Abstract

We show that by use of an array of micromechanical tunneling cantilevers we can determine the amplitude and phase profile of an incident light beam. When there is no phase jump the phase profile can be determined by the measurement of two sets of the tunneling currents that correspond to a steplike applied bias, which induces a Franz–Keldysh effect in the cantilevers. The case of phase jump is also discussed.

© 2001 Optical Society of America

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References

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  1. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).
  2. T. E. Gureyev, K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13, 1670–1682 (1996).
    [CrossRef]
  3. N. Nakajima, “Phase retrieval from Fresnel zone intensity measurements by use of Gaussian filtering,” Appl. Opt. 37, 6219–6226 (1998).
    [CrossRef]
  4. S. R. Dean, The Radon Transform and Some of Its Applications (Wiley, New York, 1993).
  5. D. Dragoman, M. Dragoman, “Optical actuation of micromechanical tunneling structures with applications in spectrum analysis and optical computing,” Appl. Opt. 38, 6773–6778 (1999).
    [CrossRef]
  6. M. A. McCord, A. Dana, R. F. W. Pease, “The micromechanical tunneling transistor,” J. Micromech. Microeng. 8, 209–212 (1998).
    [CrossRef]
  7. M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, UK, 1970), Chap. 3.
  8. P. Tayebati, L. Jauniskis, “Novel self-electro-optic device using bulk Franz–Keldysh effects in an n+-GaAlAs-GaAs-Ag asymmetric Fabry–Perot device structure,” Appl. Phys. Lett. 64, 898–900 (1994).
    [CrossRef]
  9. C. F. Klingshirn, Semiconductor Optics (Springer, Berlin, 1997), Chap. 3.
  10. O. Marti, A. Ruf, M. Hipp, H. Bielefeldt, J. Colchero, J. Mlynek, “Micromechanical and thermal effects on force microscope cantilevers,” Ultramicroscopy 42–44, 345–349 (1992).
    [CrossRef]
  11. K. Y. Yasumura, T. D. Stowe, E. M. Chow, T. Pfafman, T. W. Kenny, B. C. Stipe, D. Rugar, “Quality factors in micron- and submicron-thick cantilevers,” IEEE J. Quantum Electron. 9, 117–125 (2000).
  12. M. Grabherr, M. Miller, R. Jäger, R. Michalzik, U. Martin, H. J. Unold, K. J. Ebeling, “High-power VCSEL’s: single devices and densely packed 2D arrays,” IEEE J. Sel. Top. Quantum Electron. 5, 495–502 (1999).
    [CrossRef]

2000 (1)

K. Y. Yasumura, T. D. Stowe, E. M. Chow, T. Pfafman, T. W. Kenny, B. C. Stipe, D. Rugar, “Quality factors in micron- and submicron-thick cantilevers,” IEEE J. Quantum Electron. 9, 117–125 (2000).

1999 (2)

M. Grabherr, M. Miller, R. Jäger, R. Michalzik, U. Martin, H. J. Unold, K. J. Ebeling, “High-power VCSEL’s: single devices and densely packed 2D arrays,” IEEE J. Sel. Top. Quantum Electron. 5, 495–502 (1999).
[CrossRef]

D. Dragoman, M. Dragoman, “Optical actuation of micromechanical tunneling structures with applications in spectrum analysis and optical computing,” Appl. Opt. 38, 6773–6778 (1999).
[CrossRef]

1998 (2)

M. A. McCord, A. Dana, R. F. W. Pease, “The micromechanical tunneling transistor,” J. Micromech. Microeng. 8, 209–212 (1998).
[CrossRef]

N. Nakajima, “Phase retrieval from Fresnel zone intensity measurements by use of Gaussian filtering,” Appl. Opt. 37, 6219–6226 (1998).
[CrossRef]

1996 (1)

1994 (1)

P. Tayebati, L. Jauniskis, “Novel self-electro-optic device using bulk Franz–Keldysh effects in an n+-GaAlAs-GaAs-Ag asymmetric Fabry–Perot device structure,” Appl. Phys. Lett. 64, 898–900 (1994).
[CrossRef]

1992 (1)

O. Marti, A. Ruf, M. Hipp, H. Bielefeldt, J. Colchero, J. Mlynek, “Micromechanical and thermal effects on force microscope cantilevers,” Ultramicroscopy 42–44, 345–349 (1992).
[CrossRef]

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Bielefeldt, H.

O. Marti, A. Ruf, M. Hipp, H. Bielefeldt, J. Colchero, J. Mlynek, “Micromechanical and thermal effects on force microscope cantilevers,” Ultramicroscopy 42–44, 345–349 (1992).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, UK, 1970), Chap. 3.

Chow, E. M.

K. Y. Yasumura, T. D. Stowe, E. M. Chow, T. Pfafman, T. W. Kenny, B. C. Stipe, D. Rugar, “Quality factors in micron- and submicron-thick cantilevers,” IEEE J. Quantum Electron. 9, 117–125 (2000).

Colchero, J.

O. Marti, A. Ruf, M. Hipp, H. Bielefeldt, J. Colchero, J. Mlynek, “Micromechanical and thermal effects on force microscope cantilevers,” Ultramicroscopy 42–44, 345–349 (1992).
[CrossRef]

Dana, A.

M. A. McCord, A. Dana, R. F. W. Pease, “The micromechanical tunneling transistor,” J. Micromech. Microeng. 8, 209–212 (1998).
[CrossRef]

Dean, S. R.

S. R. Dean, The Radon Transform and Some of Its Applications (Wiley, New York, 1993).

Dragoman, D.

Dragoman, M.

Ebeling, K. J.

M. Grabherr, M. Miller, R. Jäger, R. Michalzik, U. Martin, H. J. Unold, K. J. Ebeling, “High-power VCSEL’s: single devices and densely packed 2D arrays,” IEEE J. Sel. Top. Quantum Electron. 5, 495–502 (1999).
[CrossRef]

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Grabherr, M.

M. Grabherr, M. Miller, R. Jäger, R. Michalzik, U. Martin, H. J. Unold, K. J. Ebeling, “High-power VCSEL’s: single devices and densely packed 2D arrays,” IEEE J. Sel. Top. Quantum Electron. 5, 495–502 (1999).
[CrossRef]

Gureyev, T. E.

Hipp, M.

O. Marti, A. Ruf, M. Hipp, H. Bielefeldt, J. Colchero, J. Mlynek, “Micromechanical and thermal effects on force microscope cantilevers,” Ultramicroscopy 42–44, 345–349 (1992).
[CrossRef]

Jäger, R.

M. Grabherr, M. Miller, R. Jäger, R. Michalzik, U. Martin, H. J. Unold, K. J. Ebeling, “High-power VCSEL’s: single devices and densely packed 2D arrays,” IEEE J. Sel. Top. Quantum Electron. 5, 495–502 (1999).
[CrossRef]

Jauniskis, L.

P. Tayebati, L. Jauniskis, “Novel self-electro-optic device using bulk Franz–Keldysh effects in an n+-GaAlAs-GaAs-Ag asymmetric Fabry–Perot device structure,” Appl. Phys. Lett. 64, 898–900 (1994).
[CrossRef]

Kenny, T. W.

K. Y. Yasumura, T. D. Stowe, E. M. Chow, T. Pfafman, T. W. Kenny, B. C. Stipe, D. Rugar, “Quality factors in micron- and submicron-thick cantilevers,” IEEE J. Quantum Electron. 9, 117–125 (2000).

Klingshirn, C. F.

C. F. Klingshirn, Semiconductor Optics (Springer, Berlin, 1997), Chap. 3.

Marti, O.

O. Marti, A. Ruf, M. Hipp, H. Bielefeldt, J. Colchero, J. Mlynek, “Micromechanical and thermal effects on force microscope cantilevers,” Ultramicroscopy 42–44, 345–349 (1992).
[CrossRef]

Martin, U.

M. Grabherr, M. Miller, R. Jäger, R. Michalzik, U. Martin, H. J. Unold, K. J. Ebeling, “High-power VCSEL’s: single devices and densely packed 2D arrays,” IEEE J. Sel. Top. Quantum Electron. 5, 495–502 (1999).
[CrossRef]

McCord, M. A.

M. A. McCord, A. Dana, R. F. W. Pease, “The micromechanical tunneling transistor,” J. Micromech. Microeng. 8, 209–212 (1998).
[CrossRef]

Michalzik, R.

M. Grabherr, M. Miller, R. Jäger, R. Michalzik, U. Martin, H. J. Unold, K. J. Ebeling, “High-power VCSEL’s: single devices and densely packed 2D arrays,” IEEE J. Sel. Top. Quantum Electron. 5, 495–502 (1999).
[CrossRef]

Miller, M.

M. Grabherr, M. Miller, R. Jäger, R. Michalzik, U. Martin, H. J. Unold, K. J. Ebeling, “High-power VCSEL’s: single devices and densely packed 2D arrays,” IEEE J. Sel. Top. Quantum Electron. 5, 495–502 (1999).
[CrossRef]

Mlynek, J.

O. Marti, A. Ruf, M. Hipp, H. Bielefeldt, J. Colchero, J. Mlynek, “Micromechanical and thermal effects on force microscope cantilevers,” Ultramicroscopy 42–44, 345–349 (1992).
[CrossRef]

Nakajima, N.

N. Nakajima, “Phase retrieval from Fresnel zone intensity measurements by use of Gaussian filtering,” Appl. Opt. 37, 6219–6226 (1998).
[CrossRef]

Nugent, K. A.

Pease, R. F. W.

M. A. McCord, A. Dana, R. F. W. Pease, “The micromechanical tunneling transistor,” J. Micromech. Microeng. 8, 209–212 (1998).
[CrossRef]

Pfafman, T.

K. Y. Yasumura, T. D. Stowe, E. M. Chow, T. Pfafman, T. W. Kenny, B. C. Stipe, D. Rugar, “Quality factors in micron- and submicron-thick cantilevers,” IEEE J. Quantum Electron. 9, 117–125 (2000).

Ruf, A.

O. Marti, A. Ruf, M. Hipp, H. Bielefeldt, J. Colchero, J. Mlynek, “Micromechanical and thermal effects on force microscope cantilevers,” Ultramicroscopy 42–44, 345–349 (1992).
[CrossRef]

Rugar, D.

K. Y. Yasumura, T. D. Stowe, E. M. Chow, T. Pfafman, T. W. Kenny, B. C. Stipe, D. Rugar, “Quality factors in micron- and submicron-thick cantilevers,” IEEE J. Quantum Electron. 9, 117–125 (2000).

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Stipe, B. C.

K. Y. Yasumura, T. D. Stowe, E. M. Chow, T. Pfafman, T. W. Kenny, B. C. Stipe, D. Rugar, “Quality factors in micron- and submicron-thick cantilevers,” IEEE J. Quantum Electron. 9, 117–125 (2000).

Stowe, T. D.

K. Y. Yasumura, T. D. Stowe, E. M. Chow, T. Pfafman, T. W. Kenny, B. C. Stipe, D. Rugar, “Quality factors in micron- and submicron-thick cantilevers,” IEEE J. Quantum Electron. 9, 117–125 (2000).

Tayebati, P.

P. Tayebati, L. Jauniskis, “Novel self-electro-optic device using bulk Franz–Keldysh effects in an n+-GaAlAs-GaAs-Ag asymmetric Fabry–Perot device structure,” Appl. Phys. Lett. 64, 898–900 (1994).
[CrossRef]

Unold, H. J.

M. Grabherr, M. Miller, R. Jäger, R. Michalzik, U. Martin, H. J. Unold, K. J. Ebeling, “High-power VCSEL’s: single devices and densely packed 2D arrays,” IEEE J. Sel. Top. Quantum Electron. 5, 495–502 (1999).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, UK, 1970), Chap. 3.

Yasumura, K. Y.

K. Y. Yasumura, T. D. Stowe, E. M. Chow, T. Pfafman, T. W. Kenny, B. C. Stipe, D. Rugar, “Quality factors in micron- and submicron-thick cantilevers,” IEEE J. Quantum Electron. 9, 117–125 (2000).

Appl. Opt. (1)

N. Nakajima, “Phase retrieval from Fresnel zone intensity measurements by use of Gaussian filtering,” Appl. Opt. 37, 6219–6226 (1998).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

P. Tayebati, L. Jauniskis, “Novel self-electro-optic device using bulk Franz–Keldysh effects in an n+-GaAlAs-GaAs-Ag asymmetric Fabry–Perot device structure,” Appl. Phys. Lett. 64, 898–900 (1994).
[CrossRef]

IEEE J. Quantum Electron. (1)

K. Y. Yasumura, T. D. Stowe, E. M. Chow, T. Pfafman, T. W. Kenny, B. C. Stipe, D. Rugar, “Quality factors in micron- and submicron-thick cantilevers,” IEEE J. Quantum Electron. 9, 117–125 (2000).

IEEE J. Sel. Top. Quantum Electron. (1)

M. Grabherr, M. Miller, R. Jäger, R. Michalzik, U. Martin, H. J. Unold, K. J. Ebeling, “High-power VCSEL’s: single devices and densely packed 2D arrays,” IEEE J. Sel. Top. Quantum Electron. 5, 495–502 (1999).
[CrossRef]

J. Micromech. Microeng. (1)

M. A. McCord, A. Dana, R. F. W. Pease, “The micromechanical tunneling transistor,” J. Micromech. Microeng. 8, 209–212 (1998).
[CrossRef]

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

Optik (Stuttgart) (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Ultramicroscopy (1)

O. Marti, A. Ruf, M. Hipp, H. Bielefeldt, J. Colchero, J. Mlynek, “Micromechanical and thermal effects on force microscope cantilevers,” Ultramicroscopy 42–44, 345–349 (1992).
[CrossRef]

Other (3)

S. R. Dean, The Radon Transform and Some of Its Applications (Wiley, New York, 1993).

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, UK, 1970), Chap. 3.

C. F. Klingshirn, Semiconductor Optics (Springer, Berlin, 1997), Chap. 3.

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Figures (6)

Fig. 1
Fig. 1

GaAs tunneling cantilever actuated by light. The lateral contacts are used for the production of the Franz–Keldysh effect.

Fig. 2
Fig. 2

Working principle of the amplitude–phase determination.

Fig. 3
Fig. 3

Dependence of the ratio of tunneling currents in the presence and absence of the Franz–Keldysh effect on the angle of incidence on the cantilever.

Fig. 4
Fig. 4

F| as a function of the normalized coordinate x/ x 0 for the cases of (a) one phase jump with a = 0.5x 0, θ = π (solid curve); a = 0, θ = 2π/3 (dotted curve); a = 0, θ = π (dashed–dotted curve); and (b) two phase jumps with a 1 = -0.5x 0, a 2 = 0.5x 0, θ1 = π, θ2 = 0 (solid curve); a 1 = -0.5x 0, a 2 = x 0, θ1 = π, θ2 = 0 (dotted curve); a 1 = -0.5x 0, a 2 = 0.5x 0, θ1 = π, θ2 = π/2 (dashed–dotted curve).

Fig. 5
Fig. 5

Class of solutions (a 1, a 2) of the equation ΔF(0)/ΔF(x 1) = F 1(0; a 1, a 2)/F 1(x 1; a 1, a 2) for x 1 = 0.5x 0 (solid curve) and x 1= x 0 (dotted curve).

Fig. 6
Fig. 6

F/ F 1| as a function of the normalized coordinate x/ x 0 for a 1 = -0.5x 0, a 2 = 0.5x 0 and θ2 = 0 (solid curve), θ2 = π/2 (dotted curve).

Equations (13)

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δ opt L = 2 RPL 3 / cEWt 3 ,
R = 1 - exp - α t + 4   exp - α t sin 2   δ / 1 - R int   exp - α t 2 / R int + 4   exp - α t sin 2   δ ,
R int ,   TE = cos   φ - n 2   -   sin 2   φ 2 cos   φ + n 2   -   sin 2   φ 2 ,
R int ,   TM = n   cos   φ - 1   -   sin 2   φ / n 2 2 n   cos   φ + 1   -   sin 2   φ / n 2 2
F x =   f x H a exp i θ exp ib x - x 2 d x ,
H a = 0 , x < a 1 , x a .
F calc x =   f x exp ib x - x 2 d x ,
Δ F x = F x - F calc x = exp i θ - 1 a   f x exp ib x - x 2 d x .
Δ F 0 Δ F x 1 = a   f x exp ibx 2 d x a   f x exp ib x 1 - x 2 d x
Δ F x = exp i θ 1 - 1 a 1 a 2   f x exp ib x - x 2 d x = exp i θ 1 - 1 F 1 x ;   a 1 ,   a 2 .
Δ F x = exp i θ 1 - 1 a 1 a 2   f x exp ib x - x 2 d x + exp i θ 2 - 1 a 2   f x exp ib x - x 2 d x = exp i θ 1 - 1 F 1 x ;   a 1 ,   a 2 + exp i θ 2 - 1 F 2 x ;   a 2 .
exp i θ 1 - 1 = F 2 0 ;   a 2 Δ F x 1 - Δ F 0 F 2 x 1 ;   a 2 F 2 0 ;   a 2 F 1 x 1 ;   a 1 ,   a 2 - F 1 0 ;   a 1 ,   a 2 F 2 x 1 ;   a 2 ,
exp i θ 2 - 1 = F 1 0 ;   a 1 ,   a 2 Δ F x 1 - Δ F 0 F 1 x 1 ;   a 1 ,   a 2 F 1 0 ;   a 1 ,   a 2 F 2 x 1 ;   a 2 - F 2 0 ;   a 2 F 1 x 1 ;   a 1 ,   a 2

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