Abstract

An adaptive optical system based on incoherent digital holography is described. Theoretical and experimental studies show that wavefront sensing and compensation can be achieved by numerical processing of digital holograms of incoherent objects and a guide star, thereby dispensing with the hardware components of conventional adaptive optics systems, such as lenslet arrays and deformable mirrors. The incoherent digital holographic adaptive optics (IDHAO) process is seen to be robust and effective under various ranges of parameters, such as aberration type and strength. Furthermore, low and noisy image signals can be extracted by IDHAO to yield high-quality images with good contrast and resolution, both for point-like and continuous extended objects, illuminated with common incoherent light. Potential applications in astronomical and other imaging systems appear plausible.

© 2013 Optical Society of America

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2008

J. Rosen and G. Brooker, “Non-scanning motionless fluorescence three-dimensional holographic microscopy,” Nat. Photonics 2, 190–195 (2008).
[CrossRef]

J. Kuhn, F. Charriere, T. Colomb, E. Cuche, F. Montfort, Y. Emery, P. Marquet, and C. Depeursinge, “Axial sub-nanometer accuracy in digital holographic microscopy,” Meas. Sci. Technol. 19, 074007 (2008).
[CrossRef]

2007

B. Rappaz, A. Barbul, F. Charriere, J. Kuhn, P. Marquet, R. Korenstein, C. Depeursinge, and P. J. Magistretti, “Erythrocytes analysis with a digital holographic microscope,” Proc. SPIE 6631, 66310H (2007).
[CrossRef]

L. Miccio, D. Alfieri, S. Grilli, P. Ferraro, A. Finizio, L. De Petrocellis, and S. D. Nicola, “Direct full compensation of the aberrations in quantitative phase microscopy of thin objects by a single digital hologram,” Appl. Phys. Lett. 90, 041104 (2007).
[CrossRef]

J. Rosen and G. Brooker, “Fluorescence incoherent color holography,” Opt. Express 15, 2244–2250 (2007).
[CrossRef]

J. Rosen and G. Brooker, “Digital spatially incoherent Fresnel holography,” Opt. Lett. 32, 912–914 (2007).
[CrossRef]

2006

2005

C. J. Mann, L. F. Yu, C. M. Lo, and M. K. Kim, “High-resolution quantitative phase-contrast microscopy by digital holography,” Opt. Express 13, 8693–8698 (2005).
[CrossRef]

S. Avino, E. Calloni, J. T. Baker, F. Barone, R. DeRosa, L. DiFiore, L. Milano, and S. R. Restaino, “First adaptive optics control of laser beam based on interferometric phase-front detection,” Rev. Sci. Instrum. 76, 083119 (2005).
[CrossRef]

1997

1993

1990

H. W. Babcock, “Adaptive optics revisited,” Science 249, 253–257 (1990).
[CrossRef]

1985

1967

1966

1964

A. V. Lugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory 10, 139–145 (1964).
[CrossRef]

1953

H. W. Babcock, “The possibility of compensating astronomical seeing,” Publ. Astron. Soc. Pac. 65, 229–236 (1953).
[CrossRef]

Alfieri, D.

L. Miccio, D. Alfieri, S. Grilli, P. Ferraro, A. Finizio, L. De Petrocellis, and S. D. Nicola, “Direct full compensation of the aberrations in quantitative phase microscopy of thin objects by a single digital hologram,” Appl. Phys. Lett. 90, 041104 (2007).
[CrossRef]

Aspert, N.

Avino, S.

S. Avino, E. Calloni, J. T. Baker, F. Barone, R. DeRosa, L. DiFiore, L. Milano, and S. R. Restaino, “First adaptive optics control of laser beam based on interferometric phase-front detection,” Rev. Sci. Instrum. 76, 083119 (2005).
[CrossRef]

Babcock, H. W.

H. W. Babcock, “Adaptive optics revisited,” Science 249, 253–257 (1990).
[CrossRef]

H. W. Babcock, “The possibility of compensating astronomical seeing,” Publ. Astron. Soc. Pac. 65, 229–236 (1953).
[CrossRef]

Baker, J. T.

S. Avino, E. Calloni, J. T. Baker, F. Barone, R. DeRosa, L. DiFiore, L. Milano, and S. R. Restaino, “First adaptive optics control of laser beam based on interferometric phase-front detection,” Rev. Sci. Instrum. 76, 083119 (2005).
[CrossRef]

Barbul, A.

B. Rappaz, A. Barbul, F. Charriere, J. Kuhn, P. Marquet, R. Korenstein, C. Depeursinge, and P. J. Magistretti, “Erythrocytes analysis with a digital holographic microscope,” Proc. SPIE 6631, 66310H (2007).
[CrossRef]

Barone, F.

S. Avino, E. Calloni, J. T. Baker, F. Barone, R. DeRosa, L. DiFiore, L. Milano, and S. R. Restaino, “First adaptive optics control of laser beam based on interferometric phase-front detection,” Rev. Sci. Instrum. 76, 083119 (2005).
[CrossRef]

Bourquin, S.

Brooker, G.

Callens, N.

Calloni, E.

S. Avino, E. Calloni, J. T. Baker, F. Barone, R. DeRosa, L. DiFiore, L. Milano, and S. R. Restaino, “First adaptive optics control of laser beam based on interferometric phase-front detection,” Rev. Sci. Instrum. 76, 083119 (2005).
[CrossRef]

Charlot, D.

Charriere, F.

J. Kuhn, F. Charriere, T. Colomb, E. Cuche, F. Montfort, Y. Emery, P. Marquet, and C. Depeursinge, “Axial sub-nanometer accuracy in digital holographic microscopy,” Meas. Sci. Technol. 19, 074007 (2008).
[CrossRef]

B. Rappaz, A. Barbul, F. Charriere, J. Kuhn, P. Marquet, R. Korenstein, C. Depeursinge, and P. J. Magistretti, “Erythrocytes analysis with a digital holographic microscope,” Proc. SPIE 6631, 66310H (2007).
[CrossRef]

T. Colomb, F. Montfort, J. Kuhn, N. Aspert, E. Cuche, A. Marian, F. Charriere, S. Bourquin, P. Marquet, and C. Depeursinge, “Numerical parametric lens for shifting, magnification, and complete aberration compensation in digital holographic microscopy,” J. Opt. Soc. Am. A 23, 3177–3190 (2006).
[CrossRef]

Cochran, G.

Colomb, T.

Cuche, E.

De Petrocellis, L.

L. Miccio, D. Alfieri, S. Grilli, P. Ferraro, A. Finizio, L. De Petrocellis, and S. D. Nicola, “Direct full compensation of the aberrations in quantitative phase microscopy of thin objects by a single digital hologram,” Appl. Phys. Lett. 90, 041104 (2007).
[CrossRef]

Depeursinge, C.

J. Kuhn, F. Charriere, T. Colomb, E. Cuche, F. Montfort, Y. Emery, P. Marquet, and C. Depeursinge, “Axial sub-nanometer accuracy in digital holographic microscopy,” Meas. Sci. Technol. 19, 074007 (2008).
[CrossRef]

B. Rappaz, A. Barbul, F. Charriere, J. Kuhn, P. Marquet, R. Korenstein, C. Depeursinge, and P. J. Magistretti, “Erythrocytes analysis with a digital holographic microscope,” Proc. SPIE 6631, 66310H (2007).
[CrossRef]

T. Colomb, F. Montfort, J. Kuhn, N. Aspert, E. Cuche, A. Marian, F. Charriere, S. Bourquin, P. Marquet, and C. Depeursinge, “Numerical parametric lens for shifting, magnification, and complete aberration compensation in digital holographic microscopy,” J. Opt. Soc. Am. A 23, 3177–3190 (2006).
[CrossRef]

DeRosa, R.

S. Avino, E. Calloni, J. T. Baker, F. Barone, R. DeRosa, L. DiFiore, L. Milano, and S. R. Restaino, “First adaptive optics control of laser beam based on interferometric phase-front detection,” Rev. Sci. Instrum. 76, 083119 (2005).
[CrossRef]

DiFiore, L.

S. Avino, E. Calloni, J. T. Baker, F. Barone, R. DeRosa, L. DiFiore, L. Milano, and S. R. Restaino, “First adaptive optics control of laser beam based on interferometric phase-front detection,” Rev. Sci. Instrum. 76, 083119 (2005).
[CrossRef]

Dubois, F.

Emery, Y.

J. Kuhn, F. Charriere, T. Colomb, E. Cuche, F. Montfort, Y. Emery, P. Marquet, and C. Depeursinge, “Axial sub-nanometer accuracy in digital holographic microscopy,” Meas. Sci. Technol. 19, 074007 (2008).
[CrossRef]

Ferraro, P.

L. Miccio, D. Alfieri, S. Grilli, P. Ferraro, A. Finizio, L. De Petrocellis, and S. D. Nicola, “Direct full compensation of the aberrations in quantitative phase microscopy of thin objects by a single digital hologram,” Appl. Phys. Lett. 90, 041104 (2007).
[CrossRef]

Finizio, A.

L. Miccio, D. Alfieri, S. Grilli, P. Ferraro, A. Finizio, L. De Petrocellis, and S. D. Nicola, “Direct full compensation of the aberrations in quantitative phase microscopy of thin objects by a single digital hologram,” Appl. Phys. Lett. 90, 041104 (2007).
[CrossRef]

Fisher, R. A.

R. A. Fisher, Optical Phase Conjugation (Academic, 1983).

Fugate, R. Q.

M. C. Roggemann, B. M. Welsh, and R. Q. Fugate, “Improving the resolution of ground-based telescopes,” Rev. Mod. Phys. 69, 437–506 (1997).
[CrossRef]

Grilli, S.

L. Miccio, D. Alfieri, S. Grilli, P. Ferraro, A. Finizio, L. De Petrocellis, and S. D. Nicola, “Direct full compensation of the aberrations in quantitative phase microscopy of thin objects by a single digital hologram,” Appl. Phys. Lett. 90, 041104 (2007).
[CrossRef]

Hardy, J. W.

J. W. Hardy, Adaptive Optics for Astronomical Telescopes(Oxford University, 1998).

Hoyos, M.

Katz, B.

Kim, E. S.

Kim, M. K.

Kim, S. G.

Korenstein, R.

B. Rappaz, A. Barbul, F. Charriere, J. Kuhn, P. Marquet, R. Korenstein, C. Depeursinge, and P. J. Magistretti, “Erythrocytes analysis with a digital holographic microscope,” Proc. SPIE 6631, 66310H (2007).
[CrossRef]

Kuhn, J.

J. Kuhn, F. Charriere, T. Colomb, E. Cuche, F. Montfort, Y. Emery, P. Marquet, and C. Depeursinge, “Axial sub-nanometer accuracy in digital holographic microscopy,” Meas. Sci. Technol. 19, 074007 (2008).
[CrossRef]

B. Rappaz, A. Barbul, F. Charriere, J. Kuhn, P. Marquet, R. Korenstein, C. Depeursinge, and P. J. Magistretti, “Erythrocytes analysis with a digital holographic microscope,” Proc. SPIE 6631, 66310H (2007).
[CrossRef]

T. Colomb, F. Montfort, J. Kuhn, N. Aspert, E. Cuche, A. Marian, F. Charriere, S. Bourquin, P. Marquet, and C. Depeursinge, “Numerical parametric lens for shifting, magnification, and complete aberration compensation in digital holographic microscopy,” J. Opt. Soc. Am. A 23, 3177–3190 (2006).
[CrossRef]

Kurowski, P.

Lee, B.

Leith, E. N.

Liu, C. G.

Lo, C. M.

Lugt, A. V.

J. Upatnieks, A. V. Lugt, and E. N. Leith, “Correction of lens aberrations by means of holograms,” Appl. Opt. 5, 589–593 (1966).
[CrossRef]

A. V. Lugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory 10, 139–145 (1964).
[CrossRef]

Magistretti, P. J.

B. Rappaz, A. Barbul, F. Charriere, J. Kuhn, P. Marquet, R. Korenstein, C. Depeursinge, and P. J. Magistretti, “Erythrocytes analysis with a digital holographic microscope,” Proc. SPIE 6631, 66310H (2007).
[CrossRef]

Mann, C. J.

Marian, A.

Marquet, P.

J. Kuhn, F. Charriere, T. Colomb, E. Cuche, F. Montfort, Y. Emery, P. Marquet, and C. Depeursinge, “Axial sub-nanometer accuracy in digital holographic microscopy,” Meas. Sci. Technol. 19, 074007 (2008).
[CrossRef]

B. Rappaz, A. Barbul, F. Charriere, J. Kuhn, P. Marquet, R. Korenstein, C. Depeursinge, and P. J. Magistretti, “Erythrocytes analysis with a digital holographic microscope,” Proc. SPIE 6631, 66310H (2007).
[CrossRef]

T. Colomb, F. Montfort, J. Kuhn, N. Aspert, E. Cuche, A. Marian, F. Charriere, S. Bourquin, P. Marquet, and C. Depeursinge, “Numerical parametric lens for shifting, magnification, and complete aberration compensation in digital holographic microscopy,” J. Opt. Soc. Am. A 23, 3177–3190 (2006).
[CrossRef]

Miccio, L.

L. Miccio, D. Alfieri, S. Grilli, P. Ferraro, A. Finizio, L. De Petrocellis, and S. D. Nicola, “Direct full compensation of the aberrations in quantitative phase microscopy of thin objects by a single digital hologram,” Appl. Phys. Lett. 90, 041104 (2007).
[CrossRef]

Milano, L.

S. Avino, E. Calloni, J. T. Baker, F. Barone, R. DeRosa, L. DiFiore, L. Milano, and S. R. Restaino, “First adaptive optics control of laser beam based on interferometric phase-front detection,” Rev. Sci. Instrum. 76, 083119 (2005).
[CrossRef]

Monnom, O.

Montfort, F.

Mugnier, L. M.

Nicola, S. D.

L. Miccio, D. Alfieri, S. Grilli, P. Ferraro, A. Finizio, L. De Petrocellis, and S. D. Nicola, “Direct full compensation of the aberrations in quantitative phase microscopy of thin objects by a single digital hologram,” Appl. Phys. Lett. 90, 041104 (2007).
[CrossRef]

Poon, T. C.

Porter, J.

J. Porter, Adaptive Optics for Vision Science: Principles, Practices, Design, and Applications (Wiley, 2006).

Psaltis, D.

Rappaz, B.

B. Rappaz, A. Barbul, F. Charriere, J. Kuhn, P. Marquet, R. Korenstein, C. Depeursinge, and P. J. Magistretti, “Erythrocytes analysis with a digital holographic microscope,” Proc. SPIE 6631, 66310H (2007).
[CrossRef]

Restaino, S. R.

S. Avino, E. Calloni, J. T. Baker, F. Barone, R. DeRosa, L. DiFiore, L. Milano, and S. R. Restaino, “First adaptive optics control of laser beam based on interferometric phase-front detection,” Rev. Sci. Instrum. 76, 083119 (2005).
[CrossRef]

Roggemann, M. C.

M. C. Roggemann, B. M. Welsh, and R. Q. Fugate, “Improving the resolution of ground-based telescopes,” Rev. Mod. Phys. 69, 437–506 (1997).
[CrossRef]

Rosen, J.

Siegel, N.

Sirat, G.

Sirat, G. Y.

Tyson, R. K.

R. K. Tyson, Introduction to Adaptive Optics (SPIE, 2000).

R. K. Tyson, Principles of Adaptive Optics (CRC Press, 2011).

Upatniek, J.

Upatnieks, J.

Wang, V.

Welsh, B. M.

M. C. Roggemann, B. M. Welsh, and R. Q. Fugate, “Improving the resolution of ground-based telescopes,” Rev. Mod. Phys. 69, 437–506 (1997).
[CrossRef]

Wulich, D.

Yamaguchi, I.

Yourassowsky, C.

Yu, L. F.

Zhang, T.

Appl. Opt.

Appl. Phys. Lett.

L. Miccio, D. Alfieri, S. Grilli, P. Ferraro, A. Finizio, L. De Petrocellis, and S. D. Nicola, “Direct full compensation of the aberrations in quantitative phase microscopy of thin objects by a single digital hologram,” Appl. Phys. Lett. 90, 041104 (2007).
[CrossRef]

IEEE Trans. Inf. Theory

A. V. Lugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory 10, 139–145 (1964).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Korea

Meas. Sci. Technol.

J. Kuhn, F. Charriere, T. Colomb, E. Cuche, F. Montfort, Y. Emery, P. Marquet, and C. Depeursinge, “Axial sub-nanometer accuracy in digital holographic microscopy,” Meas. Sci. Technol. 19, 074007 (2008).
[CrossRef]

Nat. Photonics

J. Rosen and G. Brooker, “Non-scanning motionless fluorescence three-dimensional holographic microscopy,” Nat. Photonics 2, 190–195 (2008).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. SPIE

B. Rappaz, A. Barbul, F. Charriere, J. Kuhn, P. Marquet, R. Korenstein, C. Depeursinge, and P. J. Magistretti, “Erythrocytes analysis with a digital holographic microscope,” Proc. SPIE 6631, 66310H (2007).
[CrossRef]

Publ. Astron. Soc. Pac.

H. W. Babcock, “The possibility of compensating astronomical seeing,” Publ. Astron. Soc. Pac. 65, 229–236 (1953).
[CrossRef]

Rev. Mod. Phys.

M. C. Roggemann, B. M. Welsh, and R. Q. Fugate, “Improving the resolution of ground-based telescopes,” Rev. Mod. Phys. 69, 437–506 (1997).
[CrossRef]

Rev. Sci. Instrum.

S. Avino, E. Calloni, J. T. Baker, F. Barone, R. DeRosa, L. DiFiore, L. Milano, and S. R. Restaino, “First adaptive optics control of laser beam based on interferometric phase-front detection,” Rev. Sci. Instrum. 76, 083119 (2005).
[CrossRef]

Science

H. W. Babcock, “Adaptive optics revisited,” Science 249, 253–257 (1990).
[CrossRef]

SPIE Rev.

M. K. Kim, “Principles and techniques of digital holographic microscopy,” SPIE Rev. 1, 018005 (2010).
[CrossRef]

Other

R. A. Fisher, Optical Phase Conjugation (Academic, 1983).

J. W. Hardy, Adaptive Optics for Astronomical Telescopes(Oxford University, 1998).

R. K. Tyson, Principles of Adaptive Optics (CRC Press, 2011).

J. Porter, Adaptive Optics for Vision Science: Principles, Practices, Design, and Applications (Wiley, 2006).

R. K. Tyson, Introduction to Adaptive Optics (SPIE, 2000).

M. K. Kim, Digital Holographic Microscopy: Principles, Techniques, and Applications (Springer, 2011).

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Figures (15)

Fig. 1.
Fig. 1.

Basic configuration of IDH. S, source point; Ψ, phase aberrator; BS, beam splitter; MA (MB), curved mirror with focal length fA (fB).

Fig. 2.
Fig. 2.

Optical configuration used in simulation and experiments. MA, piezo-mounted plane mirror; MB, curved mirror with focal length fB; L’s, lenses; IF, interference filter.

Fig. 3.
Fig. 3.

Simulation of IDH process: (a) Object field, Io; (b) one of raw interferograms hφ on the camera plane; (c) amplitude; (d) phase of the complex hologram H0; (e) amplitude; and (f) phase of reconstructed image Io.

Fig. 4.
Fig. 4.

Simulation of the AO process by IDH: (a) the object field Io, (b) the assumed phase aberration Ψ=aΨ(Z3+1+Z51) with aΨ=0.5, (c) one of the guide star interferograms gφ, (d) and (e) the guide star complex hologram GΨ, (f) one of the full-field interferograms hφ, (g) and (h) the full-field complex hologram HΨ, (i) direct image IB without holographic process, (j) and (k) IDH image IΨ without aberration compensation, (l) and (m) complex hologram H^Ψ with aberration compensation (n) and (o) IDH image I^Ψ with aberration compensation. (Note the somewhat irregular arrangement of subpanel labels.)

Fig. 5.
Fig. 5.

Resolution of reconstructed images I^Ψ versus focal length: (a) without aberration aΨ=0.0 and (b) with aberration aΨ=0.5. The focal length of MB is varied as fB=(i)5000, (ii) 3000, (iii) 2000, and (iv) 1000 mm, while zi=fBzc.

Fig. 6.
Fig. 6.

IDHAO versus strength of aberration. In each row the phase aberration Ψ, the uncompensated image IΨ, the compensated image I^Ψ, and the direct image IB are shown. Strength of the aberration Ψ=aΨ(Z3+1+Z51) is varied as (a) aΨ=0.1, (b) aΨ=0.2, (c) aΨ=0.5, and (d) aΨ=1.0.

Fig. 7.
Fig. 7.

IDHAO versus various types of aberration: (a) Ψ=(0.5)×Z1+1+(0.5)×Z20, (b) Ψ=(0.5)×Z1+1+(0.5)×Z3+1, (c) Ψ=(0.5)×Z3+3+(0.5)×Z55, and (d) Ψ=(0.7)×Z20+(0.5)×Z55.

Fig. 8.
Fig. 8.

IDHAO of extended gray-scale object: (a) the assumed phase aberration Ψ=(0.5)×(Z5+3+Z42), (b) object Io, (c) uncompensated image IΨ, and (d) compensated image I^Ψ.

Fig. 9.
Fig. 9.

Effect of noisy hologram on IDHAO: (a) aΨ=0, aΦ=5; (b) aΨ=0, aΦ=10; (c) aΨ=0.2, aΦ=5; (d) aΨ=0.2, aΦ=10. In each row the phase of GΨ and HΨ and the amplitude of IΨ and I are shown.

Fig. 10.
Fig. 10.

IDH without aberration: (a) one of the raw holograms hφ; (b) complex hologram H0; (c) reconstructed image at zi=22,000mm;, corresponding to z1=650mm; (d) reconstructed image at zi=14,870mm; corresponding to z1=440mm.

Fig. 11.
Fig. 11.

IDHAO with a phase aberrator placed just behind the objective lens: (a) one of the raw holograms hφ; (b) complex hologram HΨ, reconstructed images IΨ; (c) at zi; (d) at zi; (e) guide star hologram GΨ of an LED at z1; (f) corrected hologram H^Ψ, corrected images I^Ψ; (g) at zi; and (h) at zi.

Fig. 12.
Fig. 12.

IDHAO versus noise: (a)–(c) varying brightness of the object field, consisting of all four LEDs; (i) IDH image IΨ without AO; (ii)–(iv) IDHAO images I^Ψ, with varying brightness of the guide star.

Fig. 13.
Fig. 13.

IDH of extended objects, a resolution target (top) and a knight chess piece (bottom), illuminated with a miniature halogen lamp. In each case, a raw hologram hφ and the amplitude and phase of the complex hologram HΨ are shown.

Fig. 14.
Fig. 14.

IDHAO versus noise: (a)–(c) varying brightness of the resolution target, (i) IDH image IΨ without AO; (ii)–(iv) IDHAO images I^Ψ, with varying brightness of the guide star.

Fig. 15.
Fig. 15.

3D focusing of IDHAO of extended object: (a) uncorrected image IΨ reconstructed at zi=17,500mm, corresponding to z1=450mm; (b) reconstructed image at zi=22,000mm, corresponding to z1=650mm; (c) and (d) corresponding images I^Ψ with compensation.

Equations (30)

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EA(xc)=dxmEoQzo(xmxo)QfA(xm)Qzc(xcxm)=EoQzofA(xo)QzA+zc(xcxA),
Qz(x)exp[ik2zx2],
xA=fAzofAxo;zA=fAzofAzo.
Qz1(xx1)Qz2(xx2)=Qz1+z2(x1x2)Qz12(xx12),
z12=z1z2z1+z2;x12=z2x1+z1x2z1+z2.
G0(xc;xo)=EAEB*=Io(xo)QzAB(xcαxo),
g(xc;xo)=|EA+EB|2=2Io(xo)[1+cosk(xcαxo)22zAB],
α=zczozAB=(zA+zc)(zB+zc)zAzB.
h(xc)=dxog(xc;xo),
hφ(xc)=dxogφ(xc;xo)=dxo|EAeiφ+EB|2,
H0(xc)=(1/4){[h0hπ]i[hπ/2h3π/2]}=dxoG0(xc;xo)=dxoIo(xo)QzAB(xcαxo)=IoQzAB(xc),
Io(xc)=Io(xcα)
Io(xc)=H0QzAB(xc).
EA(xc)=dxdxmEoQz(xxo)Ψ(x)Qz(xmx)QfA(xm)Qzc(xcxm)=EoQzofA(xo)QzA+zc(xcxA)ΦA(xcαAxo),
ΦA(x)=[ΨQζA](βAx)=dxΨ(x)QζA(xβAx),
αA=zzzA+zczA+α,βA=zz+zzAzA+zc,ζA=zz+z[z+zz+zzAzczA+zc].
GΨ(xc;xo)=EAEB*=Io(xo)QzofA(xo)QzofB*(xo)QzA+zc(xcxA)QzB+zc*(xcxB)ΦA(xcαAxo)ΦB*(xcαBxo)=Io(xo)QzAB(xcαxo)ΦA(xcαAxo)ΦB*(xcαBxo),
HΨ(xc)=dxoGΨ(xc;xo)=dxoIo(xo)QzAB(xcαxo)ΦA(xcαAxo)ΦB*(xcαBxo).
GΨ(xc;xo)=Io(xo)[QzABΦAΦB*](xcαxo),
HΨ(xc)=Io[QzABΦAΦB*](xc).
GΨ(xc)GΨ(xc;0)=[QzABΦAΦB*](xc).
HΨ(xc)=IoGΨ(xc).
IΨHΨG0*=HΨG0=Io[GΨG0],
I^ΨHΨGΨ=Io[GΨGΨ]Ioδ=Io.
I^Ψ=H^ΨG0*,
H^Ψ=HΨ(GΨG0).
IB(xc)=dxo|EB(xc;xo)|2=dxoIo(xo)|ΦB(xcαxo)|2.
|ΦB(xc)|2=|Ψ˜(xc)|2,
Ψ˜(xc)=dxΨ(x)exp[ikxczcx],
IB(xc)=[Io|Ψ˜|2](xc).

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