1d2r¯t2,
Hnm=ϕ¯n*(r¯t2, z)H¯¯(r¯1, r¯2)ϕ¯m(r¯t1, z)d2r¯t1d2r¯t2,
Mnm=ψ¯n*(r¯t2, z)M¯¯(r¯1, r¯2)ϕ¯m(r¯t1, z)d2r¯t1d2r¯t2,
Nnm=ϕ¯n*(r¯t2, z)N¯¯(r¯1, r¯2)ψ¯m(r¯t1, z)d2r¯t1d2r¯t2,
η=S(E¯t×H¯t-E¯t×H¯t)zˆdS216PfPf,
η=nanbn*+an*bn216PfPf.
η=n,manam*bn*bm+anbm*bn*am+bnam*an*bm+bnbm*an*am/16PfPf.
η=n,manam*bn*bm+anbmbn*am+bnam*an*bm+bnbm*an*am/16PfPf,
η=116PfPfTr[EH+MM+NN+HE],
Ynm=Ynδnm,
M=EY,
N=YE,
H=YEY.
η=Tr[EYEY]4PfPf.
η=Tr[EYEY]Tr[EY]Tr[EY]
η=Tr[MM]Tr[M]Tr[M].
η=nmYnYmEnmEmnnmYnYmEnnEmm,
η=nmMnmMmnnmMnnMmm.
UEU=Λ,
Pf=12Tr[EY]=12Tr[UΛUY].
η=nmynymanam*aman*nmynymanan*amam*=1.
η=Tr[EE]Tr[E]Tr[E].
Tr[EE]=iλi(UEU)ii,
Tr[E]=iλi,
Tr[E]=Tr[UEU]=i(UEU)ii.
η=iλi(UEU)iiiλij(UEU)jj,
η=iλi(UEU)iiiλi(UEU)ii+ijiλi(UEU)jj.
η=Tr[ΛUEU]Tr[E]Tr[E],
η=Tr[ΛSΛS]Tr[E]Tr[E],
η=iλijλj|Sij|2.

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