Abstract

The liquid-crystal-adaptive lens (LCAL) is an electro-optical device that utilizes a graded index of refraction to bring light to focus. A set of electrodes controls the index variation in a liquid-crystal thin film. One can vary the focal length of the LCAL by changing the voltages applied to the device. The discrete nature of the electrodes causes phase aberrations. We introduce a novel electrode architecture, called conductive ladder meshing (CLM), that we developed to greatly reduce the static phase aberration (caused by the electrode structure). To reduce the dynamic phase aberration (associated with inaccurate voltages), we used a simulated-annealing voltage-dithering technique. The coherent transfer function of the LCAL was derived so that the performance of the CLM LCAL could be predicted theoretically. Theoretical analysis indicates that the CLM LCAL scatters less than 30% of incident light compared with scattering of 65% in the previous version. The focal-spot performance of the spherical LCAL was measured under coherent illumination for plane-wave illumination. Because of the improved quality of the spherical LCAL, true imaging experiments are demonstrated for a single incoming polarization under white-light illumination. Images formed by the spherical LCAL are comparable with those formed by a fixed lens in terms of resolution, although the contrast is worse.

© 1997 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).
  2. S. T. Kowel, D. S. Cleverly, P. G. Kornreich, “Focusing by electrical modulation of refraction in a liquid crystal cell,” Appl. Opt. 23, 278–289 (1984).
    [CrossRef]
  3. A. Nouhi, S. T. Kowel, “Adaptive spherical lens,” Appl. Opt. 23, 2774–2777 (1984).
    [CrossRef] [PubMed]
  4. D. A. Parthenopoulos, P. M. Rentzepis, “Three dimensional optical storage memory,” Science 245, 843–845 (1989).
    [CrossRef] [PubMed]
  5. M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964).
  6. P. P. Banerjee, T. C. Poon, Principles of Applied Optics (Aksen, Homewood, Ill., 1991).
  7. S. T. Kowel, P. Kornreich, D. Cleverly, “Adaptive liquid lens,” U.S. patent4,572,616 (25Feb.1986).
  8. P. F. Brinkley, “Diffraction and refraction behavior in a liquid crystal adaptive lens,” Ph.D. dissertation (University of California, Davis, Calif., 1992).
  9. W. W. Chan, L. Q. Ning, S. T. Kowel, P. F. Brinkley, “Liquid crystal adaptive lens: aberration correction,” in Photonics for Computers, Neural Networks, and Memories, W. J. Miceli, J. A. Neff, S. T. Kowel, eds., Proc. SPIE1773, 468–475 (1993).
    [CrossRef]
  10. P. F. Brinkley, S. T. Kowel, “Liquid crystal adaptive lens: operations and aberration,” in Photonics for Computers, Neural Networks, and Memories, W. J. Miceli, J. A. Neff, S. T. Kowel, eds., Proc. SPIE1773, 449–458 (1993).
    [CrossRef]
  11. L. Q. Ning, “Aberration adjustment in the liquid crystal adaptive lens,” M. S. thesis (University of Alabama in Huntsville, Huntsville, Ala., 1992).
  12. J. D. Kraus, K. R. Carver, Electromagnetics, 2nd ed. (McGraw-Hill, New York, 1973).
  13. W. J. Duffin, Electricity and Magnetism, 3rd ed. (McGraw-Hill, New York, 1990).
  14. H. H. William, Engineering Electromagnetics, 4th ed. (McGraw-Hill, New York, 1981).
  15. N. J. Powell, R. W. Kelsall, G. D. Love, A. Purvis, “Investigation of fringing fields in liquid-crystal devices,” in International Conference on the Application and Theory of Periodic Structures, J. M. Lerner, W. R. McKinney, eds., Proc. SPIE1545, 19–30 (1991).
    [CrossRef]
  16. G. Hass, M. W. Fritsch, H. Wohler, D. A. Mlynski, “Simulation of reverse tilt disclinations in liquid crystal displays,” in Proceedings of the 10th International Display Research Conference (VDE - Verlag, Berlin, 1990), pp. 252–255.
  17. G. Hass, H. Wohler, M. W. Fritsch, D. A. Mlynski, “Simulation of two-dimensional nematic director structures in inhomogeneous electric field,” Mol. Liq. Cryst. 198, 15–28 (1991).
    [CrossRef]
  18. W. W. Chan, “Liquid crystal adaptive lens: high density electrode configuration,” Ph.D. dissertation (University of Alabama in Huntsville, Huntsville, Ala., 1997).
  19. S. Satom, A. Sugiyamam, R. Sato, “Variable-focus liquid-crystal Fresnel lens,” Jpn. J. Appl. Phys. 24, L626–L628 (1985).
    [CrossRef]
  20. B. Dance, “Liquid crystal used in switchable Fresnel lens,” Laser Focus World 28(4), 34 (1992).
  21. N. A. Riza, M. C. DeJule, “Three-terminal adaptive nematic liquid-crystal lens device,” Opt. Lett. 19, 1013–1015 (1994).
    [CrossRef] [PubMed]
  22. S. Chandrasekhar, Liquid Crystal, 2nd ed. (Cambridge U. Press, New York, 1992).
    [CrossRef]
  23. F. J. Kahn, “Special issue: liquid crystals,” Phys. Today 35(5), 66–74 (1992).
    [CrossRef]
  24. G. Labrunie, J. Robert, “Transient behavior of the electrically controlled birefringence in a nematic liquid crystal,” J. Appl. Phys. 44, 4869–4874 (1973).
    [CrossRef]
  25. labview Version 2.0, National Instruments, Austin, Tex., 1989.
  26. E. B. Priestley, P. J. Wojtowicz, P. Sheng, Introduction to Liquid Crystals (Plenum, New York, 1975).

1994 (1)

1992 (2)

B. Dance, “Liquid crystal used in switchable Fresnel lens,” Laser Focus World 28(4), 34 (1992).

F. J. Kahn, “Special issue: liquid crystals,” Phys. Today 35(5), 66–74 (1992).
[CrossRef]

1991 (1)

G. Hass, H. Wohler, M. W. Fritsch, D. A. Mlynski, “Simulation of two-dimensional nematic director structures in inhomogeneous electric field,” Mol. Liq. Cryst. 198, 15–28 (1991).
[CrossRef]

1989 (1)

D. A. Parthenopoulos, P. M. Rentzepis, “Three dimensional optical storage memory,” Science 245, 843–845 (1989).
[CrossRef] [PubMed]

1985 (1)

S. Satom, A. Sugiyamam, R. Sato, “Variable-focus liquid-crystal Fresnel lens,” Jpn. J. Appl. Phys. 24, L626–L628 (1985).
[CrossRef]

1984 (2)

1973 (1)

G. Labrunie, J. Robert, “Transient behavior of the electrically controlled birefringence in a nematic liquid crystal,” J. Appl. Phys. 44, 4869–4874 (1973).
[CrossRef]

Banerjee, P. P.

P. P. Banerjee, T. C. Poon, Principles of Applied Optics (Aksen, Homewood, Ill., 1991).

Born, M.

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964).

Brinkley, P. F.

W. W. Chan, L. Q. Ning, S. T. Kowel, P. F. Brinkley, “Liquid crystal adaptive lens: aberration correction,” in Photonics for Computers, Neural Networks, and Memories, W. J. Miceli, J. A. Neff, S. T. Kowel, eds., Proc. SPIE1773, 468–475 (1993).
[CrossRef]

P. F. Brinkley, S. T. Kowel, “Liquid crystal adaptive lens: operations and aberration,” in Photonics for Computers, Neural Networks, and Memories, W. J. Miceli, J. A. Neff, S. T. Kowel, eds., Proc. SPIE1773, 449–458 (1993).
[CrossRef]

P. F. Brinkley, “Diffraction and refraction behavior in a liquid crystal adaptive lens,” Ph.D. dissertation (University of California, Davis, Calif., 1992).

Carver, K. R.

J. D. Kraus, K. R. Carver, Electromagnetics, 2nd ed. (McGraw-Hill, New York, 1973).

Chan, W. W.

W. W. Chan, L. Q. Ning, S. T. Kowel, P. F. Brinkley, “Liquid crystal adaptive lens: aberration correction,” in Photonics for Computers, Neural Networks, and Memories, W. J. Miceli, J. A. Neff, S. T. Kowel, eds., Proc. SPIE1773, 468–475 (1993).
[CrossRef]

W. W. Chan, “Liquid crystal adaptive lens: high density electrode configuration,” Ph.D. dissertation (University of Alabama in Huntsville, Huntsville, Ala., 1997).

Chandrasekhar, S.

S. Chandrasekhar, Liquid Crystal, 2nd ed. (Cambridge U. Press, New York, 1992).
[CrossRef]

Cleverly, D.

S. T. Kowel, P. Kornreich, D. Cleverly, “Adaptive liquid lens,” U.S. patent4,572,616 (25Feb.1986).

Cleverly, D. S.

Dance, B.

B. Dance, “Liquid crystal used in switchable Fresnel lens,” Laser Focus World 28(4), 34 (1992).

DeJule, M. C.

Duffin, W. J.

W. J. Duffin, Electricity and Magnetism, 3rd ed. (McGraw-Hill, New York, 1990).

Fritsch, M. W.

G. Hass, H. Wohler, M. W. Fritsch, D. A. Mlynski, “Simulation of two-dimensional nematic director structures in inhomogeneous electric field,” Mol. Liq. Cryst. 198, 15–28 (1991).
[CrossRef]

G. Hass, M. W. Fritsch, H. Wohler, D. A. Mlynski, “Simulation of reverse tilt disclinations in liquid crystal displays,” in Proceedings of the 10th International Display Research Conference (VDE - Verlag, Berlin, 1990), pp. 252–255.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Hass, G.

G. Hass, H. Wohler, M. W. Fritsch, D. A. Mlynski, “Simulation of two-dimensional nematic director structures in inhomogeneous electric field,” Mol. Liq. Cryst. 198, 15–28 (1991).
[CrossRef]

G. Hass, M. W. Fritsch, H. Wohler, D. A. Mlynski, “Simulation of reverse tilt disclinations in liquid crystal displays,” in Proceedings of the 10th International Display Research Conference (VDE - Verlag, Berlin, 1990), pp. 252–255.

Kahn, F. J.

F. J. Kahn, “Special issue: liquid crystals,” Phys. Today 35(5), 66–74 (1992).
[CrossRef]

Kelsall, R. W.

N. J. Powell, R. W. Kelsall, G. D. Love, A. Purvis, “Investigation of fringing fields in liquid-crystal devices,” in International Conference on the Application and Theory of Periodic Structures, J. M. Lerner, W. R. McKinney, eds., Proc. SPIE1545, 19–30 (1991).
[CrossRef]

Kornreich, P.

S. T. Kowel, P. Kornreich, D. Cleverly, “Adaptive liquid lens,” U.S. patent4,572,616 (25Feb.1986).

Kornreich, P. G.

Kowel, S. T.

S. T. Kowel, D. S. Cleverly, P. G. Kornreich, “Focusing by electrical modulation of refraction in a liquid crystal cell,” Appl. Opt. 23, 278–289 (1984).
[CrossRef]

A. Nouhi, S. T. Kowel, “Adaptive spherical lens,” Appl. Opt. 23, 2774–2777 (1984).
[CrossRef] [PubMed]

P. F. Brinkley, S. T. Kowel, “Liquid crystal adaptive lens: operations and aberration,” in Photonics for Computers, Neural Networks, and Memories, W. J. Miceli, J. A. Neff, S. T. Kowel, eds., Proc. SPIE1773, 449–458 (1993).
[CrossRef]

W. W. Chan, L. Q. Ning, S. T. Kowel, P. F. Brinkley, “Liquid crystal adaptive lens: aberration correction,” in Photonics for Computers, Neural Networks, and Memories, W. J. Miceli, J. A. Neff, S. T. Kowel, eds., Proc. SPIE1773, 468–475 (1993).
[CrossRef]

S. T. Kowel, P. Kornreich, D. Cleverly, “Adaptive liquid lens,” U.S. patent4,572,616 (25Feb.1986).

Kraus, J. D.

J. D. Kraus, K. R. Carver, Electromagnetics, 2nd ed. (McGraw-Hill, New York, 1973).

Labrunie, G.

G. Labrunie, J. Robert, “Transient behavior of the electrically controlled birefringence in a nematic liquid crystal,” J. Appl. Phys. 44, 4869–4874 (1973).
[CrossRef]

Love, G. D.

N. J. Powell, R. W. Kelsall, G. D. Love, A. Purvis, “Investigation of fringing fields in liquid-crystal devices,” in International Conference on the Application and Theory of Periodic Structures, J. M. Lerner, W. R. McKinney, eds., Proc. SPIE1545, 19–30 (1991).
[CrossRef]

Mlynski, D. A.

G. Hass, H. Wohler, M. W. Fritsch, D. A. Mlynski, “Simulation of two-dimensional nematic director structures in inhomogeneous electric field,” Mol. Liq. Cryst. 198, 15–28 (1991).
[CrossRef]

G. Hass, M. W. Fritsch, H. Wohler, D. A. Mlynski, “Simulation of reverse tilt disclinations in liquid crystal displays,” in Proceedings of the 10th International Display Research Conference (VDE - Verlag, Berlin, 1990), pp. 252–255.

Ning, L. Q.

W. W. Chan, L. Q. Ning, S. T. Kowel, P. F. Brinkley, “Liquid crystal adaptive lens: aberration correction,” in Photonics for Computers, Neural Networks, and Memories, W. J. Miceli, J. A. Neff, S. T. Kowel, eds., Proc. SPIE1773, 468–475 (1993).
[CrossRef]

L. Q. Ning, “Aberration adjustment in the liquid crystal adaptive lens,” M. S. thesis (University of Alabama in Huntsville, Huntsville, Ala., 1992).

Nouhi, A.

Parthenopoulos, D. A.

D. A. Parthenopoulos, P. M. Rentzepis, “Three dimensional optical storage memory,” Science 245, 843–845 (1989).
[CrossRef] [PubMed]

Poon, T. C.

P. P. Banerjee, T. C. Poon, Principles of Applied Optics (Aksen, Homewood, Ill., 1991).

Powell, N. J.

N. J. Powell, R. W. Kelsall, G. D. Love, A. Purvis, “Investigation of fringing fields in liquid-crystal devices,” in International Conference on the Application and Theory of Periodic Structures, J. M. Lerner, W. R. McKinney, eds., Proc. SPIE1545, 19–30 (1991).
[CrossRef]

Priestley, E. B.

E. B. Priestley, P. J. Wojtowicz, P. Sheng, Introduction to Liquid Crystals (Plenum, New York, 1975).

Purvis, A.

N. J. Powell, R. W. Kelsall, G. D. Love, A. Purvis, “Investigation of fringing fields in liquid-crystal devices,” in International Conference on the Application and Theory of Periodic Structures, J. M. Lerner, W. R. McKinney, eds., Proc. SPIE1545, 19–30 (1991).
[CrossRef]

Rentzepis, P. M.

D. A. Parthenopoulos, P. M. Rentzepis, “Three dimensional optical storage memory,” Science 245, 843–845 (1989).
[CrossRef] [PubMed]

Riza, N. A.

Robert, J.

G. Labrunie, J. Robert, “Transient behavior of the electrically controlled birefringence in a nematic liquid crystal,” J. Appl. Phys. 44, 4869–4874 (1973).
[CrossRef]

Sato, R.

S. Satom, A. Sugiyamam, R. Sato, “Variable-focus liquid-crystal Fresnel lens,” Jpn. J. Appl. Phys. 24, L626–L628 (1985).
[CrossRef]

Satom, S.

S. Satom, A. Sugiyamam, R. Sato, “Variable-focus liquid-crystal Fresnel lens,” Jpn. J. Appl. Phys. 24, L626–L628 (1985).
[CrossRef]

Sheng, P.

E. B. Priestley, P. J. Wojtowicz, P. Sheng, Introduction to Liquid Crystals (Plenum, New York, 1975).

Sugiyamam, A.

S. Satom, A. Sugiyamam, R. Sato, “Variable-focus liquid-crystal Fresnel lens,” Jpn. J. Appl. Phys. 24, L626–L628 (1985).
[CrossRef]

William, H. H.

H. H. William, Engineering Electromagnetics, 4th ed. (McGraw-Hill, New York, 1981).

Wohler, H.

G. Hass, H. Wohler, M. W. Fritsch, D. A. Mlynski, “Simulation of two-dimensional nematic director structures in inhomogeneous electric field,” Mol. Liq. Cryst. 198, 15–28 (1991).
[CrossRef]

G. Hass, M. W. Fritsch, H. Wohler, D. A. Mlynski, “Simulation of reverse tilt disclinations in liquid crystal displays,” in Proceedings of the 10th International Display Research Conference (VDE - Verlag, Berlin, 1990), pp. 252–255.

Wojtowicz, P. J.

E. B. Priestley, P. J. Wojtowicz, P. Sheng, Introduction to Liquid Crystals (Plenum, New York, 1975).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964).

Appl. Opt. (2)

J. Appl. Phys. (1)

G. Labrunie, J. Robert, “Transient behavior of the electrically controlled birefringence in a nematic liquid crystal,” J. Appl. Phys. 44, 4869–4874 (1973).
[CrossRef]

Jpn. J. Appl. Phys. (1)

S. Satom, A. Sugiyamam, R. Sato, “Variable-focus liquid-crystal Fresnel lens,” Jpn. J. Appl. Phys. 24, L626–L628 (1985).
[CrossRef]

Laser Focus World (1)

B. Dance, “Liquid crystal used in switchable Fresnel lens,” Laser Focus World 28(4), 34 (1992).

Mol. Liq. Cryst. (1)

G. Hass, H. Wohler, M. W. Fritsch, D. A. Mlynski, “Simulation of two-dimensional nematic director structures in inhomogeneous electric field,” Mol. Liq. Cryst. 198, 15–28 (1991).
[CrossRef]

Opt. Lett. (1)

Phys. Today (1)

F. J. Kahn, “Special issue: liquid crystals,” Phys. Today 35(5), 66–74 (1992).
[CrossRef]

Science (1)

D. A. Parthenopoulos, P. M. Rentzepis, “Three dimensional optical storage memory,” Science 245, 843–845 (1989).
[CrossRef] [PubMed]

Other (17)

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964).

P. P. Banerjee, T. C. Poon, Principles of Applied Optics (Aksen, Homewood, Ill., 1991).

S. T. Kowel, P. Kornreich, D. Cleverly, “Adaptive liquid lens,” U.S. patent4,572,616 (25Feb.1986).

P. F. Brinkley, “Diffraction and refraction behavior in a liquid crystal adaptive lens,” Ph.D. dissertation (University of California, Davis, Calif., 1992).

W. W. Chan, L. Q. Ning, S. T. Kowel, P. F. Brinkley, “Liquid crystal adaptive lens: aberration correction,” in Photonics for Computers, Neural Networks, and Memories, W. J. Miceli, J. A. Neff, S. T. Kowel, eds., Proc. SPIE1773, 468–475 (1993).
[CrossRef]

P. F. Brinkley, S. T. Kowel, “Liquid crystal adaptive lens: operations and aberration,” in Photonics for Computers, Neural Networks, and Memories, W. J. Miceli, J. A. Neff, S. T. Kowel, eds., Proc. SPIE1773, 449–458 (1993).
[CrossRef]

L. Q. Ning, “Aberration adjustment in the liquid crystal adaptive lens,” M. S. thesis (University of Alabama in Huntsville, Huntsville, Ala., 1992).

J. D. Kraus, K. R. Carver, Electromagnetics, 2nd ed. (McGraw-Hill, New York, 1973).

W. J. Duffin, Electricity and Magnetism, 3rd ed. (McGraw-Hill, New York, 1990).

H. H. William, Engineering Electromagnetics, 4th ed. (McGraw-Hill, New York, 1981).

N. J. Powell, R. W. Kelsall, G. D. Love, A. Purvis, “Investigation of fringing fields in liquid-crystal devices,” in International Conference on the Application and Theory of Periodic Structures, J. M. Lerner, W. R. McKinney, eds., Proc. SPIE1545, 19–30 (1991).
[CrossRef]

G. Hass, M. W. Fritsch, H. Wohler, D. A. Mlynski, “Simulation of reverse tilt disclinations in liquid crystal displays,” in Proceedings of the 10th International Display Research Conference (VDE - Verlag, Berlin, 1990), pp. 252–255.

W. W. Chan, “Liquid crystal adaptive lens: high density electrode configuration,” Ph.D. dissertation (University of Alabama in Huntsville, Huntsville, Ala., 1997).

S. Chandrasekhar, Liquid Crystal, 2nd ed. (Cambridge U. Press, New York, 1992).
[CrossRef]

labview Version 2.0, National Instruments, Austin, Tex., 1989.

E. B. Priestley, P. J. Wojtowicz, P. Sheng, Introduction to Liquid Crystals (Plenum, New York, 1975).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

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Figures (22)

Fig. 1
Fig. 1

LCAL structure.

Fig. 2
Fig. 2

Four LCAL’s cascaded to operate as a spherical lens under random polarized-light illumination.

Fig. 3
Fig. 3

Electromagnetic field inside the LCAL.

Fig. 4
Fig. 4

General form of static aberrations in the LCAL.

Fig. 5
Fig. 5

Linear voltage–phase interpolation.

Fig. 6
Fig. 6

Section (Fresnel zone) of the CLM LCAL.

Fig. 7
Fig. 7

Plot of the f-number for the continuous phase profile and the Fresnel phase profile.

Fig. 8
Fig. 8

Analysis of the conventional design (electrode period, 60 µm). The focal length of the CLM LCAL is 0.5 m.

Fig. 9
Fig. 9

Computed focal-line intensity of the LCAL (a.w., electrode width and electrode spacing). The focal length of the CLM LCAL is 0.5 m.

Fig. 10
Fig. 10

Schematic birefringence measurement of CM.

Fig. 11
Fig. 11

Birefringence data.

Fig. 12
Fig. 12

Layout of the Fresnel zones of the CLM LCAL. [Owing to symmetry, only the right-hand side (RHS) of the electrode plate is shown.]

Fig. 13
Fig. 13

Computed focal-plane intensity profile (CLM LCAL) for a normally incident plane wave. The focal length of the CLM LCAL is 0.5 m.

Fig. 14
Fig. 14

Measurement of the change of refractive index versus applied voltage.

Fig. 15
Fig. 15

Calculated modified Fresnel phase profile. (Owing to symmetry, only the RHS of the Fresnel zones is shown.)

Fig. 16
Fig. 16

Bias voltage of a modified Fresnel voltage profile: (a) f = 1.0 m and (b) f = 0.7 m.

Fig. 17
Fig. 17

Experimental setup for measuring the intensity of the focal line.

Fig. 18
Fig. 18

Focal line of the CLM LCAL (f = 0.7 m) without application of the voltage-dithering technique.

Fig. 19
Fig. 19

Focal line formed by the CLM LCAL (focal length, 0.7 m).

Fig. 20
Fig. 20

Imaging experiment of the spherical LCAL under white-light illumination.

Fig. 21
Fig. 21

Imaging performance of the CLM LCAL under incoherent-light illumination (focal length of the CLM LCAL, 0.7 m).

Fig. 22
Fig. 22

Rise time of the CLM LCAL.

Tables (2)

Tables Icon

Table 1 Number of Interpolating Electrodes in Each Fresnel Zonea

Tables Icon

Table 2 Focal-Line Peak Intensity Demonstrating Improvement Owing to Dithering

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

OPLr=ngr2/2f,
OPL=nd,
n=ngr2/2fd.
tLCALx=rectx/WexpjkdnLCALx,
nLCALx=nelectrodex+nlensletx.
hxi=1λ2dodi-+rectxWexpjkΦQ+ΦM×exp-jkxodo+xidixdx,
ΦQ=mL2-x22f,  mL-a2xmL+a2,
ΦM=nlensletx-x22f,  mL+a2xm+1L-a2.
Ixi=hxi2.
nx=ne+ni-nexo2x2,
Tx=expjkdneexp-jkdne-nixo2x2,
Tlensx=expjkdngexp-jkx2/2f,
f=xo22dne-ni.
fxo22dne-no.
1fsystem=1flens+1fLCAL,
fsystem=11flens+1fLCAL,
expjβ=expjmodN2πβ,  modAb=b±intb/A*A,
Tlensx=exp-jkx2/2f-N2π.
f=12dne-nixo2+4Nπkx2.
kx24Nπf12dne-nixo2+4Nπkx2.
kx24NWπfW12Wdne-ni 2+4NWπkx2.

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