Abstract

The use of optical interconnects for communication between points on a microchip is motivated by system-level interconnect modeling showing the saturation of metal wire capacity at the global layer. Free-space optical solutions are analyzed for intrachip communication at the global layer. A multiscale solution comprising microlenses, etched compound slope microprisms, and a curved mirror is shown to outperform a single-scale alternative. Microprisms are designed and fabricated and inserted into an optical setup apparatus to experimentally validate the concept. The multiscale free-space system is shown to have the potential to provide the bandwidth density and configuration flexibility required for global communication in future generations of microchips.

© 2006 Optical Society of America

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2004

D. Neilson, R. Frahm, P. Kolodner, C. Bolle, R. Ryf, J. Kim, A. Papazian, C. Nuzman, A. Gasparyan, N. Basavanhally, V. Aksyuk, and J. Gates, "256 × 256 port optical cross-connect subsystem," J. Lightwave Technol. 22, 1499-1509 (2004).
[CrossRef]

M. Lung Mui and K. Banerjee, "A global interconnect optimization scheme for nanometer scale VLSI with implications for latency, bandwidth, and power dissipation," IEEE Trans. Electron Devices 51, 195-203 (2004).
[CrossRef]

T. Dillon, A. Balcha, J. Murakowiski, and D. Prather, "Continuous-tone grayscale mask fabrication using high-energy-beam-sensitive glass," J. Microlithogr. Microfabr. Microsyst. 3, 550-554 (2004).
[CrossRef]

2003

E. Cassan, S. Laval, S. Lardenois, and A. Koster, "On-chip optical interconnects with compact and low-loss light distribution in silicon-on-insulator rib waveguides," IEEE J. Sel. Top. Quantum Electron. 9, 460-464 (2003).
[CrossRef]

C. Debaes, M. Vervaeke, V. Baukens, H. Ottevaere, P. Vynck, P. Tuteleers, B. Volckaerts, W. Meeus, M. Brunfaut, J. Van Campenhout, A. Hermanne, and H. Thienpont, "Low-cost micro-optical modules for MCM level optical interconnects," IEEE J. Sel. Top. Quantum Electron. 9, 518-530 (2003).
[CrossRef]

2000

P. Chiristie and D. Stroobandt, "The interpretation and application of Rent's rule," IEEE Trans. Very Large Scale Integr. (VSLI) Syst., 8, 639-648 (2000).
[CrossRef]

D. A. B. Miller, "Rationale and challenges for optical interconnects to silicon chips," Proc. IEEE 88, 728-749 (2000).
[CrossRef]

1999

1998

1996

M. W. Haney, M. P. Christensen, P. Milojkovic, G. J. Fokken, M. Vickberg, B. K. Gilbert, J. Rieve, J. Ekman, P. Chandramani, and F. Kiamilev, "Description and evaluation of the FAST-net smart pixel-based optical interconnection prototype," Proc. IEEE 88, 819-828 (1996).
[CrossRef]

N. Vasseghi, K. Yeager, E. Sarto, and M. Seddighnezhad, "A 200 Mhz superscaler RISC microprocessor," IEEE J. Solid-State Circuits 31, 1675-1686 (1996).
[CrossRef]

1983

1981

W. E. Donath, "Wire length distribution for placements of computer logic," IBM J. Res. Dev. 25, 152-155 (1981).
[CrossRef]

1971

B. S. Landman and R. L. Russo, "On a pin versus block relationship for partitions of logic graphs," IEEE Trans. Comput. C-20, 1496-1479 (1971).
[CrossRef]

1965

G. E. Moore, "Cramming more components onto integrated circuits," Fairchild Semiconductors' Electronics 38, 114-117 (1965).

Agarwal, V.

V. Agarwal, M. S. Hrishikesh, S. W. Keckler, and D. Burger, "Clock rate versus IPC: the end of the road of conventional microarchitectures," in Proceedings of the 27th ACM Annual International Symposium on Computer Architecture (2000).

Aksyuk, V.

Balcha, A.

T. Dillon, A. Balcha, J. Murakowiski, and D. Prather, "Continuous-tone grayscale mask fabrication using high-energy-beam-sensitive glass," J. Microlithogr. Microfabr. Microsyst. 3, 550-554 (2004).
[CrossRef]

Banerjee, K.

M. Lung Mui and K. Banerjee, "A global interconnect optimization scheme for nanometer scale VLSI with implications for latency, bandwidth, and power dissipation," IEEE Trans. Electron Devices 51, 195-203 (2004).
[CrossRef]

Basavanhally, N.

Baukens, V.

C. Debaes, M. Vervaeke, V. Baukens, H. Ottevaere, P. Vynck, P. Tuteleers, B. Volckaerts, W. Meeus, M. Brunfaut, J. Van Campenhout, A. Hermanne, and H. Thienpont, "Low-cost micro-optical modules for MCM level optical interconnects," IEEE J. Sel. Top. Quantum Electron. 9, 518-530 (2003).
[CrossRef]

Bolle, C.

Bridgen, J. M.

Brunfaut, M.

C. Debaes, M. Vervaeke, V. Baukens, H. Ottevaere, P. Vynck, P. Tuteleers, B. Volckaerts, W. Meeus, M. Brunfaut, J. Van Campenhout, A. Hermanne, and H. Thienpont, "Low-cost micro-optical modules for MCM level optical interconnects," IEEE J. Sel. Top. Quantum Electron. 9, 518-530 (2003).
[CrossRef]

Burger, D.

V. Agarwal, M. S. Hrishikesh, S. W. Keckler, and D. Burger, "Clock rate versus IPC: the end of the road of conventional microarchitectures," in Proceedings of the 27th ACM Annual International Symposium on Computer Architecture (2000).

Cassan, E.

E. Cassan, S. Laval, S. Lardenois, and A. Koster, "On-chip optical interconnects with compact and low-loss light distribution in silicon-on-insulator rib waveguides," IEEE J. Sel. Top. Quantum Electron. 9, 460-464 (2003).
[CrossRef]

Chandramani, P.

M. W. Haney, M. P. Christensen, P. Milojkovic, G. J. Fokken, M. Vickberg, B. K. Gilbert, J. Rieve, J. Ekman, P. Chandramani, and F. Kiamilev, "Description and evaluation of the FAST-net smart pixel-based optical interconnection prototype," Proc. IEEE 88, 819-828 (1996).
[CrossRef]

Chiarulli, D. M.

Chiristie, P.

P. Chiristie and D. Stroobandt, "The interpretation and application of Rent's rule," IEEE Trans. Very Large Scale Integr. (VSLI) Syst., 8, 639-648 (2000).
[CrossRef]

Christensen, M. P.

M. W. Haney, M. P. Christensen, P. Milojkovic, G. J. Fokken, M. Vickberg, B. K. Gilbert, J. Rieve, J. Ekman, P. Chandramani, and F. Kiamilev, "Description and evaluation of the FAST-net smart pixel-based optical interconnection prototype," Proc. IEEE 88, 819-828 (1996).
[CrossRef]

Debaes, C.

C. Debaes, M. Vervaeke, V. Baukens, H. Ottevaere, P. Vynck, P. Tuteleers, B. Volckaerts, W. Meeus, M. Brunfaut, J. Van Campenhout, A. Hermanne, and H. Thienpont, "Low-cost micro-optical modules for MCM level optical interconnects," IEEE J. Sel. Top. Quantum Electron. 9, 518-530 (2003).
[CrossRef]

Dillon, T.

T. Dillon, A. Balcha, J. Murakowiski, and D. Prather, "Continuous-tone grayscale mask fabrication using high-energy-beam-sensitive glass," J. Microlithogr. Microfabr. Microsyst. 3, 550-554 (2004).
[CrossRef]

M. Iqbal, T. Dillon, M. J. McFadden, D. Prather, and M. W. Haney, "Fabrication of beam steering elements, couplers and lenses with grayscale lithography using HEBS glass," in Proceedings of 89th OSA Annual Meeting (Optical Society of America, 2005).

Donath, W. E.

W. E. Donath, "Wire length distribution for placements of computer logic," IBM J. Res. Dev. 25, 152-155 (1981).
[CrossRef]

Ekman, J.

M. W. Haney, M. P. Christensen, P. Milojkovic, G. J. Fokken, M. Vickberg, B. K. Gilbert, J. Rieve, J. Ekman, P. Chandramani, and F. Kiamilev, "Description and evaluation of the FAST-net smart pixel-based optical interconnection prototype," Proc. IEEE 88, 819-828 (1996).
[CrossRef]

Fan, C.

Fokken, G. J.

M. W. Haney, M. P. Christensen, P. Milojkovic, G. J. Fokken, M. Vickberg, B. K. Gilbert, J. Rieve, J. Ekman, P. Chandramani, and F. Kiamilev, "Description and evaluation of the FAST-net smart pixel-based optical interconnection prototype," Proc. IEEE 88, 819-828 (1996).
[CrossRef]

Frahm, R.

Gasparyan, A.

Gates, J.

Gilbert, B. K.

M. W. Haney, M. P. Christensen, P. Milojkovic, G. J. Fokken, M. Vickberg, B. K. Gilbert, J. Rieve, J. Ekman, P. Chandramani, and F. Kiamilev, "Description and evaluation of the FAST-net smart pixel-based optical interconnection prototype," Proc. IEEE 88, 819-828 (1996).
[CrossRef]

Gimkiewicz, C.

Hagedorn, D.

Haney, M. W.

M. W. Haney, M. P. Christensen, P. Milojkovic, G. J. Fokken, M. Vickberg, B. K. Gilbert, J. Rieve, J. Ekman, P. Chandramani, and F. Kiamilev, "Description and evaluation of the FAST-net smart pixel-based optical interconnection prototype," Proc. IEEE 88, 819-828 (1996).
[CrossRef]

M. Iqbal, M. J. McFadden, and M. W. Haney are preparing a manuscript with the title "Interconnect prediction for performance evaluation of intrachip global communication".

M. Iqbal, M. J. McFadden, and M. W. Haney, "Intrachip global interconnects and the saturation of Moore's law," in Proceedings of IEEE-LEOS Summer Topical: Optical Interconnects and VLSI Photonics (IEEE, 2004).
[CrossRef]

M. J. McFadden, M. Iqbal, and M. W. Haney, "Multi-scale optical interconnects for intrachip global communication," in Proceedings of IEEE-LEOS Summer Topical: Optical Interconnects and VLSI Photonics (IEEE, 2004), pp. 71-72.
[CrossRef]

M. Iqbal, T. Dillon, M. J. McFadden, D. Prather, and M. W. Haney, "Fabrication of beam steering elements, couplers and lenses with grayscale lithography using HEBS glass," in Proceedings of 89th OSA Annual Meeting (Optical Society of America, 2005).

M. W. Haney, M. J. McFadden, and M. Iqbal, "An arbitrarily configurable optical interconnect fabric for intra-chip global communication," in Proceedings of OSA Optics in Computing (Optical Society of America, 2003).

Hermanne, A.

C. Debaes, M. Vervaeke, V. Baukens, H. Ottevaere, P. Vynck, P. Tuteleers, B. Volckaerts, W. Meeus, M. Brunfaut, J. Van Campenhout, A. Hermanne, and H. Thienpont, "Low-cost micro-optical modules for MCM level optical interconnects," IEEE J. Sel. Top. Quantum Electron. 9, 518-530 (2003).
[CrossRef]

Hrishikesh, M. S.

V. Agarwal, M. S. Hrishikesh, S. W. Keckler, and D. Burger, "Clock rate versus IPC: the end of the road of conventional microarchitectures," in Proceedings of the 27th ACM Annual International Symposium on Computer Architecture (2000).

Iqbal, M.

M. Iqbal, M. J. McFadden, and M. W. Haney, "Intrachip global interconnects and the saturation of Moore's law," in Proceedings of IEEE-LEOS Summer Topical: Optical Interconnects and VLSI Photonics (IEEE, 2004).
[CrossRef]

M. Iqbal, T. Dillon, M. J. McFadden, D. Prather, and M. W. Haney, "Fabrication of beam steering elements, couplers and lenses with grayscale lithography using HEBS glass," in Proceedings of 89th OSA Annual Meeting (Optical Society of America, 2005).

M. J. McFadden, M. Iqbal, and M. W. Haney, "Multi-scale optical interconnects for intrachip global communication," in Proceedings of IEEE-LEOS Summer Topical: Optical Interconnects and VLSI Photonics (IEEE, 2004), pp. 71-72.
[CrossRef]

M. W. Haney, M. J. McFadden, and M. Iqbal, "An arbitrarily configurable optical interconnect fabric for intra-chip global communication," in Proceedings of OSA Optics in Computing (Optical Society of America, 2003).

M. Iqbal, M. J. McFadden, and M. W. Haney are preparing a manuscript with the title "Interconnect prediction for performance evaluation of intrachip global communication".

Jahns, J.

Keckler, S. W.

V. Agarwal, M. S. Hrishikesh, S. W. Keckler, and D. Burger, "Clock rate versus IPC: the end of the road of conventional microarchitectures," in Proceedings of the 27th ACM Annual International Symposium on Computer Architecture (2000).

Keutzer, K.

D. Sylvester and K. Keutzer, "Getting to the bottom of deep submicron II: a global wiring paradigm," in Proceedings of the International Symposium on Physical Design (1999).

Kiamilev, F.

M. W. Haney, M. P. Christensen, P. Milojkovic, G. J. Fokken, M. Vickberg, B. K. Gilbert, J. Rieve, J. Ekman, P. Chandramani, and F. Kiamilev, "Description and evaluation of the FAST-net smart pixel-based optical interconnection prototype," Proc. IEEE 88, 819-828 (1996).
[CrossRef]

Kim, J.

Kley, E.-B.

Kolodner, P.

Koster, A.

E. Cassan, S. Laval, S. Lardenois, and A. Koster, "On-chip optical interconnects with compact and low-loss light distribution in silicon-on-insulator rib waveguides," IEEE J. Sel. Top. Quantum Electron. 9, 460-464 (2003).
[CrossRef]

Kurzweg, T. P.

Landman, B. S.

B. S. Landman and R. L. Russo, "On a pin versus block relationship for partitions of logic graphs," IEEE Trans. Comput. C-20, 1496-1479 (1971).
[CrossRef]

Lardenois, S.

E. Cassan, S. Laval, S. Lardenois, and A. Koster, "On-chip optical interconnects with compact and low-loss light distribution in silicon-on-insulator rib waveguides," IEEE J. Sel. Top. Quantum Electron. 9, 460-464 (2003).
[CrossRef]

Laval, S.

E. Cassan, S. Laval, S. Lardenois, and A. Koster, "On-chip optical interconnects with compact and low-loss light distribution in silicon-on-insulator rib waveguides," IEEE J. Sel. Top. Quantum Electron. 9, 460-464 (2003).
[CrossRef]

Levitan, S. P.

Marchand, P. J.

Martinez, J. A.

McCormick, F. B.

McFadden, M. J.

M. Iqbal, M. J. McFadden, and M. W. Haney, "Intrachip global interconnects and the saturation of Moore's law," in Proceedings of IEEE-LEOS Summer Topical: Optical Interconnects and VLSI Photonics (IEEE, 2004).
[CrossRef]

M. J. McFadden, "Multi-scale free-space optical interconnects for intrachip global communication," Ph.D. dissertation (University of Delaware, 2005).

M. W. Haney, M. J. McFadden, and M. Iqbal, "An arbitrarily configurable optical interconnect fabric for intra-chip global communication," in Proceedings of OSA Optics in Computing (Optical Society of America, 2003).

M. Iqbal, T. Dillon, M. J. McFadden, D. Prather, and M. W. Haney, "Fabrication of beam steering elements, couplers and lenses with grayscale lithography using HEBS glass," in Proceedings of 89th OSA Annual Meeting (Optical Society of America, 2005).

M. J. McFadden, M. Iqbal, and M. W. Haney, "Multi-scale optical interconnects for intrachip global communication," in Proceedings of IEEE-LEOS Summer Topical: Optical Interconnects and VLSI Photonics (IEEE, 2004), pp. 71-72.
[CrossRef]

M. Iqbal, M. J. McFadden, and M. W. Haney are preparing a manuscript with the title "Interconnect prediction for performance evaluation of intrachip global communication".

Meeus, W.

C. Debaes, M. Vervaeke, V. Baukens, H. Ottevaere, P. Vynck, P. Tuteleers, B. Volckaerts, W. Meeus, M. Brunfaut, J. Van Campenhout, A. Hermanne, and H. Thienpont, "Low-cost micro-optical modules for MCM level optical interconnects," IEEE J. Sel. Top. Quantum Electron. 9, 518-530 (2003).
[CrossRef]

Miller, D. A. B.

D. A. B. Miller, "Rationale and challenges for optical interconnects to silicon chips," Proc. IEEE 88, 728-749 (2000).
[CrossRef]

Milojkovic, P.

M. W. Haney, M. P. Christensen, P. Milojkovic, G. J. Fokken, M. Vickberg, B. K. Gilbert, J. Rieve, J. Ekman, P. Chandramani, and F. Kiamilev, "Description and evaluation of the FAST-net smart pixel-based optical interconnection prototype," Proc. IEEE 88, 819-828 (1996).
[CrossRef]

Monnig, K.

K. Monnig, "The coming "red brick walls" in the ITRS interconnect roadmap," in Proceedings of IEEE-LEOS Summer Topical: Optical Interconnects and VLSI Photonics (IEEE, 2004).

Moore, G. E.

G. E. Moore, "Cramming more components onto integrated circuits," Fairchild Semiconductors' Electronics 38, 114-117 (1965).

Mui, M. Lung

M. Lung Mui and K. Banerjee, "A global interconnect optimization scheme for nanometer scale VLSI with implications for latency, bandwidth, and power dissipation," IEEE Trans. Electron Devices 51, 195-203 (2004).
[CrossRef]

Murakowiski, J.

T. Dillon, A. Balcha, J. Murakowiski, and D. Prather, "Continuous-tone grayscale mask fabrication using high-energy-beam-sensitive glass," J. Microlithogr. Microfabr. Microsyst. 3, 550-554 (2004).
[CrossRef]

Neilson, D.

Nuzman, C.

Ottevaere, H.

C. Debaes, M. Vervaeke, V. Baukens, H. Ottevaere, P. Vynck, P. Tuteleers, B. Volckaerts, W. Meeus, M. Brunfaut, J. Van Campenhout, A. Hermanne, and H. Thienpont, "Low-cost micro-optical modules for MCM level optical interconnects," IEEE J. Sel. Top. Quantum Electron. 9, 518-530 (2003).
[CrossRef]

Papazian, A.

Prather, D.

T. Dillon, A. Balcha, J. Murakowiski, and D. Prather, "Continuous-tone grayscale mask fabrication using high-energy-beam-sensitive glass," J. Microlithogr. Microfabr. Microsyst. 3, 550-554 (2004).
[CrossRef]

M. Iqbal, T. Dillon, M. J. McFadden, D. Prather, and M. W. Haney, "Fabrication of beam steering elements, couplers and lenses with grayscale lithography using HEBS glass," in Proceedings of 89th OSA Annual Meeting (Optical Society of America, 2005).

Rempel, M. A.

Rieve, J.

M. W. Haney, M. P. Christensen, P. Milojkovic, G. J. Fokken, M. Vickberg, B. K. Gilbert, J. Rieve, J. Ekman, P. Chandramani, and F. Kiamilev, "Description and evaluation of the FAST-net smart pixel-based optical interconnection prototype," Proc. IEEE 88, 819-828 (1996).
[CrossRef]

Russo, R. L.

B. S. Landman and R. L. Russo, "On a pin versus block relationship for partitions of logic graphs," IEEE Trans. Comput. C-20, 1496-1479 (1971).
[CrossRef]

Ryf, R.

Sarto, E.

N. Vasseghi, K. Yeager, E. Sarto, and M. Seddighnezhad, "A 200 Mhz superscaler RISC microprocessor," IEEE J. Solid-State Circuits 31, 1675-1686 (1996).
[CrossRef]

Seddighnezhad, M.

N. Vasseghi, K. Yeager, E. Sarto, and M. Seddighnezhad, "A 200 Mhz superscaler RISC microprocessor," IEEE J. Solid-State Circuits 31, 1675-1686 (1996).
[CrossRef]

Self, S. A.

Stroobandt, D.

P. Chiristie and D. Stroobandt, "The interpretation and application of Rent's rule," IEEE Trans. Very Large Scale Integr. (VSLI) Syst., 8, 639-648 (2000).
[CrossRef]

D. Stroobandt, A Priori Wire Length Estimation for Digital Design (Kluwer, 2001).
[CrossRef]

Sylvester, D.

D. Sylvester and K. Keutzer, "Getting to the bottom of deep submicron II: a global wiring paradigm," in Proceedings of the International Symposium on Physical Design (1999).

Thienpont, H.

C. Debaes, M. Vervaeke, V. Baukens, H. Ottevaere, P. Vynck, P. Tuteleers, B. Volckaerts, W. Meeus, M. Brunfaut, J. Van Campenhout, A. Hermanne, and H. Thienpont, "Low-cost micro-optical modules for MCM level optical interconnects," IEEE J. Sel. Top. Quantum Electron. 9, 518-530 (2003).
[CrossRef]

Thoma, F.

Tuteleers, P.

C. Debaes, M. Vervaeke, V. Baukens, H. Ottevaere, P. Vynck, P. Tuteleers, B. Volckaerts, W. Meeus, M. Brunfaut, J. Van Campenhout, A. Hermanne, and H. Thienpont, "Low-cost micro-optical modules for MCM level optical interconnects," IEEE J. Sel. Top. Quantum Electron. 9, 518-530 (2003).
[CrossRef]

Van Campenhout, J.

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Figures (20)

Fig. 1
Fig. 1

Schematic of the three-tier interconnect-centric design paradigm for chip layout.

Fig. 2
Fig. 2

Comparison of global wire length estimates with published statistics for a reduced instruction-set computing processor.

Fig. 3
Fig. 3

Global bandwidth density available in metal wires, compared with the communication capacity required.

Fig. 4
Fig. 4

Schematic of the micro-optical system with all the relevant parameters indicated.

Fig. 5
Fig. 5

Schematic of the multiscale optical system with all the relevant parameters indicated. The macromirror is represented by a thin macrolens.

Fig. 6
Fig. 6

Diagram of the folded micro-optical system geometry with a microprism used to perform the necessary beam steering.

Fig. 7
Fig. 7

Schematic of the multiscale system geometry, which uses a curved macromirror in the place of a planar reflector.

Fig. 8
Fig. 8

Plot showing the density of point-to-point channels that can be achieved over a particular chip size for both the micro-optical system and the multiscale system.

Fig. 9
Fig. 9

Bandwidth capacity of free-space optical interconnects compared to the capacity of metal wires and the requirements for global communication in several generations of microchips.

Fig. 10
Fig. 10

Inherent misalignment tolerance resulting from returning beam to normal with receiving-end microprism and collection by a microlens.

Fig. 11
Fig. 11

Simplified flow diagram of an algorithm to obtain the microprism surface profile angles, performed for every transmit- and receive-end microprism pair. The input is a list of coordinates and tolerances, and the output is a list of surface profile angles.

Fig. 12
Fig. 12

Angles α and β of the compound-slope prism are transformed into elevation and rotation for encoding as an electron-beam dose profile.

Fig. 13
Fig. 13

Photograph of a HEBS glass mask for an array of microprisms showing the variation in optical density.

Fig. 14
Fig. 14

Scanning electron micrograph of a fabricated silicon microprism array showing distinct compound-slope profiles.

Fig. 15
Fig. 15

Schematic of the optical system setup that includes a relay and observation system and the system under observation, in this case the multiscale system. The diagram is not to scale.

Fig. 16
Fig. 16

Photograph of the optical system consisting of a relay and observation system and the system under observation.

Fig. 17
Fig. 17

Spot photographs (left) and schematic (right) of (a) aligned microlens and (b) microprism steering. The schematics are not to scale.

Fig. 18
Fig. 18

Spot photographs (left) and schematic (right) of (a) aligned microlens, and (b) and (c) microprism steering in the multiscale system. The schematics are not to scale.

Fig. 19
Fig. 19

Schematic showing the interconnection topology and dimensions of the experimental setup using silicon microprisms.

Fig. 20
Fig. 20

Composite photograph of the beam spots showing optical channels in the multiscale system between the transmitter array (bottom left) and the receiver arrays (top center and bottom right). The pattern and physical dimensions are shown in Fig. 19.

Equations (10)

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

L max = 2 f 1 ( 1 + f 1 2 z R 0 ) .
d 1 = 2.12 × 2 ω 0 1 + ( f 1 + z R 0 z R 0 ) 2 .
D micro-optical = 1 2 d 1 2 .
D multiscale = 1 ( 2.12 × 2 ) 2 { ω 4     2 [ 1 + ( x 5 z R 2 ) 2 ] + ω 0 2 [ 1 + ( f 1 + z R 0 z R 0 ) 2 ] } .
A micro-optical = ( L max   sin   θ max ) 2 2 .
A multiscale = 2 ( L max   tan   θ max ) 2 .
θ s = sin - 1 ( n   sin   ψ ) ψ .
θ max 0.65 [ 90 ° sin - 1 ( 1 n ) ] .
orientation : φ = tan - 1 ( tan   α tan   β ) ,
gradient : θ = tan - 1 [ ( tan   α + tan   φ   tan   β ) cos   φ ] .

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