Abstract

Attention is called to the analogy between linear optical systems and linear two-port electrical networks. For both, the transformation of a pair of oscillating quantities between input and output is of interest. The mapping of polarization by an optical system and of impedance (admittance) by a two-port network is described by a bilinear transformation. Therefore for each transfer property of a system of one type, there is a similar property for the system of the other type. Two-port electrical networks are synthesized whose impedance-(or admittance-) mapping properties are the same as the polarization-mapping properties of a given optical system. The opposite problem of finding the optical analogs of two-port networks is also considered. Besides unifying the methods of handling these two different kinds of systems, the analogy appears fruitful if used reciprocally to simulate electrical networks by optical systems, and vice versa. Linear mechanoacoustic systems have optical analogs besides their well-known electrical analogs.

© 1972 Optical Society of America

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References

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  1. When there is more than one incident beam or when a single incident beam generates a number of emergent beams (e.g., in the presence of a diffraction grating), the electrical analog of the optical system becomes a multiport network. There is no loss of generality, however, when a two-port network is used to represent the relation between one incident and one emergent beam.
  2. W. A. Shurcliff, Polarized Light (Harvard U.P., Cambridge, 1962).
  3. R. M. A. Azzam, N. M. Bashara, J. Opt. Soc. Am. 62, 222 (1972).
    [CrossRef]
  4. R. M. A. Azzam, N. M. Bashara, Optics Commun. 4, 203 (1971).
    [CrossRef]
  5. R. M. A. Azzam, N. M. Bashara, J. Opt. Soc. Am. 62, 36 (1972); Opt. Commun. 5, 5 (1972).
  6. R. M. A. Azzam, N. M. Bashara, Appl. Opt. 12, January (1973).
    [PubMed]
  7. Large-signal equivalent circuits are sometimes used to relate voltages and currents of the same frequency in the presence of nonlinear effects [see, e.g., M.A.H. El-Said, IEEE Trans. Circuit Theory 17, 8 (1970)]. Such circuits might simulate nonlinear optical systems if light waves of the same frequency are coherent at input and output.
    [CrossRef]
  8. Conventionally voltages are listed before currents in the input and output vectors.
  9. V. H. Rumsey, Proc. IRE 39, 535 (1951).
    [CrossRef]
  10. G. A. Deschamps in Proc. Symp. Modern Network SynthesisJ. Fox, Ed. (Polytechnic Press, Brooklyn, 1952).
  11. E. A. Guillemin, Mathematics of Circuit Analysis, (Massachusetts Institute of Technology Press, Cambridge, 1949).
  12. When the orthogonal left and right circular polarizations are used as basis states, the azimuth θ and ellipticity e of the ellipse of polarization are obtained from the complex polarization variable χ [Eq. (3)] by the relations θ=12 Arg(χ) and e = (|χ| − 1)/(|χ| + 1). These lead to the equiazimuth-equiellipticity chart of Fig. 2 (left).

1973

R. M. A. Azzam, N. M. Bashara, Appl. Opt. 12, January (1973).
[PubMed]

1972

R. M. A. Azzam, N. M. Bashara, J. Opt. Soc. Am. 62, 36 (1972); Opt. Commun. 5, 5 (1972).

R. M. A. Azzam, N. M. Bashara, J. Opt. Soc. Am. 62, 222 (1972).
[CrossRef]

1971

R. M. A. Azzam, N. M. Bashara, Optics Commun. 4, 203 (1971).
[CrossRef]

1970

Large-signal equivalent circuits are sometimes used to relate voltages and currents of the same frequency in the presence of nonlinear effects [see, e.g., M.A.H. El-Said, IEEE Trans. Circuit Theory 17, 8 (1970)]. Such circuits might simulate nonlinear optical systems if light waves of the same frequency are coherent at input and output.
[CrossRef]

1951

V. H. Rumsey, Proc. IRE 39, 535 (1951).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Appl. Opt. 12, January (1973).
[PubMed]

R. M. A. Azzam, N. M. Bashara, J. Opt. Soc. Am. 62, 222 (1972).
[CrossRef]

R. M. A. Azzam, N. M. Bashara, J. Opt. Soc. Am. 62, 36 (1972); Opt. Commun. 5, 5 (1972).

R. M. A. Azzam, N. M. Bashara, Optics Commun. 4, 203 (1971).
[CrossRef]

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Appl. Opt. 12, January (1973).
[PubMed]

R. M. A. Azzam, N. M. Bashara, J. Opt. Soc. Am. 62, 222 (1972).
[CrossRef]

R. M. A. Azzam, N. M. Bashara, J. Opt. Soc. Am. 62, 36 (1972); Opt. Commun. 5, 5 (1972).

R. M. A. Azzam, N. M. Bashara, Optics Commun. 4, 203 (1971).
[CrossRef]

Deschamps, G. A.

G. A. Deschamps in Proc. Symp. Modern Network SynthesisJ. Fox, Ed. (Polytechnic Press, Brooklyn, 1952).

El-Said, M.A.H.

Large-signal equivalent circuits are sometimes used to relate voltages and currents of the same frequency in the presence of nonlinear effects [see, e.g., M.A.H. El-Said, IEEE Trans. Circuit Theory 17, 8 (1970)]. Such circuits might simulate nonlinear optical systems if light waves of the same frequency are coherent at input and output.
[CrossRef]

Guillemin, E. A.

E. A. Guillemin, Mathematics of Circuit Analysis, (Massachusetts Institute of Technology Press, Cambridge, 1949).

Rumsey, V. H.

V. H. Rumsey, Proc. IRE 39, 535 (1951).
[CrossRef]

Shurcliff, W. A.

W. A. Shurcliff, Polarized Light (Harvard U.P., Cambridge, 1962).

Appl. Opt.

R. M. A. Azzam, N. M. Bashara, Appl. Opt. 12, January (1973).
[PubMed]

IEEE Trans. Circuit Theory

Large-signal equivalent circuits are sometimes used to relate voltages and currents of the same frequency in the presence of nonlinear effects [see, e.g., M.A.H. El-Said, IEEE Trans. Circuit Theory 17, 8 (1970)]. Such circuits might simulate nonlinear optical systems if light waves of the same frequency are coherent at input and output.
[CrossRef]

J. Opt. Soc. Am.

R. M. A. Azzam, N. M. Bashara, J. Opt. Soc. Am. 62, 36 (1972); Opt. Commun. 5, 5 (1972).

R. M. A. Azzam, N. M. Bashara, J. Opt. Soc. Am. 62, 222 (1972).
[CrossRef]

Optics Commun.

R. M. A. Azzam, N. M. Bashara, Optics Commun. 4, 203 (1971).
[CrossRef]

Proc. IRE

V. H. Rumsey, Proc. IRE 39, 535 (1951).
[CrossRef]

Other

G. A. Deschamps in Proc. Symp. Modern Network SynthesisJ. Fox, Ed. (Polytechnic Press, Brooklyn, 1952).

E. A. Guillemin, Mathematics of Circuit Analysis, (Massachusetts Institute of Technology Press, Cambridge, 1949).

When the orthogonal left and right circular polarizations are used as basis states, the azimuth θ and ellipticity e of the ellipse of polarization are obtained from the complex polarization variable χ [Eq. (3)] by the relations θ=12 Arg(χ) and e = (|χ| − 1)/(|χ| + 1). These lead to the equiazimuth-equiellipticity chart of Fig. 2 (left).

When there is more than one incident beam or when a single incident beam generates a number of emergent beams (e.g., in the presence of a diffraction grating), the electrical analog of the optical system becomes a multiport network. There is no loss of generality, however, when a two-port network is used to represent the relation between one incident and one emergent beam.

W. A. Shurcliff, Polarized Light (Harvard U.P., Cambridge, 1962).

Conventionally voltages are listed before currents in the input and output vectors.

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Figures (4)

Fig. 1
Fig. 1

(a) Linear optical system S; (b) linear two-port network W. For S and W the transformation of a pair of oscillating quantities between input and output is of interest.

Fig. 2
Fig. 2

The two orthogonal families of straight lines (—) and concentric circles (—) through and around the origin in the polarization–impedance (χZ) plane (left) represent polarization states of equiazimuth and equiellipticity and impedances of constant angle and constant magnitude, respectively. These are mapped by the optical system S and the two-port netowrk W into the orthogonal families of circles passing through and enclosing the two points A′ and B′ (right). A′ and B′ in the (χ′–Z′) plane are the images of the origin (A) and the point at infinity (B) of the (χZ) plane.

Fig. 3
Fig. 3

Two-port electric networks with impedance-mapping properties identical to the polarization-mapping properties of a given optical system. The parameters of the networks (a) and (b) are given by Eqs. (25) and (27), respectively.

Fig. 4
Fig. 4

The circuit analogs of some of the simple optical devices: (a) ideal polarizer, (b) imperfect polarizer with leakage |Z2| ≪ |Z0|, (c) purely dichroic plate, and (d) optical rotator.

Equations (32)

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( E 1 E 2 ) = ( T 11 T 12 T 21 T 22 ) ( E 1 E 2 ) ,
( E x E y ) = ( T 11 x y T 12 x y T 21 x y T 22 x y ) ( E x E y ) ,
χ = E 2 / E 1 ,             χ = E 2 / E 1 .
χ = ( T 22 χ + T 21 ) / ( T 12 χ + T 11 ) ,
( I V ) = ( W 11 W 12 W 21 W 22 ) ( I V ) ,
Z = V / I ,             Z = V / I ,
Z = ( W 22 Z + W 21 ) / ( W 12 Z + W 11 ) ,
Y = ( W 11 Y + W 12 ) / ( W 21 Y + W 22 ) .
( η - η 1 ) / ( η - η 2 ) = [ ( η 3 - η 1 ) ( ζ 3 - ζ 2 ) / ( η 3 - η 2 ) ( ζ 3 - ζ 1 ) ] × [ ( ζ - ζ 1 ) / ( ζ - ζ 2 ) ] .
χ e 1 , 2 = ( 1 / 2 T 12 ) × { ( T 22 - T 11 ) ± [ ( T 22 - T 11 ) 2 + 4 T 12 T 21 ] 1 2 } .
Z i 1 , 2 = ( 1 / 2 W 12 ) × { ( W 22 - W 11 ) ± [ ( W 22 - W 11 ) 2 + 4 W 12 W 21 ] 1 2 } .
η = ( A ζ + B ) / ( C ζ + D ) ,
η = ζ ,
( A ζ + B ) / ( C ζ + D ) = ζ .
Arg ( η ) = Arg ( ζ ) ,
Arg [ ( A ζ + B ) / ( C ζ + D ) ] = Arg ( ζ ) .
( x 2 + y 2 ) Q 1 - Q 2 = 0 ,
x Q 3 - y Q 4 = 0 ,
ζ = x + j y ,
Q 1 = ( C · C ) ( x 2 + y 2 ) + ( C · D ) x + 2 ( C × D ) y + ( D · D ) , Q 2 = ( A · A ) ( x 2 + y 2 ) + 2 ( A · B ) x + 2 ( A × B ) y + ( B · B ) , Q 3 = ( C × A ) ( x 2 + y 2 ) - ( B × C - D × A ) x - ( B · C - D · A ) y + ( D × B ) , Q 4 = ( C · A ) ( x 2 + y 2 ) + ( B · C + D · A ) x - ( B × C + D × A ) y + ( D · B ) .
T = T N T 2 T 1 ,
Z = ( T 22 Z + T 21 ) / ( T 12 Z + T 11 ) ,
Y = ( T 22 Y + T 21 ) / ( T 12 Y + T 11 ) .
Z = Z 3 + [ Z 2 2 / ( Z + Z 1 ) ] ,
Z 1 = T 11 / T 12 ,             Z 2 = ( det T ) 1 2 / T 12 ,             Z 3 = T 22 / T 12 ,
det T = T 11 T 22 - T 12 T 21 .
Z 1 = ( T 11 / T 12 ) - Z 2 ,             Z 2 = ( det T ) 1 2 / T 12 ,             Z 3 = ( T 22 / T 12 ) - Z 2 .
χ = χ 0 = T 22 / T 12 = T 21 / T 11 ,
χ = K χ ,
T r = ( cos α sin α - sin α cos α ) ,
χ = ( χ cos α - sin α ) / ( χ sin α + cos α ) .
Z 1 = - tan ( α / 2 ) ,             Z 2 = csc α ,             Z 3 = - tan ( α / 2 ) ,

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