Abstract

The fabrication and characterization of an optically addressable deformable mirror for a spatial light modulator are described. Device operation utilizes an electrostatically driven pixelated aluminized polymeric membrane mirror supported above an optically controlled photoconductive GaAs substrate. A 5  μm thick grid of patterned photoresist supports the 2  μm thick aluminized Mylar membrane. A conductive ZnO layer is placed on the backside of the GaAs wafer. Similar devices were also fabricated with InP. A standard Michelson interferometer is used to measure mirror deformation data as a function of illumination, applied voltage, and frequency. The device operates as an impedance distribution between two cascaded impedances of deformable membrane substrate, substrate, and electrode. An analysis of device's operation under several bias conditions, which relates membrane deformation to operating parameters, is presented.

© 2006 Optical Society of America

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  1. C. Warde, J. McCann, J. M. V. Shrauger, H. Ieong, A. Ersen, X.-Y. Wang, and J. Hubbard, "Membrane mirror light modulator technology," in Diffractive/Holographic Technologies and Spatial Light Modulators VII, I.V.Cindrich, S.H.Lee, and R.L.Sutherland, eds., Proc. SPIE 3951,191-199 (2000).
  2. C. Warde, J. E. Hubbard, Jr., G. Genetti, L. Lerman, and W. Loizides, "Membrane mirror light valve for high-definition projection display," in Electroluminescent Materials, Devices, and Large-Screen Displays, E.M.Conwell and M.R.Miller, eds., Proc. SPIE 1910,270-279 (1993).
  3. C. Warde and J. E. Hubbard, Jr., "Membrane-mirror-light-valve-based infrared scene projector," in Characterization and Propagation of Sources and Backgrounds, W.R.Watkins and D.Clement, eds., Proc. SPIE 2223,544-557 (1994).
  4. L. E. Somers, "The photoemitter membrane light modulator image transducer," in Advances in Electronics and Electron Physics, J. D. McGee, ed. (Academic, 1972), Vol. 33A, p. 493.
    [CrossRef]
  5. F. Reizman, "An optical spatial phase modulator array activated by optical signals," in Proceedings of the 1969 Electro-Optical Systems Design Conference (Institute of Electrical and Electronics Engineers, 1969), pp. 225-230.
  6. A. D. Fisher, L. C. Ling, J. N. Lee, and R. C. Fukuda, "Photoemitter membrane light modulator," Opt. Eng. 25, 261-268 (1986).
  7. D. R. Pape, "Optically addressed membrane spatial light modulator," Opt. Eng. 24, 107-110 (1985).
  8. K. Preston, Jr., "An array optical spatial phase modulator," in Proceedings of the IEEE International Solid State Circuits Conference (Institute of Electrical and Electronics Engineers, 1968), p. 100.
  9. D. R. Pape and L. J. Hornbeck, "Characteristics of the deformable mirror device for optical information processing," Opt. Eng. 22, 675-681 (1983).
  10. L. J. Hornbeck, Deformable-Mirror Spatial Light Modulators, Vol. 1150 of SPIE Critical Reviews (SPIE, 1989), pp. 86-102.
  11. N. Clark, "A silicon eye using MEMS micromirrors," MRS Bull. 26,320-324 (2001).
    [CrossRef]
  12. C. Quan, S. H. Wang, C. J. Tay, A. Q. Liu, and H. M. Shang, "Deformation measurement of a micro-rf capacitive switch membrane using laser interferometry," Opt. Eng. 42, 92-97 (2003).
    [CrossRef]

2003 (1)

C. Quan, S. H. Wang, C. J. Tay, A. Q. Liu, and H. M. Shang, "Deformation measurement of a micro-rf capacitive switch membrane using laser interferometry," Opt. Eng. 42, 92-97 (2003).
[CrossRef]

2001 (1)

N. Clark, "A silicon eye using MEMS micromirrors," MRS Bull. 26,320-324 (2001).
[CrossRef]

1986 (1)

A. D. Fisher, L. C. Ling, J. N. Lee, and R. C. Fukuda, "Photoemitter membrane light modulator," Opt. Eng. 25, 261-268 (1986).

1985 (1)

D. R. Pape, "Optically addressed membrane spatial light modulator," Opt. Eng. 24, 107-110 (1985).

1983 (1)

D. R. Pape and L. J. Hornbeck, "Characteristics of the deformable mirror device for optical information processing," Opt. Eng. 22, 675-681 (1983).

Clark, N.

N. Clark, "A silicon eye using MEMS micromirrors," MRS Bull. 26,320-324 (2001).
[CrossRef]

Ersen, A.

C. Warde, J. McCann, J. M. V. Shrauger, H. Ieong, A. Ersen, X.-Y. Wang, and J. Hubbard, "Membrane mirror light modulator technology," in Diffractive/Holographic Technologies and Spatial Light Modulators VII, I.V.Cindrich, S.H.Lee, and R.L.Sutherland, eds., Proc. SPIE 3951,191-199 (2000).

Fisher, A. D.

A. D. Fisher, L. C. Ling, J. N. Lee, and R. C. Fukuda, "Photoemitter membrane light modulator," Opt. Eng. 25, 261-268 (1986).

Fukuda, R. C.

A. D. Fisher, L. C. Ling, J. N. Lee, and R. C. Fukuda, "Photoemitter membrane light modulator," Opt. Eng. 25, 261-268 (1986).

Genetti, G.

C. Warde, J. E. Hubbard, Jr., G. Genetti, L. Lerman, and W. Loizides, "Membrane mirror light valve for high-definition projection display," in Electroluminescent Materials, Devices, and Large-Screen Displays, E.M.Conwell and M.R.Miller, eds., Proc. SPIE 1910,270-279 (1993).

Hornbeck, L. J.

D. R. Pape and L. J. Hornbeck, "Characteristics of the deformable mirror device for optical information processing," Opt. Eng. 22, 675-681 (1983).

L. J. Hornbeck, Deformable-Mirror Spatial Light Modulators, Vol. 1150 of SPIE Critical Reviews (SPIE, 1989), pp. 86-102.

Hubbard, J.

C. Warde, J. McCann, J. M. V. Shrauger, H. Ieong, A. Ersen, X.-Y. Wang, and J. Hubbard, "Membrane mirror light modulator technology," in Diffractive/Holographic Technologies and Spatial Light Modulators VII, I.V.Cindrich, S.H.Lee, and R.L.Sutherland, eds., Proc. SPIE 3951,191-199 (2000).

Hubbard, J. E.

C. Warde and J. E. Hubbard, Jr., "Membrane-mirror-light-valve-based infrared scene projector," in Characterization and Propagation of Sources and Backgrounds, W.R.Watkins and D.Clement, eds., Proc. SPIE 2223,544-557 (1994).

C. Warde, J. E. Hubbard, Jr., G. Genetti, L. Lerman, and W. Loizides, "Membrane mirror light valve for high-definition projection display," in Electroluminescent Materials, Devices, and Large-Screen Displays, E.M.Conwell and M.R.Miller, eds., Proc. SPIE 1910,270-279 (1993).

Ieong, H.

C. Warde, J. McCann, J. M. V. Shrauger, H. Ieong, A. Ersen, X.-Y. Wang, and J. Hubbard, "Membrane mirror light modulator technology," in Diffractive/Holographic Technologies and Spatial Light Modulators VII, I.V.Cindrich, S.H.Lee, and R.L.Sutherland, eds., Proc. SPIE 3951,191-199 (2000).

Lee, J. N.

A. D. Fisher, L. C. Ling, J. N. Lee, and R. C. Fukuda, "Photoemitter membrane light modulator," Opt. Eng. 25, 261-268 (1986).

Lerman, L.

C. Warde, J. E. Hubbard, Jr., G. Genetti, L. Lerman, and W. Loizides, "Membrane mirror light valve for high-definition projection display," in Electroluminescent Materials, Devices, and Large-Screen Displays, E.M.Conwell and M.R.Miller, eds., Proc. SPIE 1910,270-279 (1993).

Ling, L. C.

A. D. Fisher, L. C. Ling, J. N. Lee, and R. C. Fukuda, "Photoemitter membrane light modulator," Opt. Eng. 25, 261-268 (1986).

Liu, A. Q.

C. Quan, S. H. Wang, C. J. Tay, A. Q. Liu, and H. M. Shang, "Deformation measurement of a micro-rf capacitive switch membrane using laser interferometry," Opt. Eng. 42, 92-97 (2003).
[CrossRef]

Loizides, W.

C. Warde, J. E. Hubbard, Jr., G. Genetti, L. Lerman, and W. Loizides, "Membrane mirror light valve for high-definition projection display," in Electroluminescent Materials, Devices, and Large-Screen Displays, E.M.Conwell and M.R.Miller, eds., Proc. SPIE 1910,270-279 (1993).

McCann, J.

C. Warde, J. McCann, J. M. V. Shrauger, H. Ieong, A. Ersen, X.-Y. Wang, and J. Hubbard, "Membrane mirror light modulator technology," in Diffractive/Holographic Technologies and Spatial Light Modulators VII, I.V.Cindrich, S.H.Lee, and R.L.Sutherland, eds., Proc. SPIE 3951,191-199 (2000).

Pape, D. R.

D. R. Pape, "Optically addressed membrane spatial light modulator," Opt. Eng. 24, 107-110 (1985).

D. R. Pape and L. J. Hornbeck, "Characteristics of the deformable mirror device for optical information processing," Opt. Eng. 22, 675-681 (1983).

Preston, K.

K. Preston, Jr., "An array optical spatial phase modulator," in Proceedings of the IEEE International Solid State Circuits Conference (Institute of Electrical and Electronics Engineers, 1968), p. 100.

Quan, C.

C. Quan, S. H. Wang, C. J. Tay, A. Q. Liu, and H. M. Shang, "Deformation measurement of a micro-rf capacitive switch membrane using laser interferometry," Opt. Eng. 42, 92-97 (2003).
[CrossRef]

Reizman, F.

F. Reizman, "An optical spatial phase modulator array activated by optical signals," in Proceedings of the 1969 Electro-Optical Systems Design Conference (Institute of Electrical and Electronics Engineers, 1969), pp. 225-230.

Shang, H. M.

C. Quan, S. H. Wang, C. J. Tay, A. Q. Liu, and H. M. Shang, "Deformation measurement of a micro-rf capacitive switch membrane using laser interferometry," Opt. Eng. 42, 92-97 (2003).
[CrossRef]

Shrauger, J. M. V.

C. Warde, J. McCann, J. M. V. Shrauger, H. Ieong, A. Ersen, X.-Y. Wang, and J. Hubbard, "Membrane mirror light modulator technology," in Diffractive/Holographic Technologies and Spatial Light Modulators VII, I.V.Cindrich, S.H.Lee, and R.L.Sutherland, eds., Proc. SPIE 3951,191-199 (2000).

Somers, L. E.

L. E. Somers, "The photoemitter membrane light modulator image transducer," in Advances in Electronics and Electron Physics, J. D. McGee, ed. (Academic, 1972), Vol. 33A, p. 493.
[CrossRef]

Tay, C. J.

C. Quan, S. H. Wang, C. J. Tay, A. Q. Liu, and H. M. Shang, "Deformation measurement of a micro-rf capacitive switch membrane using laser interferometry," Opt. Eng. 42, 92-97 (2003).
[CrossRef]

Wang, S. H.

C. Quan, S. H. Wang, C. J. Tay, A. Q. Liu, and H. M. Shang, "Deformation measurement of a micro-rf capacitive switch membrane using laser interferometry," Opt. Eng. 42, 92-97 (2003).
[CrossRef]

Wang, X.-Y.

C. Warde, J. McCann, J. M. V. Shrauger, H. Ieong, A. Ersen, X.-Y. Wang, and J. Hubbard, "Membrane mirror light modulator technology," in Diffractive/Holographic Technologies and Spatial Light Modulators VII, I.V.Cindrich, S.H.Lee, and R.L.Sutherland, eds., Proc. SPIE 3951,191-199 (2000).

Warde, C.

C. Warde, J. McCann, J. M. V. Shrauger, H. Ieong, A. Ersen, X.-Y. Wang, and J. Hubbard, "Membrane mirror light modulator technology," in Diffractive/Holographic Technologies and Spatial Light Modulators VII, I.V.Cindrich, S.H.Lee, and R.L.Sutherland, eds., Proc. SPIE 3951,191-199 (2000).

C. Warde and J. E. Hubbard, Jr., "Membrane-mirror-light-valve-based infrared scene projector," in Characterization and Propagation of Sources and Backgrounds, W.R.Watkins and D.Clement, eds., Proc. SPIE 2223,544-557 (1994).

C. Warde, J. E. Hubbard, Jr., G. Genetti, L. Lerman, and W. Loizides, "Membrane mirror light valve for high-definition projection display," in Electroluminescent Materials, Devices, and Large-Screen Displays, E.M.Conwell and M.R.Miller, eds., Proc. SPIE 1910,270-279 (1993).

MRS Bull. (1)

N. Clark, "A silicon eye using MEMS micromirrors," MRS Bull. 26,320-324 (2001).
[CrossRef]

Opt. Eng. (4)

C. Quan, S. H. Wang, C. J. Tay, A. Q. Liu, and H. M. Shang, "Deformation measurement of a micro-rf capacitive switch membrane using laser interferometry," Opt. Eng. 42, 92-97 (2003).
[CrossRef]

A. D. Fisher, L. C. Ling, J. N. Lee, and R. C. Fukuda, "Photoemitter membrane light modulator," Opt. Eng. 25, 261-268 (1986).

D. R. Pape, "Optically addressed membrane spatial light modulator," Opt. Eng. 24, 107-110 (1985).

D. R. Pape and L. J. Hornbeck, "Characteristics of the deformable mirror device for optical information processing," Opt. Eng. 22, 675-681 (1983).

Other (7)

L. J. Hornbeck, Deformable-Mirror Spatial Light Modulators, Vol. 1150 of SPIE Critical Reviews (SPIE, 1989), pp. 86-102.

K. Preston, Jr., "An array optical spatial phase modulator," in Proceedings of the IEEE International Solid State Circuits Conference (Institute of Electrical and Electronics Engineers, 1968), p. 100.

C. Warde, J. McCann, J. M. V. Shrauger, H. Ieong, A. Ersen, X.-Y. Wang, and J. Hubbard, "Membrane mirror light modulator technology," in Diffractive/Holographic Technologies and Spatial Light Modulators VII, I.V.Cindrich, S.H.Lee, and R.L.Sutherland, eds., Proc. SPIE 3951,191-199 (2000).

C. Warde, J. E. Hubbard, Jr., G. Genetti, L. Lerman, and W. Loizides, "Membrane mirror light valve for high-definition projection display," in Electroluminescent Materials, Devices, and Large-Screen Displays, E.M.Conwell and M.R.Miller, eds., Proc. SPIE 1910,270-279 (1993).

C. Warde and J. E. Hubbard, Jr., "Membrane-mirror-light-valve-based infrared scene projector," in Characterization and Propagation of Sources and Backgrounds, W.R.Watkins and D.Clement, eds., Proc. SPIE 2223,544-557 (1994).

L. E. Somers, "The photoemitter membrane light modulator image transducer," in Advances in Electronics and Electron Physics, J. D. McGee, ed. (Academic, 1972), Vol. 33A, p. 493.
[CrossRef]

F. Reizman, "An optical spatial phase modulator array activated by optical signals," in Proceedings of the 1969 Electro-Optical Systems Design Conference (Institute of Electrical and Electronics Engineers, 1969), pp. 225-230.

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Figures (9)

Fig. 1
Fig. 1

Photoconductive optically controlled spatial light modulator.

Fig. 2
Fig. 2

Device parameters for a single pixel of the MEMS device.

Fig. 3
Fig. 3

(Color online) MEMS device driven by ac voltage.

Fig. 4
Fig. 4

(Color online) A, transmittivity and reflectivity of InP versus wavelength; A′, absorption coefficient α versus wavelength for InP; B, B′, transmission of light versus wavelength for GaAs; C, absorption coefficient α versus wavelength for GaAs; C′, inverse of absorption coefficient 1∕α versus wavelength for GaAs.

Fig. 5
Fig. 5

(Color online) Experimental arrangement.

Fig. 6
Fig. 6

(Color online) Results for the 2 mm × 2 mm sample: A, displacement versus frequency for constant dc and different ac; B, displacement versus frequency for constant ac and different dc; C, displacement versus ac + dc; D, displacement versus intensity of light.

Fig. 7
Fig. 7

(Color online) Punch-hole results: A, displacement versus ac + dc; B, displacement versus frequency for constant dc and different ac; C, displacement versus frequency for constant ac and different dc.

Fig. 8
Fig. 8

Fringes of the 2 mm × 2 mm sample focused on 1 pixel by a lens and before the following kinds of light illumination on the back: A, flashlight; A′, diode laser; A″, HeNe laser. The same fringes after B, flashlight; B′, diode laser; B″, HeNe laser. C–C″, the differences.

Fig. 9
Fig. 9

Fringes of the punch-hole sample focused on the center by a lens and before the following kinds of light illumination on the back: A, flashlight; A′, diode laser; A″, HeNe laser. The same fringes after B, flashlight; B, diode laser; B″, HeNe laser. C–C″, the differences.

Equations (49)

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c 1 π ε 0 r 1     2 s + π ε 0 r 1     2 h 2 s 2 ,
h = ε 0 r 1     2 V 2 32 T s 2 ,
V 1 = V Z 1 Z 1 + Z 2 ,
Z 2 = L q μ n ( n + Δ n ) A ,
V 1 = V 1 + [ L j ω c 1 / q μ n ( n + Δ n ) A ] .
| V 1 | 2        q 2 μ n 2 ( n + Δ n ) 2 A 2 V 2 q 2 μ n 2 ( n + Δ n ) 2 A 2 + L 2 ω 2 ( π ε 0 r 1     2 / s ) 2 .
Δ h = ε 0 3 r 1     6 q 2 μ n 2 A 2 V 2 L 2 ω 2 π 2 ( 2 n Δ n + Δ n 2 ) [ s 2 q 2 μ n 2 ( n + Δ n ) 2 A 2 + L 2 ω 2 π 2 ε 0 2 r 1     4 ] ( s 2 q 2 μ n 2 n 2 A 2 + L 2 ω 2 π 2 ε 0 2 r 1     4 ) .
y = h r 1     2 x 2 + s h .
c = 0 r 1 ε 0 2 π r d r ( h r 2 / r 1     2 ) + s h
=  2 π ε 0 r 1     2 0 r 1 r d r h r 2 + ( s h ) r 1     2
=  2 π ε 0 r 1     2 0 r 1 ( 1 / 2 ) d ( r 2 ) h r 2 + ( s h ) r 1 2
=  π ε 0 r 1     2 h 0 r 1 d ( h r 2 ) h r 2 + ( s h ) r 1     2
=  π ε 0 r 1     2 h 0 r 1 d [ h r 2 + ( s h ) r 1     2 ] h r 2 + ( s h ) r 1     2
=  π ε 0 r 1     2 h ln [ hr 2 + ( s h ) r 1     2 ] | 0 r 1
=  π ε 0 r 1     2 h ln [ h r 2 + ( s h ) r 1     2 ( s h ) r 1     2 ]
⇒  c = π ε 0 r 1     2 h ln ( s s h ) .
c = π ε 0 r 1     2 ln ( s s h ) 1 / h = π ε 0 r 1     2 s ln ( s s h ) s / h = π ε 0 r 1     2 s ln [ 1 1 ( h / s ) ] s / h π ε 0 r 1     2 s ln ( 1 + h s ) s / h .
lim x ( 1 + 1 x ) x = e ,
c = π ε 0 r 1 2 s ln ( 1 + 1 s / h ) s / h = π ε 0 r 1 2 s ln ( e ) c = π ε 0 r 1 2 s
c = π ε 0 r 1 2 h ln ( s s h ) = π ε 0 r 1 2 h ln [ 1 1 ( h / s ) ] = π ε 0 r 1 2 h ( h s + h 2 2 s 2 ) ,
c π ε 0 r 1 2 s + π ε 0 r 1 2 h 2 s 2 ,
h = ε 0 r 1 2 V 2 8 T s 2 ,
V 1 = V Z 1 Z 1 + Z 2 ,
Z 2 = ρ L A = L q μ n n A .
Z 2 = L q μ n ( n + Δ n ) A .
V 1 = V 1 / j ω c 1 1 j ω c 1 + L q μ n ( n + Δ n ) A = V 1 + L j ω c 1 q μ n ( n + Δ n ) A .
c 1 = π ε 0 r 1 2 s + π ε 0 r 1 2 ( ε 0 r 1 2 V 1 2 / 8 T s 2 ) 2 s 2 .
c 1 = b + a V 1 2 .
V 1 = V 1 + [ L j ω ( b + a V 1 2 ) / q μ n ( n + Δ n ) A ] = V m m + L j ω ( b + a V 1 2 ) ,
| V 1 | 2 =  V 2 m 2 m 2 + L 2 ω 2 ( b + a V 1 2 ) 2
⇒  m 2 | V 1 | 2 + L 2 ω 2 ( b + a V 1 2 ) 2 | V 1 | 2 = V 2 m 2
⇒  m 2 | V 1 | 2 + L 2 ω 2 ( a 2 | V 1 | 4 + b 2 + 2 a b | V 1 | 2 ) × | V 1 | 2 =  V 2 m 2
⇒  m 2 | V 1 | 2 + L 2 ω 2 a 2 | V 1 | 6 + L 2 ω 2 b 2 | V 1 | 2 + 2 L 2 ω 2 a b | V 1 | 4 = V 2 m 2
⇒  L 2 ω 2 a 2 | V 1 | 6 + 2 L 2 ω 2 a b | V 1 | 4 + ( m 2 + L 2 ω 2 b 2 ) | V 1 | 2 = V 2 m 2 ,
( m 2 + L 2 ω 2 b 2 ) | V 1 | 2 V 2 m 2 | V 1 | 2 V 2 m 2 ( m 2 + L 2 ω 2 b 2 )
q 2 μ n 2 ( n + Δ n ) 2 A 2 V 2 q 2 μ n 2 ( n + Δ n ) 2 A 2 + L 2 ω 2 ( π ε 0 r 1 2 / s ) 2 .
h = ε 0 r 1 2 q 2 μ n 2 ( n + Δ n ) 2 A 2 V 2 8 T s 2 [ q 2 μ n 2 ( n + Δ n ) 2 A 2 + L 2 ω 2 ( π ε 0 r 1 2 / s ) 2 ] = ε 0 r 1 2 q 2 μ n 2 ( n + Δ n ) 2 A 2 V 2 8 T s 2 q 2 μ n 2 ( n + Δ n ) 2 A 2 + 8 T L 2 ω 2 π 2 ε 0 2 r 1 4 ;
h = ε 0 r 1 2 q 2 μ n 2 n 2 A 2 V 2 8 T s 2 q 2 μ n 2 n 2 A 2 + 8 T L 2 ω 2 π 2 ε 0 2 r 1 4 ,
Δ h = h h = ε 0 r 1 2 q 2 μ n 2 ( n + Δ n ) 2 A 2 V 2 8 T s 2 q 2 μ n 2 ( n + Δ n ) 2 A 2 + 8 T L 2 ω 2 π 2 ε 0 2 r 1 4 ε 0 r 1 2 q 2 μ n 2 n 2 A 2 V 2 8 T s 2 q 2 μ n 2 n 2 A 2 + 8 T L 2 ω 2 π 2 ε 0 2 r 1 4
= 8 ε 0 3 r 1 6 q 2 μ n 2 A 2 V 2 T L 2 ω 2 π 2 ( 2 n Δ n + Δ n 2 ) [ 8 T s 2 q 2 μ n 2 ( n + Δ n ) 2 A 2 + 8 T L 2 ω 2 π 2 ε 0 2 r 1 4 ] ( 8 T s 2 q 2 μ n 2 n 2 A 2 + 8 T L 2 ω 2 π 2 ε 0 2 r 1 4 )
Δ h = ε 0 3 r 1 6 q 2 μ n 2 A 2 V 2 L 2 ω 2 π 2 ( 2 n Δ n + Δ n 2 ) [ s 2 q 2 μ n 2 ( n + Δ n ) 2 A 2 + L 2 ω 2 π 2 ε 0 2 r 1 4 ] ( s 2 q 2 μ n 2 n 2 A 2 + L 2 ω 2 π 2 ε 0 2 r 1 4 ) .
Δ h = x ( a x 2 b + c x 2 ) Δ x , x n ,
Δ x = Δ n ,
x = n + Δ n ,
a = ε 0 r 1 2 q 2 μ n     2 A 2 V 2 ,
b = 8 T s π ε 0 r 1 2 L 2 ω 2 ,
c = 8 T s 2 q 2 μ n     2 A 2 ,
Δ h = 2 a n ( b + c n 2 ) 2 a c n 2 ( b + c n 2 ) 2 Δ n = 2 a b n Δ n ( b + c n 2 ) 2
Δ h = 2 ε 0 2 r 1 4 q 2 μ n     2 A 2 V 2 L 2 ω 2 π n Δ n ( s q 2 μ n     2 n 2 A 2 + L 2 ω 2 π ε 0 r 1 2 ) .

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