Abstract

Interference lithography holds the promise of fabricating large-area, defect-free photonic structures on the submicrometer scale both rapidly and cheaply. There is a need for a procedure to establish a connection between the structures that are formed and the parameters of the interfering beams. There is also a need to produce self-supporting three-dimensional bicontinuous structures. A generic technique correlating parameters of the interfering beams with the symmetry elements present in the resultant structures by a level-set approach is developed. A particular space group is ensured by equating terms of the intensity equation to a representative level surface of the desired space group. Single- and multiple-exposure techniques are discussed. The beam parameters for certain cubic bicontinuous structures relevant to photonic crystals, viz., the diamond (D), the simple cubic (P), and the chiral gyroid (G) are derived by utilizing either linear or elliptically polarized light.

© 2003 Optical Society of America

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References

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  1. M. Maldovan, A. M. Urbas, N. Yufa, W. C. Carter, E. L. Thomas, “Photonic properties of bicontinuous cubic microphases,” Phys. Rev. B 65, 165123:1–165123:5 (2002).
    [CrossRef]
  2. J. C. Shah, Y. Sadhale, D. M. Chilukuri, “Cubic phase gels as drug delivery systems,” Adv. Drug Deliver. Rev. 47, 229–250 (2001).
    [CrossRef]
  3. E. L. Thomas, D. B. Alward, D. J. Kinning, D. C. Martin, D. L. Handlin, L. J. Fetters, “Ordered bicontinuous double-diamond structure of star block copolymers: a new equilibrium microdomain morphology,” Macromolecules 19, 2197–2202 (1986).
    [CrossRef]
  4. J. Wijnhoven, W. L. Vos, “Preparation of photonic crystals made of air spheres in titania,” Science 281, 802–804 (1998).
    [CrossRef]
  5. P. Alexandridis, U. Olsson, B. Lindman, “A record nine different phases (four cubic, two hexagonal, and one lamellar lyotropic liquid crystalline and two micellar solutions) in a ternary isothermal system of an amphiphilic block copolymer and selective solvents (water and oil),” Langmuir 14, 2627–2638 (1998).
    [CrossRef]
  6. M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53–56 (2000).
    [CrossRef] [PubMed]
  7. A. J. Turberfield, “Photonic crystals made by holographic lithography,” MRS Bull. 26, 632–636 (2001).
    [CrossRef]
  8. L. Z. Cai, X. L. Yang, Y. R. Wang, “All fourteen Bravais lattices can be formed by interference of four noncoplanar beams,” Opt. Lett. 27, 900–902 (2002).
    [CrossRef]
  9. M. Buerger, Elementary Crystallography (MIT, Cambridge, Mass., 1978).
  10. K. I. Petsas, A. B. Coates, G. Grynberg, “Crystallography of optical lattices,” Phys. Rev. A 50, 5173–5189 (1994).
    [CrossRef] [PubMed]
  11. L. Z. Cai, X. L. Yang, Y. R. Wang, “Formation of three-dimensional periodic microstructures by interference of four noncoplanar beams,” J. Opt. Soc. Am. A 19, 2238–2244 (2002).
    [CrossRef]
  12. S. Yang, M. Megens, J. Aizenberg, P. Wiltzius, P. M. Chaikin, W. B. Russel, “Creating periodic three-dimensional structures by multibeam interference of visible laser,” Chem. Mater. 14, 2831–2833 (2002).
    [CrossRef]
  13. A. Chelnokov, S. Rowson, J. M. Lourtioz, V. Berger, J. Y. Courtois, “An optical drill for the fabrication of photonic crystals,” J. Opt. A Pure Appl. Opt. 1, L3–L6 (1999).
    [CrossRef]
  14. U. Shmueli, ed., Reciprocal Space, Vol. B of International Tables for Crystallography (Kluwer Academic, Dordrecht, The Netherlands, 1996).
  15. Structures without a center of inversion are achieved by use of elliptically polarized light.
  16. M. Wohlgemuth, N. Yufa, J. Hoffman, E. L. Thomas, “Triply periodic bicontinuous cubic microdomain morphologies by symmetries,” Macromolecules 34, 6083–6089 (2001).
    [CrossRef]
  17. D. N. Sharp, M. Campbell, E. R. Dedman, M. T. Harrison, R. G. Denning, A. J. Turberfield, “Photonic crystals for the visible spectrum by holographic lithography,” Opt. Quantum Electron. 34, 3–12 (2002).
    [CrossRef]
  18. R. Sutherland, V. Tondiglia, L. Natarajan, S. Chandra, D. Tomlin, T. Bunning, “Switchable orthorhombic F photonic crystals formed by holographic polymerization-induced phase separation of liquid crystal,” Opt. Express 10, 1074–1082 (2002).
    [CrossRef] [PubMed]

2002 (6)

M. Maldovan, A. M. Urbas, N. Yufa, W. C. Carter, E. L. Thomas, “Photonic properties of bicontinuous cubic microphases,” Phys. Rev. B 65, 165123:1–165123:5 (2002).
[CrossRef]

L. Z. Cai, X. L. Yang, Y. R. Wang, “All fourteen Bravais lattices can be formed by interference of four noncoplanar beams,” Opt. Lett. 27, 900–902 (2002).
[CrossRef]

L. Z. Cai, X. L. Yang, Y. R. Wang, “Formation of three-dimensional periodic microstructures by interference of four noncoplanar beams,” J. Opt. Soc. Am. A 19, 2238–2244 (2002).
[CrossRef]

S. Yang, M. Megens, J. Aizenberg, P. Wiltzius, P. M. Chaikin, W. B. Russel, “Creating periodic three-dimensional structures by multibeam interference of visible laser,” Chem. Mater. 14, 2831–2833 (2002).
[CrossRef]

D. N. Sharp, M. Campbell, E. R. Dedman, M. T. Harrison, R. G. Denning, A. J. Turberfield, “Photonic crystals for the visible spectrum by holographic lithography,” Opt. Quantum Electron. 34, 3–12 (2002).
[CrossRef]

R. Sutherland, V. Tondiglia, L. Natarajan, S. Chandra, D. Tomlin, T. Bunning, “Switchable orthorhombic F photonic crystals formed by holographic polymerization-induced phase separation of liquid crystal,” Opt. Express 10, 1074–1082 (2002).
[CrossRef] [PubMed]

2001 (3)

M. Wohlgemuth, N. Yufa, J. Hoffman, E. L. Thomas, “Triply periodic bicontinuous cubic microdomain morphologies by symmetries,” Macromolecules 34, 6083–6089 (2001).
[CrossRef]

A. J. Turberfield, “Photonic crystals made by holographic lithography,” MRS Bull. 26, 632–636 (2001).
[CrossRef]

J. C. Shah, Y. Sadhale, D. M. Chilukuri, “Cubic phase gels as drug delivery systems,” Adv. Drug Deliver. Rev. 47, 229–250 (2001).
[CrossRef]

2000 (1)

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53–56 (2000).
[CrossRef] [PubMed]

1999 (1)

A. Chelnokov, S. Rowson, J. M. Lourtioz, V. Berger, J. Y. Courtois, “An optical drill for the fabrication of photonic crystals,” J. Opt. A Pure Appl. Opt. 1, L3–L6 (1999).
[CrossRef]

1998 (2)

J. Wijnhoven, W. L. Vos, “Preparation of photonic crystals made of air spheres in titania,” Science 281, 802–804 (1998).
[CrossRef]

P. Alexandridis, U. Olsson, B. Lindman, “A record nine different phases (four cubic, two hexagonal, and one lamellar lyotropic liquid crystalline and two micellar solutions) in a ternary isothermal system of an amphiphilic block copolymer and selective solvents (water and oil),” Langmuir 14, 2627–2638 (1998).
[CrossRef]

1994 (1)

K. I. Petsas, A. B. Coates, G. Grynberg, “Crystallography of optical lattices,” Phys. Rev. A 50, 5173–5189 (1994).
[CrossRef] [PubMed]

1986 (1)

E. L. Thomas, D. B. Alward, D. J. Kinning, D. C. Martin, D. L. Handlin, L. J. Fetters, “Ordered bicontinuous double-diamond structure of star block copolymers: a new equilibrium microdomain morphology,” Macromolecules 19, 2197–2202 (1986).
[CrossRef]

Aizenberg, J.

S. Yang, M. Megens, J. Aizenberg, P. Wiltzius, P. M. Chaikin, W. B. Russel, “Creating periodic three-dimensional structures by multibeam interference of visible laser,” Chem. Mater. 14, 2831–2833 (2002).
[CrossRef]

Alexandridis, P.

P. Alexandridis, U. Olsson, B. Lindman, “A record nine different phases (four cubic, two hexagonal, and one lamellar lyotropic liquid crystalline and two micellar solutions) in a ternary isothermal system of an amphiphilic block copolymer and selective solvents (water and oil),” Langmuir 14, 2627–2638 (1998).
[CrossRef]

Alward, D. B.

E. L. Thomas, D. B. Alward, D. J. Kinning, D. C. Martin, D. L. Handlin, L. J. Fetters, “Ordered bicontinuous double-diamond structure of star block copolymers: a new equilibrium microdomain morphology,” Macromolecules 19, 2197–2202 (1986).
[CrossRef]

Berger, V.

A. Chelnokov, S. Rowson, J. M. Lourtioz, V. Berger, J. Y. Courtois, “An optical drill for the fabrication of photonic crystals,” J. Opt. A Pure Appl. Opt. 1, L3–L6 (1999).
[CrossRef]

Buerger, M.

M. Buerger, Elementary Crystallography (MIT, Cambridge, Mass., 1978).

Bunning, T.

Cai, L. Z.

Campbell, M.

D. N. Sharp, M. Campbell, E. R. Dedman, M. T. Harrison, R. G. Denning, A. J. Turberfield, “Photonic crystals for the visible spectrum by holographic lithography,” Opt. Quantum Electron. 34, 3–12 (2002).
[CrossRef]

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53–56 (2000).
[CrossRef] [PubMed]

Carter, W. C.

M. Maldovan, A. M. Urbas, N. Yufa, W. C. Carter, E. L. Thomas, “Photonic properties of bicontinuous cubic microphases,” Phys. Rev. B 65, 165123:1–165123:5 (2002).
[CrossRef]

Chaikin, P. M.

S. Yang, M. Megens, J. Aizenberg, P. Wiltzius, P. M. Chaikin, W. B. Russel, “Creating periodic three-dimensional structures by multibeam interference of visible laser,” Chem. Mater. 14, 2831–2833 (2002).
[CrossRef]

Chandra, S.

Chelnokov, A.

A. Chelnokov, S. Rowson, J. M. Lourtioz, V. Berger, J. Y. Courtois, “An optical drill for the fabrication of photonic crystals,” J. Opt. A Pure Appl. Opt. 1, L3–L6 (1999).
[CrossRef]

Chilukuri, D. M.

J. C. Shah, Y. Sadhale, D. M. Chilukuri, “Cubic phase gels as drug delivery systems,” Adv. Drug Deliver. Rev. 47, 229–250 (2001).
[CrossRef]

Coates, A. B.

K. I. Petsas, A. B. Coates, G. Grynberg, “Crystallography of optical lattices,” Phys. Rev. A 50, 5173–5189 (1994).
[CrossRef] [PubMed]

Courtois, J. Y.

A. Chelnokov, S. Rowson, J. M. Lourtioz, V. Berger, J. Y. Courtois, “An optical drill for the fabrication of photonic crystals,” J. Opt. A Pure Appl. Opt. 1, L3–L6 (1999).
[CrossRef]

Dedman, E. R.

D. N. Sharp, M. Campbell, E. R. Dedman, M. T. Harrison, R. G. Denning, A. J. Turberfield, “Photonic crystals for the visible spectrum by holographic lithography,” Opt. Quantum Electron. 34, 3–12 (2002).
[CrossRef]

Denning, R. G.

D. N. Sharp, M. Campbell, E. R. Dedman, M. T. Harrison, R. G. Denning, A. J. Turberfield, “Photonic crystals for the visible spectrum by holographic lithography,” Opt. Quantum Electron. 34, 3–12 (2002).
[CrossRef]

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53–56 (2000).
[CrossRef] [PubMed]

Fetters, L. J.

E. L. Thomas, D. B. Alward, D. J. Kinning, D. C. Martin, D. L. Handlin, L. J. Fetters, “Ordered bicontinuous double-diamond structure of star block copolymers: a new equilibrium microdomain morphology,” Macromolecules 19, 2197–2202 (1986).
[CrossRef]

Grynberg, G.

K. I. Petsas, A. B. Coates, G. Grynberg, “Crystallography of optical lattices,” Phys. Rev. A 50, 5173–5189 (1994).
[CrossRef] [PubMed]

Handlin, D. L.

E. L. Thomas, D. B. Alward, D. J. Kinning, D. C. Martin, D. L. Handlin, L. J. Fetters, “Ordered bicontinuous double-diamond structure of star block copolymers: a new equilibrium microdomain morphology,” Macromolecules 19, 2197–2202 (1986).
[CrossRef]

Harrison, M. T.

D. N. Sharp, M. Campbell, E. R. Dedman, M. T. Harrison, R. G. Denning, A. J. Turberfield, “Photonic crystals for the visible spectrum by holographic lithography,” Opt. Quantum Electron. 34, 3–12 (2002).
[CrossRef]

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53–56 (2000).
[CrossRef] [PubMed]

Hoffman, J.

M. Wohlgemuth, N. Yufa, J. Hoffman, E. L. Thomas, “Triply periodic bicontinuous cubic microdomain morphologies by symmetries,” Macromolecules 34, 6083–6089 (2001).
[CrossRef]

Kinning, D. J.

E. L. Thomas, D. B. Alward, D. J. Kinning, D. C. Martin, D. L. Handlin, L. J. Fetters, “Ordered bicontinuous double-diamond structure of star block copolymers: a new equilibrium microdomain morphology,” Macromolecules 19, 2197–2202 (1986).
[CrossRef]

Lindman, B.

P. Alexandridis, U. Olsson, B. Lindman, “A record nine different phases (four cubic, two hexagonal, and one lamellar lyotropic liquid crystalline and two micellar solutions) in a ternary isothermal system of an amphiphilic block copolymer and selective solvents (water and oil),” Langmuir 14, 2627–2638 (1998).
[CrossRef]

Lourtioz, J. M.

A. Chelnokov, S. Rowson, J. M. Lourtioz, V. Berger, J. Y. Courtois, “An optical drill for the fabrication of photonic crystals,” J. Opt. A Pure Appl. Opt. 1, L3–L6 (1999).
[CrossRef]

Maldovan, M.

M. Maldovan, A. M. Urbas, N. Yufa, W. C. Carter, E. L. Thomas, “Photonic properties of bicontinuous cubic microphases,” Phys. Rev. B 65, 165123:1–165123:5 (2002).
[CrossRef]

Martin, D. C.

E. L. Thomas, D. B. Alward, D. J. Kinning, D. C. Martin, D. L. Handlin, L. J. Fetters, “Ordered bicontinuous double-diamond structure of star block copolymers: a new equilibrium microdomain morphology,” Macromolecules 19, 2197–2202 (1986).
[CrossRef]

Megens, M.

S. Yang, M. Megens, J. Aizenberg, P. Wiltzius, P. M. Chaikin, W. B. Russel, “Creating periodic three-dimensional structures by multibeam interference of visible laser,” Chem. Mater. 14, 2831–2833 (2002).
[CrossRef]

Natarajan, L.

Olsson, U.

P. Alexandridis, U. Olsson, B. Lindman, “A record nine different phases (four cubic, two hexagonal, and one lamellar lyotropic liquid crystalline and two micellar solutions) in a ternary isothermal system of an amphiphilic block copolymer and selective solvents (water and oil),” Langmuir 14, 2627–2638 (1998).
[CrossRef]

Petsas, K. I.

K. I. Petsas, A. B. Coates, G. Grynberg, “Crystallography of optical lattices,” Phys. Rev. A 50, 5173–5189 (1994).
[CrossRef] [PubMed]

Rowson, S.

A. Chelnokov, S. Rowson, J. M. Lourtioz, V. Berger, J. Y. Courtois, “An optical drill for the fabrication of photonic crystals,” J. Opt. A Pure Appl. Opt. 1, L3–L6 (1999).
[CrossRef]

Russel, W. B.

S. Yang, M. Megens, J. Aizenberg, P. Wiltzius, P. M. Chaikin, W. B. Russel, “Creating periodic three-dimensional structures by multibeam interference of visible laser,” Chem. Mater. 14, 2831–2833 (2002).
[CrossRef]

Sadhale, Y.

J. C. Shah, Y. Sadhale, D. M. Chilukuri, “Cubic phase gels as drug delivery systems,” Adv. Drug Deliver. Rev. 47, 229–250 (2001).
[CrossRef]

Shah, J. C.

J. C. Shah, Y. Sadhale, D. M. Chilukuri, “Cubic phase gels as drug delivery systems,” Adv. Drug Deliver. Rev. 47, 229–250 (2001).
[CrossRef]

Sharp, D. N.

D. N. Sharp, M. Campbell, E. R. Dedman, M. T. Harrison, R. G. Denning, A. J. Turberfield, “Photonic crystals for the visible spectrum by holographic lithography,” Opt. Quantum Electron. 34, 3–12 (2002).
[CrossRef]

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53–56 (2000).
[CrossRef] [PubMed]

Sutherland, R.

Thomas, E. L.

M. Maldovan, A. M. Urbas, N. Yufa, W. C. Carter, E. L. Thomas, “Photonic properties of bicontinuous cubic microphases,” Phys. Rev. B 65, 165123:1–165123:5 (2002).
[CrossRef]

M. Wohlgemuth, N. Yufa, J. Hoffman, E. L. Thomas, “Triply periodic bicontinuous cubic microdomain morphologies by symmetries,” Macromolecules 34, 6083–6089 (2001).
[CrossRef]

E. L. Thomas, D. B. Alward, D. J. Kinning, D. C. Martin, D. L. Handlin, L. J. Fetters, “Ordered bicontinuous double-diamond structure of star block copolymers: a new equilibrium microdomain morphology,” Macromolecules 19, 2197–2202 (1986).
[CrossRef]

Tomlin, D.

Tondiglia, V.

Turberfield, A. J.

D. N. Sharp, M. Campbell, E. R. Dedman, M. T. Harrison, R. G. Denning, A. J. Turberfield, “Photonic crystals for the visible spectrum by holographic lithography,” Opt. Quantum Electron. 34, 3–12 (2002).
[CrossRef]

A. J. Turberfield, “Photonic crystals made by holographic lithography,” MRS Bull. 26, 632–636 (2001).
[CrossRef]

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53–56 (2000).
[CrossRef] [PubMed]

Urbas, A. M.

M. Maldovan, A. M. Urbas, N. Yufa, W. C. Carter, E. L. Thomas, “Photonic properties of bicontinuous cubic microphases,” Phys. Rev. B 65, 165123:1–165123:5 (2002).
[CrossRef]

Vos, W. L.

J. Wijnhoven, W. L. Vos, “Preparation of photonic crystals made of air spheres in titania,” Science 281, 802–804 (1998).
[CrossRef]

Wang, Y. R.

Wijnhoven, J.

J. Wijnhoven, W. L. Vos, “Preparation of photonic crystals made of air spheres in titania,” Science 281, 802–804 (1998).
[CrossRef]

Wiltzius, P.

S. Yang, M. Megens, J. Aizenberg, P. Wiltzius, P. M. Chaikin, W. B. Russel, “Creating periodic three-dimensional structures by multibeam interference of visible laser,” Chem. Mater. 14, 2831–2833 (2002).
[CrossRef]

Wohlgemuth, M.

M. Wohlgemuth, N. Yufa, J. Hoffman, E. L. Thomas, “Triply periodic bicontinuous cubic microdomain morphologies by symmetries,” Macromolecules 34, 6083–6089 (2001).
[CrossRef]

Yang, S.

S. Yang, M. Megens, J. Aizenberg, P. Wiltzius, P. M. Chaikin, W. B. Russel, “Creating periodic three-dimensional structures by multibeam interference of visible laser,” Chem. Mater. 14, 2831–2833 (2002).
[CrossRef]

Yang, X. L.

Yufa, N.

M. Maldovan, A. M. Urbas, N. Yufa, W. C. Carter, E. L. Thomas, “Photonic properties of bicontinuous cubic microphases,” Phys. Rev. B 65, 165123:1–165123:5 (2002).
[CrossRef]

M. Wohlgemuth, N. Yufa, J. Hoffman, E. L. Thomas, “Triply periodic bicontinuous cubic microdomain morphologies by symmetries,” Macromolecules 34, 6083–6089 (2001).
[CrossRef]

Adv. Drug Deliver. Rev. (1)

J. C. Shah, Y. Sadhale, D. M. Chilukuri, “Cubic phase gels as drug delivery systems,” Adv. Drug Deliver. Rev. 47, 229–250 (2001).
[CrossRef]

Chem. Mater. (1)

S. Yang, M. Megens, J. Aizenberg, P. Wiltzius, P. M. Chaikin, W. B. Russel, “Creating periodic three-dimensional structures by multibeam interference of visible laser,” Chem. Mater. 14, 2831–2833 (2002).
[CrossRef]

J. Opt. A Pure Appl. Opt. (1)

A. Chelnokov, S. Rowson, J. M. Lourtioz, V. Berger, J. Y. Courtois, “An optical drill for the fabrication of photonic crystals,” J. Opt. A Pure Appl. Opt. 1, L3–L6 (1999).
[CrossRef]

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

Langmuir (1)

P. Alexandridis, U. Olsson, B. Lindman, “A record nine different phases (four cubic, two hexagonal, and one lamellar lyotropic liquid crystalline and two micellar solutions) in a ternary isothermal system of an amphiphilic block copolymer and selective solvents (water and oil),” Langmuir 14, 2627–2638 (1998).
[CrossRef]

Macromolecules (2)

E. L. Thomas, D. B. Alward, D. J. Kinning, D. C. Martin, D. L. Handlin, L. J. Fetters, “Ordered bicontinuous double-diamond structure of star block copolymers: a new equilibrium microdomain morphology,” Macromolecules 19, 2197–2202 (1986).
[CrossRef]

M. Wohlgemuth, N. Yufa, J. Hoffman, E. L. Thomas, “Triply periodic bicontinuous cubic microdomain morphologies by symmetries,” Macromolecules 34, 6083–6089 (2001).
[CrossRef]

MRS Bull. (1)

A. J. Turberfield, “Photonic crystals made by holographic lithography,” MRS Bull. 26, 632–636 (2001).
[CrossRef]

Nature (1)

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53–56 (2000).
[CrossRef] [PubMed]

Opt. Express (1)

Opt. Lett. (1)

Opt. Quantum Electron. (1)

D. N. Sharp, M. Campbell, E. R. Dedman, M. T. Harrison, R. G. Denning, A. J. Turberfield, “Photonic crystals for the visible spectrum by holographic lithography,” Opt. Quantum Electron. 34, 3–12 (2002).
[CrossRef]

Phys. Rev. A (1)

K. I. Petsas, A. B. Coates, G. Grynberg, “Crystallography of optical lattices,” Phys. Rev. A 50, 5173–5189 (1994).
[CrossRef] [PubMed]

Phys. Rev. B (1)

M. Maldovan, A. M. Urbas, N. Yufa, W. C. Carter, E. L. Thomas, “Photonic properties of bicontinuous cubic microphases,” Phys. Rev. B 65, 165123:1–165123:5 (2002).
[CrossRef]

Science (1)

J. Wijnhoven, W. L. Vos, “Preparation of photonic crystals made of air spheres in titania,” Science 281, 802–804 (1998).
[CrossRef]

Other (3)

M. Buerger, Elementary Crystallography (MIT, Cambridge, Mass., 1978).

U. Shmueli, ed., Reciprocal Space, Vol. B of International Tables for Crystallography (Kluwer Academic, Dordrecht, The Netherlands, 1996).

Structures without a center of inversion are achieved by use of elliptically polarized light.

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Figures (3)

Fig. 1
Fig. 1

Plot of the approximation to the P surface showing one and eight unit cells. The level set corresponds to space group Pm 3̅m (No. 221) and provides a filling fraction of 50%. All images courtesy of J. Hoffman http://www.msri.org/publications/sgp/jim/images/stills/indexc.html.

Fig. 2
Fig. 2

(a) Plot of the level-set approximation to the D surface with a filling fraction of 50%. (b) The insets (i)–(iii) show three different volume fractions and how the volume fraction can be changed by simply varying the constant term t in the level-set equation [in this case, Eq. (11)]. The single D network dielectric volume fractions shown are (i) 12.5%, t = - 1.8 ; (ii) 25.2%, t = - 1.2 ; (iii) 66.4%, t = - 0.8 .

Fig. 3
Fig. 3

Plot of the level-set approximation to the G surface, with a filling fraction of 50%.

Equations (59)

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

I ( r ) = m = 1 n l = 1 n l     m * exp [ i ( k l - k m )     r ] l = 1 n m = 1 n a lm exp   i G lm     r ,
I ( r ) = E 0 2 + E 1 2 + E 2 2 + 2 E 0 E 1 cos ( 2 π x / a ) + 2 E 0 E 2 cos ( 2 π y / a ) + 2 E 1 E 2 cos [ 2 π ( x - y ) / a ] .
F ( 10 ) + F ( 01 ) = cos ( x ) + cos ( y ) ,
F ( 100 ) = cos   2 π x + cos   2 π y + cos   2 π z .
k 1 - k 0 = 2 π / a [ 100 ] ,
k 2 - k 0 = 2 π / a [ 010 ] ,
k 3 - k 0 = 2 π / a [ 001 ] ,
| k i | = 2 π / λ .
k 0 = π / a [ 1 ¯ 1 ¯ 1 ¯ ] ,
k 1 = π / a [ 1   1 ¯ 1 ¯ ] ,
k 2 = π / a [ 1 ¯ 1   1 ¯ ] ,
k 3 = π / a [ 1 ¯ 1 ¯ 1 ] .
I ( r ) = E 0 2 + E 1 2 + E 2 2 + E 3 2 + 2 E 0 E 1 cos ( 2 π x / a ) + 2 E 0 E 2 cos ( 2 π y / a ) + 2 E 0 E 3 cos ( 2 π z / a ) + 2 E 1 E 2 cos [ 2 π / a ( x - y ) ] + 2 E 1 E 3 cos [ 2 π / a ( x - z ) ] + 2 E 2 E 3 cos [ 2 π / a ( y - z ) ] .
E 0 = 1.000 [ 0.0 ,   0.707 ,   - 0.707 ] ,
E 1 = 0.632 [ - 0.5 ,   0.309 ,   - 0.809 ] ,
E 2 = 0.874 [ 0.809 ,   0.5 ,   - 0.309 ] ,
E 3 = 2.288 [ - 0.309 ,   0.809 ,   0.5 ] .
I ( r ) = 7.4 + cos ( 2 π x / a ) + cos ( 2 π y / a ) + cos ( 2 π z / a ) ,
F ( 111 ) = cos   2 π x   cos   2 π xy   cos   2 π z + sin   2 π x   sin   2 π y   cos   2 π z + sin   2 π x   cos   2 π y   sin   2 π z + cos   2 π x   sin   2 π y   sin   2 π z .
F ( 111 ) = sin [ 2 π ( x + y + z ) ] + sin [ 2 π ( x + y - z ) ] + sin [ 2 π ( x - y + z ) ] + sin [ 2 π ( - x + y + z ) ] ,
k 0 = π / a [ 2   0   1 ] ; ϕ 0 = π / 2 ,
k 1 = π / a [ 2 ¯ 0   1 ] ; ϕ 1 = π / 2 ,
k 2 = π / a [ 0   2   1 ¯ ] ; ϕ 2 = 0 ,
k 3 = π / a [ 0   2 ¯ 1 ¯ ] ; ϕ 3 = 0 .
E 0 = 2.163 [ 0.116 ,   - 0.966 ,   - 0.231 ] ,
E 1 = 0.566 [ 0.442 ,   - 0.158 ,   0.883 ] ,
E 2 = 1.225 [ 0.890 ,   0.204 ,   0.408 ] ,
E 3 = 1.274 [ 0.294 ,   0.427 ,   - 0.855 ] .
I ( r ) = 8.125 + sin [ 2 π / a ( x + y + z ) ] + sin [ 2 π / a ( - x + y + z ) ] + sin [ 2 π / a ( x - y + z ) ] + sin [ 2 π / a ( x + y - z ) ] .
F ( 110 ) = sin [ 2 π ( x + y ) ] + sin [ 2 π ( x - y ) ] + sin [ 2 π ( y - z ) ] + sin [ 2 π ( y + z ) ] + sin [ 2 π ( x + z ) ] + sin [ 2 π ( z - x ) ] .
k 0 = π / a [ 1 ¯ 1 ¯ 1 ¯ ] ,
k 1 = π / a [ 1   1   1 ¯ ] ,
k 2 = π / a [ 1   1 ¯ 1 ] ,
k 3 = π / a [ 1 ¯ 1   1 ] .
I ( r ) = | E 0 | 2 + | E 1 | 2 + | E 2 | 2 + | E 3 | 2 + 2   Re ( E 0 E 1 * ) cos [ ( k 0 - k 1 )     r ] - 2   Im ( E 0 E 1 * ) sin [ ( k 0 - k 1 )     r ] + 2   Re ( E 0 E 2 * ) cos [ ( k 0 - k 2 )     r ] - 2   Im ( E 0 E 2 * ) sin [ ( k 0 - k 2 )     r ] + 2   Re ( E 0 E 3 * ) cos [ ( k 0 - k 3 )     r ] - 2   Im ( E 0 E 3 * ) sin [ ( k 0 - k 3 )     r ] + 2   Re ( E 1 E 2 * ) cos [ ( k 1 - k 2 )     r ] - 2   Im ( E 1 E 2 * ) sin [ ( k 1 - k 2 )     r ] + 2   Re ( E 1 E 3 * ) cos [ ( k 1 - k 3 )     r ] - 2   Im ( E 1 E 3 * ) sin [ ( k 1 - k 3 )     r ] + 2   Re ( E 2 E 3 * ) cos [ ( k 2 - k 3 )     r ] - 2   Im ( E 2 E 3 * ) sin [ ( k 2 - k 3 )     r ] .
Re ( E 0     E 1 * ) = 0 , Im ( E 0     E 1 * ) = c ,
Re ( E 0     E 2 * ) = 0 , Im ( E 0     E 2 * ) = c ,
Re ( E 0     E 3 * ) = 0 , Im ( E 0     E 3 * ) = c ,
Re ( E 1     E 2 * ) = 0 , Im ( E 1     E 2 * ) = c ,
Re ( E 1     E 3 * ) = 0 , Im ( E 1     E 3 * ) = - c ,
Re ( E 2     E 3 * ) = 0 , Im ( E 2     E 3 * ) = c ,
| E 0 | 2 = a 0 , | E 1 | 2 = a 1 ,
| E 2 | 2 = a 2 , | E 3 | 2 = a 3 ,
k 0     E 0 = 0 , k 1     E 1 = 0 ,
k 2     E 2 = 0 , k 3     E 3 = 0 .
E 0 = { - 0.153 - i 0.342 ,   0.520 - i 0.470 ,   - 0.367 + i 0.813 } ,
E 1 = { - 0.520 - i 0.725 ,   0.373 + i 0.039 ,   - 0.147 - i 0.686 } ,
E 2 = { 0.667 + i 0.215 ,   0.888 - i 0.088 ,   0.220 - i 0.303 } ,
E 3 = { - 0.209 + i 0.362 ,   0.209 + i 0.362 ,   0.418 } .
I ( r ) = 4.811 - sin [ 2 π / a ( y - x ) ] - sin [ 2 π / a ( x + y ) ] - sin [ 2 π / a ( x - z ) ] - sin [ 2 π / a ( z - y ) ] - sin [ 2 π / a ( x + z ) ] - sin [ 2 π / a ( y + z ) ] .
k 0 = k 4 = π / a [ 1 ¯ 1 ¯ 1 ¯ ] ,
k 1 = k 5 = π / a [ 1   1   1 ¯ ] ,
k 2 = k 6 = π / a [ 1   1 ¯ 1 ] ,
k 3 = k 7 = π / a [ 1 ¯ 1   1 ] ,
ϕ 0 = 0 , ϕ 4 = π / 2 ,
ϕ 1 = 0 , ϕ 5 = π / 2 ,
ϕ 2 = 0 , ϕ 6 = π / 2 ,
ϕ 3 = 0 , ϕ 7 = π / 2 .
I ( r ) = const + ( E 0 E 1 + E 4 E 5 ) cos [ 2 π / a ( x + y ) ] + ( E 0 E 2 + E 4 E 6 ) cos [ 2 π / a ( x + z ) ] + ( E 5 E 6 + E 1 E 2 ) cos [ 2 π / a ( y - z ) ] + ( E 0 E 3 + E 4 E 7 ) cos [ 2 π / a ( y + z ) ] + ( E 1 E 3 + E 5 E 7 ) cos [ 2 π / a ( x - z ) ] + ( E 6 E 7 + E 2 E 3 ) cos [ 2 π / a ( x - y ) ] + ( E 1 E 4 - E 0 E 5 ) sin [ 2 π / a ( x + y ) ] + ( E 2 E 4 - E 0 E 6 ) sin [ 2 π / a ( x + z ) ] + ( E 1 E 6 - E 2 E 5 ) sin [ 2 π / a ( y - z ) ] + ( E 3 E 4 - E 0 E 7 ) sin [ 2 π / a ( y + z ) ] + ( E 3 E 5 - E 1 E 7 ) sin [ 2 π / a ( x - z ) ] + ( E 2 E 7 - E 3 E 6 ) sin [ 2 π / a ( x - y ) ] .

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