Abstract

The paper describes the design and performance of an electrostatically scanned film reflector. A drive scan sensitivity of better than a wavelength per volt at 6328 Å can be achieved with the use of appropriate polarization voltages. The power requirements for both polarization and drive may be made extremely low. In addition the device is capable of good optical properties together with large linear displacements. It should find wide application in interferometer and laser systems, particularly in the infrared region of the spectrum.

© 1972 Optical Society of America

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References

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  1. L. J. Vande Kieft, E. A. Murray, W. M. Frey, P. W. Yunker, Tech. Rep. 1723, Ballistic Research Laboratories, Aberdeen Proving Ground, Maryland (Oct.1969).
  2. R. Williamson, Audio, p. 17 (July1963).
  3. V. A. J. Carbauh, Electrochem. Technol. 6, 3 (1968).
  4. C. W. Reedyk, Electrochem. Technol. 6, 6 (1968).
  5. M. M. Perlman, Electrochem. Technol. 6, 95 (1968).
  6. A. E. Robertson, Microphones (Hayden, New York; Iliffe, London, 1963).
  7. N. W. McLachlan, Loud Speakers (Dover, New York, 1960).
  8. E. A. Ballik, J. Appl. Phys. (in press).

1968 (3)

V. A. J. Carbauh, Electrochem. Technol. 6, 3 (1968).

C. W. Reedyk, Electrochem. Technol. 6, 6 (1968).

M. M. Perlman, Electrochem. Technol. 6, 95 (1968).

1963 (1)

R. Williamson, Audio, p. 17 (July1963).

Ballik, E. A.

E. A. Ballik, J. Appl. Phys. (in press).

Carbauh, V. A. J.

V. A. J. Carbauh, Electrochem. Technol. 6, 3 (1968).

Frey, W. M.

L. J. Vande Kieft, E. A. Murray, W. M. Frey, P. W. Yunker, Tech. Rep. 1723, Ballistic Research Laboratories, Aberdeen Proving Ground, Maryland (Oct.1969).

McLachlan, N. W.

N. W. McLachlan, Loud Speakers (Dover, New York, 1960).

Murray, E. A.

L. J. Vande Kieft, E. A. Murray, W. M. Frey, P. W. Yunker, Tech. Rep. 1723, Ballistic Research Laboratories, Aberdeen Proving Ground, Maryland (Oct.1969).

Perlman, M. M.

M. M. Perlman, Electrochem. Technol. 6, 95 (1968).

Reedyk, C. W.

C. W. Reedyk, Electrochem. Technol. 6, 6 (1968).

Robertson, A. E.

A. E. Robertson, Microphones (Hayden, New York; Iliffe, London, 1963).

Vande Kieft, L. J.

L. J. Vande Kieft, E. A. Murray, W. M. Frey, P. W. Yunker, Tech. Rep. 1723, Ballistic Research Laboratories, Aberdeen Proving Ground, Maryland (Oct.1969).

Williamson, R.

R. Williamson, Audio, p. 17 (July1963).

Yunker, P. W.

L. J. Vande Kieft, E. A. Murray, W. M. Frey, P. W. Yunker, Tech. Rep. 1723, Ballistic Research Laboratories, Aberdeen Proving Ground, Maryland (Oct.1969).

Audio (1)

R. Williamson, Audio, p. 17 (July1963).

Electrochem. Technol. (3)

V. A. J. Carbauh, Electrochem. Technol. 6, 3 (1968).

C. W. Reedyk, Electrochem. Technol. 6, 6 (1968).

M. M. Perlman, Electrochem. Technol. 6, 95 (1968).

Other (4)

A. E. Robertson, Microphones (Hayden, New York; Iliffe, London, 1963).

N. W. McLachlan, Loud Speakers (Dover, New York, 1960).

E. A. Ballik, J. Appl. Phys. (in press).

L. J. Vande Kieft, E. A. Murray, W. M. Frey, P. W. Yunker, Tech. Rep. 1723, Ballistic Research Laboratories, Aberdeen Proving Ground, Maryland (Oct.1969).

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Figures (13)

Fig. 1
Fig. 1

Schematic representation of a simple single-element electrostatically scanned reflector.

Fig. 2
Fig. 2

Schematic representation of a push-pull electrostatically scanned reflector system.

Fig. 3
Fig. 3

General sensitivity curve for the push-pull system calculated from Eq. (28). The dashed line corresponds to the use of a linear approximation. All parameters are normalized in terms of the collapse voltage V2 C .

Fig. 4
Fig. 4

Normalized drive signal required in push-pull operation to produce a relative displacement /D. The family of curves are for normalized values of the polarization voltage.

Fig. 5
Fig. 5

Percentage nonlinearity as a function of the relative displacement /D. The family of curves are for normalized values of the polarization voltage. The dashed curves (with values in parentheses) are used to emphasize a change in the progression of the curves.

Fig. 6
Fig. 6

Figure 5 in expanded scale.

Fig. 7
Fig. 7

Cross section of the experimental prototype.

Fig. 8
Fig. 8

Experimental arrangement for the evaluation of the electrostatically scanned reflector.

Fig. 9
Fig. 9

Prototype in single-system operation. The polarity of the drive voltage V refers to the fixed element. The membrane and the other fixed element are both at ground potential.

Fig. 10
Fig. 10

Electret behavior in single-system operation. The drive voltage |V| = 500 is applied at t ≈ 5 sec and removed at t ≈ 50 sec.

Fig. 11
Fig. 11

Prototype in push-pull operation. The magnitude of the polarization voltage on each element is given by V.

Fig. 12
Fig. 12

Measured inverse sensitivity of the push-pull system as a function of the polarization voltage.

Fig. 13
Fig. 13

Parameters used in calculating the membrane displacement.

Equations (74)

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Q = C V ,
C = 8.84 × 10 - 12 A K / D F ,
F e = Q E / 2 = Q V / 2 D = C V 2 / 2 D ,
F e = 4.42 × 10 - 12 A K V 2 / D 2 .
F e V 2 / D 2 .
F e V 2 / D 2 .
( V + v ) 2 / ( D - ) 2 .
[ 1 - ( / D ) ] 2 = ( k V 2 / D 2 ) [ 1 + ( v / V ) ] 2 .
[ 1 - ( 2 / D ) ] = k V 2 / D 2 [ 1 + ( 2 v / V ) ] .
s 1 = d / d v ,
s 1 = 2 k V / D 2 .
( d / d ) { [ 1 - ( / D ) ] 2 } = 0.
1 C = D / 3.
1 C [ 1 - ( 1 C / D ) ] 2 = k V 1 C 1 / D 2 .
V 1 C 2 = 2 D 3 / 9 k .
= k { [ ( V 1 + v ) 2 / ( D 1 - ) 2 ] - [ ( V 2 - v ) 2 / ( D 2 + ) 2 ] } .
V 1 = V 2 = V ,             D 1 = D 2 = D .
= k { [ ( V - v ) 2 / ( D - ) 2 ] - [ ( V - v ) 2 / ( D + ) 2 ] } .
{ [ 1 - ( 2 / D 2 ) ] 2 - ( 4 k / D 3 ) ( V 2 + v 2 ) } = ( 4 k / D 2 ) [ 1 + ( 2 / D 2 ) ] V v .
{ [ 1 - ( 2 2 / D 2 ) ] - 4 k V v / D 4 - ( 4 k / D 3 ) ( V 2 + v 2 ) } = 4 k V v / D 2 .
[ 1 - ( 4 k / D 3 ) ( V 2 + v 2 ) ] = 4 k V v / D 2 .
s 2 = d / d v = 4 k V / D 2 [ 1 - ( 4 k V 2 / D 3 ) ] .
( d / d ) { [ 1 - ( 4 k V 2 / D 3 ) ] } = 0 ,
V 2 C 2 = D 3 / 4 k .
V 2 C 2 / V 1 C 2 = 9 / 8.
/ D = V v { V 2 C 2 [ 1 - 2 ( 2 / D 2 ) ] - V v ( / D ) - ( V 2 + v 2 ) } .
/ D = V v / [ V 2 C 2 - ( V 2 + v 2 ) ]             for / D 1.
s 2 = ( D / V 2 C ) { ( V / V 2 C ) / [ 1 - ( V 2 / V 2 C ) ] } .
V 1 C = 640 V .
V / s 2 ( 1 - γ V 2 ) ,
V 2 C 740 V ,
d = d / k .
D = 0.013 3 cm .
2.3 7 × 10 - 3 cm             ( maximum ) .
/ D = 0.18             ( maximum ) .
f 0 = 1.01 5 kHz ,             f 0 = 500 Hz ,             f 0 = 350 Hz .
f 0 = ( 1 / π R ) ( 3 T / 2 m ) 1 2 ,
T = 2.14 × 10 5 dynes / cm .
= [ ( 2.21 × 10 - 7 V 2 ) / D 2 T ] [ ( R 2 - R 1 ) 2 + ( R 2 2 - R 1 2 ) l n ( R / R 2 ) ] ,
T = 1.55 × 10 - 3 ( V 2 / ) .
T = 2.08 × 10 5 dynes / cm ,
SPL 1 = 47 dB , SPL 2 = 80 dB , SPL 3 = 80 dB . ( free air ) ( enclosed air ) ( i n v a c u o )
T x = T sin θ .
T sin θ = T tan θ = T ( d x / d r ) .
F ρ = - 2 π r T ( d x / d r ) .
F e = C V 2 / 2 ( D - ) 2 .
d F e ( r ) = F e 2 π r d r / A ,
A = π ( R 2 2 - R 1 2 ) .
F e ( r ) = ( 2 π / A ) F e R 1 r r d r = ( π / A ) ( r 2 - R 1 2 ) F e for R 1 < r < R 2 .
F ρ + F e ( r ) = 0
2 π r T ( d x / d r ) = ( π / A ) ( r 2 - R 1 2 ) F e .
x ( r ) = β R 1 τ [ r - ( R 1 2 / r ) ] d r ,
x ( r ) = β { [ ( r 2 - R 1 2 ) / 2 ] - R 1 2 ln ( r / R 1 ) } .
Δ = r - R 1 ,
x ( Δ ) = β { [ ( Δ 2 + 2 Δ R 1 ) / 2 ] - R 1 2 ln [ 1 + ( Δ / R 1 ) ] } .
ln ( 1 + z ) = z - ( z 2 / 2 ) + ( z 3 / 3 ) - ( z 4 / 4 ) +
x ( Δ ) = β Δ 2 [ 1 - ( Δ 3 R 1 ) + ( Δ 2 / 4 R 1 2 ) - ] .
x ( Δ ) β Δ 2
X = x [ ( R 2 - R 1 ) ] β ( R 2 - R 1 ) 2 .
2 π r T ( d x / d r ) = F e .
x ( r ) = β A π R 2 r d r r .
x ( r ) = ( β A / π ) ln ( r / R 2 ) .
X = x ( R ) = ( β A / π ) ln ( R / R 2 ) .
X + X = .
d F e F e 2 π r d r / A { 1 + [ x ( r ) / ( D - ) ] } 2 ,
F e ( Δ ) F e 2 π A 0 Δ ( Δ + R 1 ) d Δ { 1 + [ β Δ 2 / ( D - ) ] } 2 .
F e ( Δ ) F e 2 π A 0 Δ ( Δ + R 1 ) { 1 - 2 [ β Δ 2 / ( D - ) ] } d Δ .
F e ( Δ ) ( F e 2 π / A ) { ( Δ 2 / 2 ) + R 1 Δ - [ 2 β R 1 Δ 3 / 3 ( D - ) ] - [ β Δ 4 / 2 ( D - ) ] } .
F e F e { 1 - [ β ( R 2 - R 1 ) 2 / ( D - ) ] [ ( 3 R 2 + R 1 ) / 3 ( R 2 + R 1 ) ] } .
F e F e > F e [ 1 - X / ( D - ) ] .
R = 3.01 cm ,             R 2 = 2.06 cm ,             R 1 = 1.43 cm .
X / X = π ( R 2 - R 1 ) 2 / A ln ( R / R 2 ) 2 = ( R 2 - R 1 ) / ( R 2 + R 1 ) ln ( R / R 2 ) .
X / 3.
F e F e F e { 1 - [ / 3 ( D - ) ] }

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