Abstract

We extend the theory of photoresistive heterodyne detectors to include the cases where two electromagnetic fields propagate in different directions and detector dimensions include many wavelengths of the heterodyned signal. Our theory shows that the amplitude of the time-varying component of photoconductivity has a [(sinA)/A (sinB)/B] dependence. The argument of each sampling function depends upon the difference in propagation constants of the two waves in the transverse and longitudinal directions, respectively. The significance of these results to the experimenter is discussed.

© 1971 Optical Society of America

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References

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  1. A. E. Siegman, Proc. IEEE 54, 1350 (1966).
    [CrossRef]
  2. G. A. Morton, RCA Rev. 26, 8(1965).
  3. Z. Shamir, R. Fox, S. G. Lipson, Appl. Opt. 8, 1 (1969).
    [CrossRef]
  4. N. H. Frank, Introduction to Electricity and Optics (McGraw-Hill, New York, 1950), p. 91.

1969 (1)

1966 (1)

A. E. Siegman, Proc. IEEE 54, 1350 (1966).
[CrossRef]

1965 (1)

G. A. Morton, RCA Rev. 26, 8(1965).

Fox, R.

Frank, N. H.

N. H. Frank, Introduction to Electricity and Optics (McGraw-Hill, New York, 1950), p. 91.

Lipson, S. G.

Morton, G. A.

G. A. Morton, RCA Rev. 26, 8(1965).

Shamir, Z.

Siegman, A. E.

A. E. Siegman, Proc. IEEE 54, 1350 (1966).
[CrossRef]

Appl. Opt. (1)

Proc. IEEE (1)

A. E. Siegman, Proc. IEEE 54, 1350 (1966).
[CrossRef]

RCA Rev. (1)

G. A. Morton, RCA Rev. 26, 8(1965).

Other (1)

N. H. Frank, Introduction to Electricity and Optics (McGraw-Hill, New York, 1950), p. 91.

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Figures (2)

Fig. 1
Fig. 1

Two laser beams impinging on the detector of length Y0 and depth Z0 at angles θ1 and θ2, respectively. Their associated vector quantities are indicated.

Fig. 2
Fig. 2

Arbitrary and ideal orientations for detector. For maximum fluctuations in resistivity, the detector should be oriented along the wavefront and be as narrow as possible relative to the conductivity wavelength.

Equations (29)

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G = a I α ,
E 1 = a x E 01 cos ( β 1 y Y - β 1 z Z + ω 1 t ) ,
E 2 = a x E 02 cos ( β 2 y Y - β 2 z Z + ω 2 t ) ,
β 1 y = β 1 sin θ 1 ,             β 2 y = β 2 sin θ 2 ,             ( 1 a ) ,
β 1 z = β 1 cos θ 1 ,             β 2 z = β 2 cos θ 2 ,             ( 1 b ) , β 1 = ω 1 / c ,             β 2 = ω 2 / c ;
E 1 = a x E 01 cos a ,             E 2 = a y E 02 cos b ,
a = β 1 y Y - β 1 z Z + ω 1 t ,             b = β 2 y Y - β 2 z Z + ω 2 t .
E 1 + E 2 = a x [ ( E 01 - E 02 ) cos a + E 02 ( cos a + cos b ) ] .
E 2 ( Y , Z , t ) = E 02 2 ( cos a + cos b ) 2 = E 0 2 [ 1 + cos ( a - b ) + cos ( a + b ) + 1 2 ( cos 2 b + cos 2 a ) ] ,
E 2 ( Y , Z , t ) = E 0 2 [ 1 + cos ( a - b ) ] , a - b = ( β 1 y - β 2 y ) Y - ( β 1 z - β 2 z ) Z + ( ω 1 - ω 2 ) t , a - b = K 1 Y + K 2 Z + K 3 t ,
E 2 ( Y , Z , t ) = E 0 2 [ 1 + cos ( K 1 Y + K 2 Z + K 3 t ) ] .
σ ( Y , Z , t ) = σ 0 + A 0 E 2 ( Y , Z , t ) ,
σ ( Y , Z , t ) = σ 0 + A 0 E 0 2 [ 1 + cos ( K 1 Y + K 2 Z + K 3 t ) ] .
R X = [ 1 σ ( Y , Z , t ) d Y d Z ] d X , 0 Y 0 σ ( Y , Z , t ) d Y = ( σ 0 + A 0 E 0 2 ) Y 0 + A 0 E 0 2 × { sin [ K 1 Y 0 + ( K 2 Z + K 3 t ) ] - sin ( K 2 Z + K 3 t ) K 1 } = [ ( σ 0 + A 0 E 0 2 ) Y 0 ] + A 0 E 0 2 Y 0 sin K 1 Y 0 2 K 1 Y 0 2 sin ( K 2 Z + K 3 t + β ) .
0 Z 0 ( 0 Y 0 σ d Y ) d Z = [ ( σ 0 + A 0 E 0 2 ) Y 0 Z 0 ] + A 0 E 0 2 Y 0 sin [ ( K 1 Y 0 ) / 2 ] K 1 Y 0 / 2 × { - cos [ K 2 Z 0 + ( K 3 t + ϕ ) ] + cos ( K 3 t + β ) } K 2 = [ ( σ 0 + A 0 E 0 2 ) Y 0 Z 0 ] + A 0 E 0 2 Y 0 Z 0 × sin [ ( K 1 Y 0 ) / 2 ] sin [ ( K 2 Z 0 ) / 2 ] ( K 1 Y 0 ) / 2 ( K 2 Z 0 ) / 2 sin ( K 3 t + β - ψ ) .
R x = 0 X 0 1 σ 0 Y 0 Z 0 + A 0 E 0 2 Y 0 Z 0 { 1 + [ sin ( K 1 Y 0 ) / 2 ] [ sin ( K 2 Z 0 ) / 2 ] [ ( K 1 Y 0 ) / 2 ] [ ( K 2 Z 0 ) / 2 ] } sin ( K 3 t + β - ψ ) d x , R x = 1 σ 0 Y 0 Z 0 X 0 + A 0 E 0 2 Y 0 Z 0 K 0 1 + sin U y sin U z U y · U z sin ( K 3 t + δ ) , R x = 1 1 R 0 + 1 R 1 { 1 + sin U y sin U z U y · U z sin [ ( ω 1 - ω 2 ) t + δ ] } ,
U y = ( K 1 Y 0 ) / 2 ,             U z = ( K 2 Z 0 ) / 2.
R p = R 1 [ 1 + ( sin U y ) U y ( sin U z ) U z sin ( ω 1 - ω 2 ) t ] - 1
R p = R 1 [ 1 + m sin ( ω 1 - ω 2 ) t ] - 1 ,
R x = R 0 R p R 0 + R p = R 0 R 1 R 0 + R 1 [ 1 - R 0 m R 1 + R 0 sin ( ω 1 - ω 2 ) t + ( R 0 m R 1 + R 0 ) 2 sin 2 ( ω 1 - ω 2 ) t ] .
R x = R 0 R 1 R 0 + R 1 [ 1 - R 0 R 1 + R 0 sin U y U y sin U z U z sin ( ω 1 - ω 2 ) t ] .
π 2 > U y = K 1 Y 0 2 = Y 0 2 ( β 1 y - β 2 y ) = Y 0 2 ( ω 1 c sin θ 1 - ω 2 c sin θ 2 ) , π 2 > Y 0 π λ 2 [ Δ f f 2 sin θ 1 + ( sin θ 2 - sin θ 1 ) ] ,
1 > Y 0 λ / 2 [ Δ f f sin θ 1 + ( sin θ 2 - sin θ 1 ) ] .
1 > [ Y 0 / ( λ / 2 ) ] Δ θ cos θ ,
π 2 > U Z = K 2 Z 0 2 = Z 0 2 ( β 1 z - β 2 z ) = Z 0 2 ( ω 1 c cos θ 1 - ω 2 c cos θ 2 ) , π 2 > Z 0 π λ [ Δ f f cos θ 1 + ( cos θ 1 - cos θ 2 ) ] , 1 > Z 0 λ / 2 [ Δ f f cos θ 1 + ( cos θ 1 - cos θ 2 ) ] .
cos θ 1 - cos θ 2 Δ θ sin θ ,
1 > [ Z 0 / ( λ / 2 ) ] ( sin θ ) Δ θ .
1 > 10 2 Δ θ sin θ .
1 > ( Z 0 / Y 0 ) tan θ .

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