Abstract

We have investigated the causes of low efficiency for optical beam steering devices based on liquid crystal decentered microlens arrays (DLAs). We show that the efficiency is effected by the relative phase of light exiting the individual lenses, the imperfect focusing of small lenses due to diffraction, the aberrations related to off-axis light going through a lens, and the diffraction spreading of light beams going through the DLA structure. A high steering efficiency of over 94.4% is demonstrated by modeling the transmitted light through the DLA with scalar diffraction theory. We also propose modified phase profiles for the lenses that are a function of angle that substantially improve the performance of these types of device over the unmodified profiles.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. E. Motamedi, M. C. Wu, and K. S. K. Pister, “Micro-opto-electro-mechanical devices and on-chip optical processing,” Opt. Eng. 36, 1282-1297 (1997).
  2. M. E. Motamedi, Park Sangtae, A. Wang, M. Dadkhah, A. P. Andrews, H. O. Marcy, M. Khoshnevisan, A. E. Chiou, R. J. Huhn, C. F. Sell, and J. G. Smits, “Development of micro-electro-mechanical optical scanner,” Opt. Eng. 36, 1346-1353(1997).
  3. T. P. Kurzweg, J. A. Martinez, S. P. Levitan, M. T. Shomsky, P. J. Marchand, and D. M. Chiarulli, “Modelling optical MEM systems,” J. Model. Simul. Microsys. 2, 21-34 (2001).
  4. V. Milanovic, M. Last, and K. S. J. Pister, “Laterally actuated torsional micromirrors for large static deflection,” IEEE Photon. Technol. Lett. 15, 245-247 (2003).
    [CrossRef]
  5. P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268-298 (1996).
    [CrossRef]
  6. D. M. Burns,V. M. Bright, S. C. Gustafson, and E. A. Watson, “Optical beam steering using surface micromachined gratings and optical phased arrays,” Proc. SPIE 3131, 99-110 (1997).
  7. D. Resler, D. Hobbs, R. Sharp, L. Friedman, and T. Dorschner, “High efficiency liquid crystal optical phased array beam steering,” Opt. Lett. 21, 689-611 (1996).
    [CrossRef]
  8. E. A. Watson, “Analysis of beam steering with decentered microlens arrays,” Opt. Eng. 32, 2665-2670 (1993).
  9. W. C. Goltsos and M. Holz, “Agile beam steering using binary optics microlens arrays,” Opt. Eng. 29, 1392-1397 (1990).
  10. J. L. Gibson, B. D. Duncan, E. A. Watson, and J. L. Loomis, “Wide-angle decentered lens beam steering for infrared countermeasures applications,” Opt. Eng. 43, 2312-2321 (2004).
  11. P. F. McManamon and E. A. Watson, “Optical beam steering using phased array technology,” Proc. SPIE 3131, 90-98(1997).
  12. R. Guenther, Modern Optics (Wiley, 1990), pp. 345-346.
  13. E. Hecht, Optics, 2nd ed. (Addison-Wesley, 1987), pp. 459-463.

2004 (1)

J. L. Gibson, B. D. Duncan, E. A. Watson, and J. L. Loomis, “Wide-angle decentered lens beam steering for infrared countermeasures applications,” Opt. Eng. 43, 2312-2321 (2004).

2003 (1)

V. Milanovic, M. Last, and K. S. J. Pister, “Laterally actuated torsional micromirrors for large static deflection,” IEEE Photon. Technol. Lett. 15, 245-247 (2003).
[CrossRef]

2001 (1)

T. P. Kurzweg, J. A. Martinez, S. P. Levitan, M. T. Shomsky, P. J. Marchand, and D. M. Chiarulli, “Modelling optical MEM systems,” J. Model. Simul. Microsys. 2, 21-34 (2001).

1997 (4)

M. E. Motamedi, M. C. Wu, and K. S. K. Pister, “Micro-opto-electro-mechanical devices and on-chip optical processing,” Opt. Eng. 36, 1282-1297 (1997).

M. E. Motamedi, Park Sangtae, A. Wang, M. Dadkhah, A. P. Andrews, H. O. Marcy, M. Khoshnevisan, A. E. Chiou, R. J. Huhn, C. F. Sell, and J. G. Smits, “Development of micro-electro-mechanical optical scanner,” Opt. Eng. 36, 1346-1353(1997).

P. F. McManamon and E. A. Watson, “Optical beam steering using phased array technology,” Proc. SPIE 3131, 90-98(1997).

D. M. Burns,V. M. Bright, S. C. Gustafson, and E. A. Watson, “Optical beam steering using surface micromachined gratings and optical phased arrays,” Proc. SPIE 3131, 99-110 (1997).

1996 (2)

D. Resler, D. Hobbs, R. Sharp, L. Friedman, and T. Dorschner, “High efficiency liquid crystal optical phased array beam steering,” Opt. Lett. 21, 689-611 (1996).
[CrossRef]

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268-298 (1996).
[CrossRef]

1993 (1)

E. A. Watson, “Analysis of beam steering with decentered microlens arrays,” Opt. Eng. 32, 2665-2670 (1993).

1990 (1)

W. C. Goltsos and M. Holz, “Agile beam steering using binary optics microlens arrays,” Opt. Eng. 29, 1392-1397 (1990).

Andrews, A. P.

M. E. Motamedi, Park Sangtae, A. Wang, M. Dadkhah, A. P. Andrews, H. O. Marcy, M. Khoshnevisan, A. E. Chiou, R. J. Huhn, C. F. Sell, and J. G. Smits, “Development of micro-electro-mechanical optical scanner,” Opt. Eng. 36, 1346-1353(1997).

Bright, V. M.

D. M. Burns,V. M. Bright, S. C. Gustafson, and E. A. Watson, “Optical beam steering using surface micromachined gratings and optical phased arrays,” Proc. SPIE 3131, 99-110 (1997).

Burns, D. M.

D. M. Burns,V. M. Bright, S. C. Gustafson, and E. A. Watson, “Optical beam steering using surface micromachined gratings and optical phased arrays,” Proc. SPIE 3131, 99-110 (1997).

Chiarulli, D. M.

T. P. Kurzweg, J. A. Martinez, S. P. Levitan, M. T. Shomsky, P. J. Marchand, and D. M. Chiarulli, “Modelling optical MEM systems,” J. Model. Simul. Microsys. 2, 21-34 (2001).

Chiou, A. E.

M. E. Motamedi, Park Sangtae, A. Wang, M. Dadkhah, A. P. Andrews, H. O. Marcy, M. Khoshnevisan, A. E. Chiou, R. J. Huhn, C. F. Sell, and J. G. Smits, “Development of micro-electro-mechanical optical scanner,” Opt. Eng. 36, 1346-1353(1997).

Corkum, D. L.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268-298 (1996).
[CrossRef]

Dadkhah, M.

M. E. Motamedi, Park Sangtae, A. Wang, M. Dadkhah, A. P. Andrews, H. O. Marcy, M. Khoshnevisan, A. E. Chiou, R. J. Huhn, C. F. Sell, and J. G. Smits, “Development of micro-electro-mechanical optical scanner,” Opt. Eng. 36, 1346-1353(1997).

Dorschner, T.

Dorschner, T. A.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268-298 (1996).
[CrossRef]

Duncan, B. D.

J. L. Gibson, B. D. Duncan, E. A. Watson, and J. L. Loomis, “Wide-angle decentered lens beam steering for infrared countermeasures applications,” Opt. Eng. 43, 2312-2321 (2004).

Friedman, L.

Friedman, L. J.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268-298 (1996).
[CrossRef]

Gibson, J. L.

J. L. Gibson, B. D. Duncan, E. A. Watson, and J. L. Loomis, “Wide-angle decentered lens beam steering for infrared countermeasures applications,” Opt. Eng. 43, 2312-2321 (2004).

Goltsos, W. C.

W. C. Goltsos and M. Holz, “Agile beam steering using binary optics microlens arrays,” Opt. Eng. 29, 1392-1397 (1990).

Guenther, R.

R. Guenther, Modern Optics (Wiley, 1990), pp. 345-346.

Gustafson, S. C.

D. M. Burns,V. M. Bright, S. C. Gustafson, and E. A. Watson, “Optical beam steering using surface micromachined gratings and optical phased arrays,” Proc. SPIE 3131, 99-110 (1997).

Hecht, E.

E. Hecht, Optics, 2nd ed. (Addison-Wesley, 1987), pp. 459-463.

Hobbs, D.

Hobbs, D. S.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268-298 (1996).
[CrossRef]

Holz, M.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268-298 (1996).
[CrossRef]

W. C. Goltsos and M. Holz, “Agile beam steering using binary optics microlens arrays,” Opt. Eng. 29, 1392-1397 (1990).

Huhn, R. J.

M. E. Motamedi, Park Sangtae, A. Wang, M. Dadkhah, A. P. Andrews, H. O. Marcy, M. Khoshnevisan, A. E. Chiou, R. J. Huhn, C. F. Sell, and J. G. Smits, “Development of micro-electro-mechanical optical scanner,” Opt. Eng. 36, 1346-1353(1997).

Khoshnevisan, M.

M. E. Motamedi, Park Sangtae, A. Wang, M. Dadkhah, A. P. Andrews, H. O. Marcy, M. Khoshnevisan, A. E. Chiou, R. J. Huhn, C. F. Sell, and J. G. Smits, “Development of micro-electro-mechanical optical scanner,” Opt. Eng. 36, 1346-1353(1997).

Kurzweg, T. P.

T. P. Kurzweg, J. A. Martinez, S. P. Levitan, M. T. Shomsky, P. J. Marchand, and D. M. Chiarulli, “Modelling optical MEM systems,” J. Model. Simul. Microsys. 2, 21-34 (2001).

Last, M.

V. Milanovic, M. Last, and K. S. J. Pister, “Laterally actuated torsional micromirrors for large static deflection,” IEEE Photon. Technol. Lett. 15, 245-247 (2003).
[CrossRef]

Levitan, S. P.

T. P. Kurzweg, J. A. Martinez, S. P. Levitan, M. T. Shomsky, P. J. Marchand, and D. M. Chiarulli, “Modelling optical MEM systems,” J. Model. Simul. Microsys. 2, 21-34 (2001).

Liberman, S.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268-298 (1996).
[CrossRef]

Loomis, J. L.

J. L. Gibson, B. D. Duncan, E. A. Watson, and J. L. Loomis, “Wide-angle decentered lens beam steering for infrared countermeasures applications,” Opt. Eng. 43, 2312-2321 (2004).

Marchand, P. J.

T. P. Kurzweg, J. A. Martinez, S. P. Levitan, M. T. Shomsky, P. J. Marchand, and D. M. Chiarulli, “Modelling optical MEM systems,” J. Model. Simul. Microsys. 2, 21-34 (2001).

Marcy, H. O.

M. E. Motamedi, Park Sangtae, A. Wang, M. Dadkhah, A. P. Andrews, H. O. Marcy, M. Khoshnevisan, A. E. Chiou, R. J. Huhn, C. F. Sell, and J. G. Smits, “Development of micro-electro-mechanical optical scanner,” Opt. Eng. 36, 1346-1353(1997).

Martinez, J. A.

T. P. Kurzweg, J. A. Martinez, S. P. Levitan, M. T. Shomsky, P. J. Marchand, and D. M. Chiarulli, “Modelling optical MEM systems,” J. Model. Simul. Microsys. 2, 21-34 (2001).

McManamon, P. F.

P. F. McManamon and E. A. Watson, “Optical beam steering using phased array technology,” Proc. SPIE 3131, 90-98(1997).

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268-298 (1996).
[CrossRef]

Milanovic, V.

V. Milanovic, M. Last, and K. S. J. Pister, “Laterally actuated torsional micromirrors for large static deflection,” IEEE Photon. Technol. Lett. 15, 245-247 (2003).
[CrossRef]

Motamedi, M. E.

M. E. Motamedi, M. C. Wu, and K. S. K. Pister, “Micro-opto-electro-mechanical devices and on-chip optical processing,” Opt. Eng. 36, 1282-1297 (1997).

M. E. Motamedi, Park Sangtae, A. Wang, M. Dadkhah, A. P. Andrews, H. O. Marcy, M. Khoshnevisan, A. E. Chiou, R. J. Huhn, C. F. Sell, and J. G. Smits, “Development of micro-electro-mechanical optical scanner,” Opt. Eng. 36, 1346-1353(1997).

Nguyen, H. Q.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268-298 (1996).
[CrossRef]

Pister, K. S. J.

V. Milanovic, M. Last, and K. S. J. Pister, “Laterally actuated torsional micromirrors for large static deflection,” IEEE Photon. Technol. Lett. 15, 245-247 (2003).
[CrossRef]

Pister, K. S. K.

M. E. Motamedi, M. C. Wu, and K. S. K. Pister, “Micro-opto-electro-mechanical devices and on-chip optical processing,” Opt. Eng. 36, 1282-1297 (1997).

Resler, D.

Resler, D. P.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268-298 (1996).
[CrossRef]

Sangtae, Park

M. E. Motamedi, Park Sangtae, A. Wang, M. Dadkhah, A. P. Andrews, H. O. Marcy, M. Khoshnevisan, A. E. Chiou, R. J. Huhn, C. F. Sell, and J. G. Smits, “Development of micro-electro-mechanical optical scanner,” Opt. Eng. 36, 1346-1353(1997).

Sell, C. F.

M. E. Motamedi, Park Sangtae, A. Wang, M. Dadkhah, A. P. Andrews, H. O. Marcy, M. Khoshnevisan, A. E. Chiou, R. J. Huhn, C. F. Sell, and J. G. Smits, “Development of micro-electro-mechanical optical scanner,” Opt. Eng. 36, 1346-1353(1997).

Sharp, R.

Sharp, R. C.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268-298 (1996).
[CrossRef]

Shomsky, M. T.

T. P. Kurzweg, J. A. Martinez, S. P. Levitan, M. T. Shomsky, P. J. Marchand, and D. M. Chiarulli, “Modelling optical MEM systems,” J. Model. Simul. Microsys. 2, 21-34 (2001).

Smits, J. G.

M. E. Motamedi, Park Sangtae, A. Wang, M. Dadkhah, A. P. Andrews, H. O. Marcy, M. Khoshnevisan, A. E. Chiou, R. J. Huhn, C. F. Sell, and J. G. Smits, “Development of micro-electro-mechanical optical scanner,” Opt. Eng. 36, 1346-1353(1997).

Wang, A.

M. E. Motamedi, Park Sangtae, A. Wang, M. Dadkhah, A. P. Andrews, H. O. Marcy, M. Khoshnevisan, A. E. Chiou, R. J. Huhn, C. F. Sell, and J. G. Smits, “Development of micro-electro-mechanical optical scanner,” Opt. Eng. 36, 1346-1353(1997).

Watson, E. A.

J. L. Gibson, B. D. Duncan, E. A. Watson, and J. L. Loomis, “Wide-angle decentered lens beam steering for infrared countermeasures applications,” Opt. Eng. 43, 2312-2321 (2004).

P. F. McManamon and E. A. Watson, “Optical beam steering using phased array technology,” Proc. SPIE 3131, 90-98(1997).

D. M. Burns,V. M. Bright, S. C. Gustafson, and E. A. Watson, “Optical beam steering using surface micromachined gratings and optical phased arrays,” Proc. SPIE 3131, 99-110 (1997).

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268-298 (1996).
[CrossRef]

E. A. Watson, “Analysis of beam steering with decentered microlens arrays,” Opt. Eng. 32, 2665-2670 (1993).

Wu, M. C.

M. E. Motamedi, M. C. Wu, and K. S. K. Pister, “Micro-opto-electro-mechanical devices and on-chip optical processing,” Opt. Eng. 36, 1282-1297 (1997).

IEEE Photon. Technol. Lett. (1)

V. Milanovic, M. Last, and K. S. J. Pister, “Laterally actuated torsional micromirrors for large static deflection,” IEEE Photon. Technol. Lett. 15, 245-247 (2003).
[CrossRef]

J. Model. Simul. Microsys. (1)

T. P. Kurzweg, J. A. Martinez, S. P. Levitan, M. T. Shomsky, P. J. Marchand, and D. M. Chiarulli, “Modelling optical MEM systems,” J. Model. Simul. Microsys. 2, 21-34 (2001).

Opt. Eng. (5)

M. E. Motamedi, M. C. Wu, and K. S. K. Pister, “Micro-opto-electro-mechanical devices and on-chip optical processing,” Opt. Eng. 36, 1282-1297 (1997).

M. E. Motamedi, Park Sangtae, A. Wang, M. Dadkhah, A. P. Andrews, H. O. Marcy, M. Khoshnevisan, A. E. Chiou, R. J. Huhn, C. F. Sell, and J. G. Smits, “Development of micro-electro-mechanical optical scanner,” Opt. Eng. 36, 1346-1353(1997).

E. A. Watson, “Analysis of beam steering with decentered microlens arrays,” Opt. Eng. 32, 2665-2670 (1993).

W. C. Goltsos and M. Holz, “Agile beam steering using binary optics microlens arrays,” Opt. Eng. 29, 1392-1397 (1990).

J. L. Gibson, B. D. Duncan, E. A. Watson, and J. L. Loomis, “Wide-angle decentered lens beam steering for infrared countermeasures applications,” Opt. Eng. 43, 2312-2321 (2004).

Opt. Lett. (1)

Proc. IEEE (1)

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268-298 (1996).
[CrossRef]

Proc. SPIE (2)

D. M. Burns,V. M. Bright, S. C. Gustafson, and E. A. Watson, “Optical beam steering using surface micromachined gratings and optical phased arrays,” Proc. SPIE 3131, 99-110 (1997).

P. F. McManamon and E. A. Watson, “Optical beam steering using phased array technology,” Proc. SPIE 3131, 90-98(1997).

Other (2)

R. Guenther, Modern Optics (Wiley, 1990), pp. 345-346.

E. Hecht, Optics, 2nd ed. (Addison-Wesley, 1987), pp. 459-463.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (20)

Fig. 1
Fig. 1

(a) All-positive three-layer DLA and (b) mixed three-layer DLA.

Fig. 2
Fig. 2

Beam steering with (a) all-positive decentered macroscopic lenses and (b) mixed decentered macroscopic lenses.

Fig. 3
Fig. 3

Off-axis aberration of a single positive lens.

Fig. 4
Fig. 4

Off-axis aberration of a single negative lens.

Fig. 5
Fig. 5

Phase-up correction for third positive microlens layer: (a) ideal phase for designed wavelength, (b) phase of E 3 b with 2 π phase-up, (c) unfolded phase of E 3 b , (d) phase profile of E 3 b , (e) third layer lens.

Fig. 6
Fig. 6

Illustration of mirror symmetry correction of the second lens phase profile.

Fig. 7
Fig. 7

Scalar integral approach of three-layer DLA.

Fig. 8
Fig. 8

Structure diagram of a single positive LC microlens.

Fig. 9
Fig. 9

Calculated OPD of a positive LC cell with respect to the applied voltage.

Fig. 10
Fig. 10

Near-field intensity profile for a nonsteered positive DLA (five lenses in each layer) as a function of OPD and diameter.

Fig. 11
Fig. 11

(a) Incident Gaussian beam profile on a system with five lenses in each layer ( r = 552 μm ) and (b) OPD range of a robust all-positive microlens array.

Fig. 12
Fig. 12

All-positive microlens array with 552 μm radius: (a) far-field intensity for nonsteering and (b) diffraction efficiency with respect to deflection angle.

Fig. 13
Fig. 13

(a) Incident square wave and first layer phase profile of mixed microlens array and (b) out-of-lens leakage right before the second layer.

Fig. 14
Fig. 14

Out-of-lens leakage with respect to the diameter of microlens.

Fig. 15
Fig. 15

(a) OPD profile of a positive LC layer and (b) director configuration of a positive LC layer.

Fig. 16
Fig. 16

(a) Voltage profile for a positive LC layer and (b) voltage profile for a negative LC layer with steering angle 0.64 ° .

Fig. 17
Fig. 17

(a) OPD profile for a negative LC lens with steering angle 0 ° and (b) OPD profile for a negative LC lens with steering angle 0.64 ° .

Fig. 18
Fig. 18

Contribution to the final DE by the three different correction approaches.

Fig. 19
Fig. 19

Schematic diagram of beam profile measurement of LC microlens: a, 632.8 nm He–Ne laser; b, beam expander; c, polarizer; d, LC microlens; e, polarizer; f, signal collect lens system; g, PULNiX camera; h, amplifier; i, computer controlled by LabVIEW program.

Fig. 20
Fig. 20

(a) Beam profile image of the 25 μm positive microlens LC layer at focal plane and (b) intensity profile with comparison of simulated results.

Equations (33)

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

Γ = ( π w 2 λ R ) 1 ,
Γ = ( π w 2 λ R ) = ( π D 2 4 λ f ) D 2 λ f 1.
Γ = D 2 λ f = 8 OPD max λ 1.
OPD Δ + = DC = OD OC = OB + B D OC = OB + B D OB .
OPD Δ + ( x ) = f 3 2 + d 2 + x sin φ 3 f 3 2 + ( x d ) 2 = x sin φ 3 ( f 3 2 + ( x + f 3 tan φ 3 ) 2 f 3 2 + ( f 3 tan φ 3 ) 2 ) .
OPD Δ = DC = A C A D = A B A B B D .
OPD Δ ( x ) = f 2 2 + ( x + d ) 2 f 2 2 + d 2 x sin φ 2 = f 2 2 + ( x + f 2 tan φ 2 ) 2 f 2 2 + ( f 2 tan φ 2 ) 2 x sin φ 2 .
OPD 0 + ( x ) = f 3 f 3 2 + x 2 ,
OPD 0 ( x ) = f 2 2 + x 2 f 2 .
OPD Δ 3 ( x 3 ) = f 3 2 + ( f 3 tan φ 3 ) 2 f 3 + x 3 sin φ 3 ( f 3 2 + ( x 3 + f 3 tan φ 3 ) 2 f 3 2 + x 3 2 ) ,
OPD Δ 3 ( x 3 ) = x 3 sin φ 3 ( f 3 2 + ( x 3 + f 3 tan φ 3 ) 2 f 3 2 + x 3 2 ) .
OPD Δ 2 ( x 2 ) = f 2 2 + ( x 2 + f 2 tan φ 2 ) 2 f 2 2 + x 2 2 x 2 sin φ 2 ) .
P 2 b ( x ) = P 2 a ( x ) + P 2 ( x ) .
P 2 a ( x ) = P 2 b ( x ) + 2 h , P 2 a ( x ) = ( P 2 a ( x ) + P 2 ( x ) ) + 2 h ,
P 2 ( x ) = 2 P 2 a ( x ) + 2 h .
OPD Δ mirror 2 ( x ) = P 2 ( x ) OPD 2 ( x ) .
OPD 1 ( x 1 ) = r 2 + f 1 2 x 1 2 + f 1 2 .
OPD 2 ( x 2 ) = x 2 2 + f 2 2 f 2 .
OPD 1 ( x 1 ) = x 1 2 2 f 1 + C ,
OPD 2 ( x 2 ) = + x 2 2 2 f 2 + C ,
OPD 1 ( x 1 ) = OPD 1 ( x 1 ) , OPD ( x 2 ) = OPD 2 ( x 2 ) + OPD Δ 2 ( x 2 ) + OPD Δ mirror 2 ( x 1 ) , OPD 3 ( x 3 ) = OPD 3 ( x 3 ) + OPD Δ 3 ( x 3 ) + OPD Δ phase _ up 3 ( x 3 ) .
E 0 ( x 1 ) = e 2 ( x 1 / w ) 2 ,
w = 2 r N lens Γ ,
E 1 ( x 1 ) = E 0 ( x 1 ) e i k ( OPD 1 ( x 1 ) ) .
E 2 a ( x 2 ) = E 1 ( x 1 ) ( 1 + cos θ 1 ( x 1 ) 2 ) e i k r 1 r 1 d x 1 .
E 2 b ( x 2 ) = E 2 a ( x 2 ) e i k ( OPD 2 ( x 2 ) ) .
E 3 a ( x 3 ) = E 2 b ( x 2 ) ( 1 + cos θ 2 ( x 2 ) 2 ) e i k r 2 r 2 d x 2 .
E 3 b ( x 3 ) = E 3 a ( x 3 ) e i k ( OPD 3 ( x 3 ) ) .
[ f G ] n i = δ f G δ n i = f G n i j = x , y , z d d j [ f G ( d n i d j ) ] , i = x , y , z .
γ d n i d t = δ f G δ n i + λ n i , i = x , y , z .
γ Δ n i Δ t = [ f G ] n i Δ n i = Δ t γ [ f G ] n i n i new = n i old Δ t γ [ f G ] n i .
OPD ( x ) = 0 d ( n eff ( x , z ) n o ) d z ,
n eff ( x , z ) = n o n e n o 2 cos 2 θ ( x , z ) + n e 2 sin 2 θ ( x , z ) ,

Metrics