Abstract

Chiral gyroid photonic crystals are fabricated in the high refractive index chalcogenide glass arsenic trisulfide with an adaptive optics enhanced direct laser writing system. The severe spherical aberration imparted when focusing into the arsenic trisulfide is mitigated with a defocus decoupled aberration compensation technique that reduces the level of aberration that must be compensated by over an order of magnitude. The fabricated gyroids are shown to have excellent uniformity after our adaptive optics method is employed, and the transmission spectra of the gyroids are shown to have good agreement with numerical simulations that are based on a uniform and diffraction limited fabrication resolution.

© 2014 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]

2013 (1)

M. D. Turner, M. Saba, Q. Zhang, B. P. Cumming, G. E. Schröder-Turk, M. Gu, “Miniature chiral beam-splitter based on Gyroid photonic crystals,” Nat. Photonics 7, 801–805 (2013).
[CrossRef]

2012 (3)

2011 (7)

B. J. Eggleton, B. Luther-Davies, K. Richardson, “Chalcogenide photonics,” Nat. Photonics 5, 141–148 (2011).

M. Saba, M. Thiel, M. D. Turner, S. T. Hyde, M. Gu, K. Grosse-Brauckmann, D. N. Neshev, K. Mecke, G. E. Schröder-Turk, “Circular dichroism in biological photonic crystals and cubic chiral nets,” Phys. Rev. Lett. 106,103902 (2011).
[CrossRef] [PubMed]

E. Nicoletti, D. Bulla, B. Luther-Davies, M. Gu, “Wide-angle stop-gap chalcogenide photonic crystals generated by direct multiple-line laser writing,” Appl. Phys. B 105, 847–850 (2011).
[CrossRef]

E. Nicoletti, D. Bulla, B. Luther-Davies, M. Gu, “Generation of λ/12 nanowires in chalcogenide glasses,” Nano Lett., 11, 4218–4221 (2011).
[CrossRef] [PubMed]

B. P. Cumming, A. Jesacher, M. J. Booth, T. Wilson, M. Gu, “Adaptive aberration compensation for three-dimensional micro-fabrication of photonic crystals in lithium niobate,” Opt. Express 19, 9419–9425 (2011).
[CrossRef] [PubMed]

M. D. Turner, G. E. Schröder-Turk, M. Gu, “Fabrication and characterization of three-dimensional biomimetic chiral composites,” Opt. Express 19, 10001–10008 (2011).
[CrossRef] [PubMed]

R. D. Simmonds, P. S. Salter, A. Jesacher, M. J. Booth, “Three dimensional laser microfabrication in diamond using a dual adaptive optics system,” Opt. Express 19, 24122–24128 (2011).
[CrossRef] [PubMed]

2010 (3)

2008 (3)

2006 (1)

S. Wong, M. Deubel, F. Pérez-Willard, S. John, G.A. Ozin, M. Wegener, G. vonFreymann, “Direct laser writing of three- dimensional photonic crystals with a complete photonic bandgap in chalcogenide glasses,” Adv. Mater. 18, 265–269 (2006).
[CrossRef]

1995 (1)

1993 (1)

H. Kobayashi, H. Kanbara, M. Koga, K. Kubodera, “Third-order nonlinear optical properties of As2S3 chalcogenide glass,” J. Appl. Phys. 74, 3683–3687 (1993).
[CrossRef]

1953 (1)

Arezki, B.

Audouard, E.

Booker, G. R.

Booth, M. J.

Bulla, D.

E. Nicoletti, D. Bulla, B. Luther-Davies, M. Gu, “Wide-angle stop-gap chalcogenide photonic crystals generated by direct multiple-line laser writing,” Appl. Phys. B 105, 847–850 (2011).
[CrossRef]

E. Nicoletti, D. Bulla, B. Luther-Davies, M. Gu, “Generation of λ/12 nanowires in chalcogenide glasses,” Nano Lett., 11, 4218–4221 (2011).
[CrossRef] [PubMed]

D. Choi, S. Madden, D. Bulla, R. Wang, A. Rode, B. Luther-Davies, “Thermal annealing of arsenic trisulphide thin film and its influence on device performance,” J. Appl. Phys. 107,053106 (2010).
[CrossRef]

E. Nicoletti, G. Zhou, B. Jia, M. J. Ventura, D. Bulla, B. Luther-Davies, M. Gu, “Observation of multiple higher-order stopgaps from three-dimensional chalcogenide glass photonic crystals,” Opt. Lett. 33, 2311–2313 (2008).
[CrossRef] [PubMed]

Busch, K.

K. Busch, S. Lölkes, R. B. Wehrspohn, H. Föll, Photonic Crystals: Advances in Design, Fabrication, and Characterization (Wiley-VCH, Weinheim, 2004).
[CrossRef]

Choi, D.

D. Choi, S. Madden, D. Bulla, R. Wang, A. Rode, B. Luther-Davies, “Thermal annealing of arsenic trisulphide thin film and its influence on device performance,” J. Appl. Phys. 107,053106 (2010).
[CrossRef]

Cumming, B. P.

M. D. Turner, M. Saba, Q. Zhang, B. P. Cumming, G. E. Schröder-Turk, M. Gu, “Miniature chiral beam-splitter based on Gyroid photonic crystals,” Nat. Photonics 7, 801–805 (2013).
[CrossRef]

B. P. Cumming, S. Debbarma, B. Luther-Davies, M. Gu, “Effect of refractive index mismatch aberration in arsenic trisulfide,” Appl. Phys. B 109, 227–232 (2012).
[CrossRef]

B. P. Cumming, A. Jesacher, M. J. Booth, T. Wilson, M. Gu, “Adaptive aberration compensation for three-dimensional micro-fabrication of photonic crystals in lithium niobate,” Opt. Express 19, 9419–9425 (2011).
[CrossRef] [PubMed]

Debbarma, S.

B. P. Cumming, S. Debbarma, B. Luther-Davies, M. Gu, “Effect of refractive index mismatch aberration in arsenic trisulfide,” Appl. Phys. B 109, 227–232 (2012).
[CrossRef]

Deubel, M.

S. Wong, M. Deubel, F. Pérez-Willard, S. John, G.A. Ozin, M. Wegener, G. vonFreymann, “Direct laser writing of three- dimensional photonic crystals with a complete photonic bandgap in chalcogenide glasses,” Adv. Mater. 18, 265–269 (2006).
[CrossRef]

Eggleton, B. J.

B. J. Eggleton, B. Luther-Davies, K. Richardson, “Chalcogenide photonics,” Nat. Photonics 5, 141–148 (2011).

Föll, H.

K. Busch, S. Lölkes, R. B. Wehrspohn, H. Föll, Photonic Crystals: Advances in Design, Fabrication, and Characterization (Wiley-VCH, Weinheim, 2004).
[CrossRef]

Frerichs, R.

Grosse-Brauckmann, K.

M. Saba, M. Thiel, M. D. Turner, S. T. Hyde, M. Gu, K. Grosse-Brauckmann, D. N. Neshev, K. Mecke, G. E. Schröder-Turk, “Circular dichroism in biological photonic crystals and cubic chiral nets,” Phys. Rev. Lett. 106,103902 (2011).
[CrossRef] [PubMed]

Gu, M.

M. D. Turner, M. Saba, Q. Zhang, B. P. Cumming, G. E. Schröder-Turk, M. Gu, “Miniature chiral beam-splitter based on Gyroid photonic crystals,” Nat. Photonics 7, 801–805 (2013).
[CrossRef]

B. P. Cumming, S. Debbarma, B. Luther-Davies, M. Gu, “Effect of refractive index mismatch aberration in arsenic trisulfide,” Appl. Phys. B 109, 227–232 (2012).
[CrossRef]

B. P. Cumming, A. Jesacher, M. J. Booth, T. Wilson, M. Gu, “Adaptive aberration compensation for three-dimensional micro-fabrication of photonic crystals in lithium niobate,” Opt. Express 19, 9419–9425 (2011).
[CrossRef] [PubMed]

M. D. Turner, G. E. Schröder-Turk, M. Gu, “Fabrication and characterization of three-dimensional biomimetic chiral composites,” Opt. Express 19, 10001–10008 (2011).
[CrossRef] [PubMed]

E. Nicoletti, D. Bulla, B. Luther-Davies, M. Gu, “Generation of λ/12 nanowires in chalcogenide glasses,” Nano Lett., 11, 4218–4221 (2011).
[CrossRef] [PubMed]

M. Saba, M. Thiel, M. D. Turner, S. T. Hyde, M. Gu, K. Grosse-Brauckmann, D. N. Neshev, K. Mecke, G. E. Schröder-Turk, “Circular dichroism in biological photonic crystals and cubic chiral nets,” Phys. Rev. Lett. 106,103902 (2011).
[CrossRef] [PubMed]

E. Nicoletti, D. Bulla, B. Luther-Davies, M. Gu, “Wide-angle stop-gap chalcogenide photonic crystals generated by direct multiple-line laser writing,” Appl. Phys. B 105, 847–850 (2011).
[CrossRef]

E. Nicoletti, G. Zhou, B. Jia, M. J. Ventura, D. Bulla, B. Luther-Davies, M. Gu, “Observation of multiple higher-order stopgaps from three-dimensional chalcogenide glass photonic crystals,” Opt. Lett. 33, 2311–2313 (2008).
[CrossRef] [PubMed]

M. Gu, Advanced Optical Imaging Theory (Springer, Heidelberg, 2000).
[CrossRef]

Huot, N.

Hyde, S. T.

M. Saba, M. Thiel, M. D. Turner, S. T. Hyde, M. Gu, K. Grosse-Brauckmann, D. N. Neshev, K. Mecke, G. E. Schröder-Turk, “Circular dichroism in biological photonic crystals and cubic chiral nets,” Phys. Rev. Lett. 106,103902 (2011).
[CrossRef] [PubMed]

S. T. Hyde, M. O’Keeffe, D. M. Proserpio, “A short history of an elusive yet ubiquitous structure in chemistry, materials, and mathematics,” Angew. Chem. Int. Ed. 47, 7996–8000 (2008).
[CrossRef]

Jesacher, A.

Jha, A.

Jia, B.

John, S.

S. Wong, M. Deubel, F. Pérez-Willard, S. John, G.A. Ozin, M. Wegener, G. vonFreymann, “Direct laser writing of three- dimensional photonic crystals with a complete photonic bandgap in chalcogenide glasses,” Adv. Mater. 18, 265–269 (2006).
[CrossRef]

Jose, G.

Kanbara, H.

H. Kobayashi, H. Kanbara, M. Koga, K. Kubodera, “Third-order nonlinear optical properties of As2S3 chalcogenide glass,” J. Appl. Phys. 74, 3683–3687 (1993).
[CrossRef]

Kar, A.

Kern, P.

Kobayashi, H.

H. Kobayashi, H. Kanbara, M. Koga, K. Kubodera, “Third-order nonlinear optical properties of As2S3 chalcogenide glass,” J. Appl. Phys. 74, 3683–3687 (1993).
[CrossRef]

Koga, M.

H. Kobayashi, H. Kanbara, M. Koga, K. Kubodera, “Third-order nonlinear optical properties of As2S3 chalcogenide glass,” J. Appl. Phys. 74, 3683–3687 (1993).
[CrossRef]

Kubodera, K.

H. Kobayashi, H. Kanbara, M. Koga, K. Kubodera, “Third-order nonlinear optical properties of As2S3 chalcogenide glass,” J. Appl. Phys. 74, 3683–3687 (1993).
[CrossRef]

Kuebler, S. M.

Labadie, L.

Laczik, Z.

Lölkes, S.

K. Busch, S. Lölkes, R. B. Wehrspohn, H. Föll, Photonic Crystals: Advances in Design, Fabrication, and Characterization (Wiley-VCH, Weinheim, 2004).
[CrossRef]

Luo, Z.

Luther-Davies, B.

B. P. Cumming, S. Debbarma, B. Luther-Davies, M. Gu, “Effect of refractive index mismatch aberration in arsenic trisulfide,” Appl. Phys. B 109, 227–232 (2012).
[CrossRef]

B. J. Eggleton, B. Luther-Davies, K. Richardson, “Chalcogenide photonics,” Nat. Photonics 5, 141–148 (2011).

E. Nicoletti, D. Bulla, B. Luther-Davies, M. Gu, “Generation of λ/12 nanowires in chalcogenide glasses,” Nano Lett., 11, 4218–4221 (2011).
[CrossRef] [PubMed]

E. Nicoletti, D. Bulla, B. Luther-Davies, M. Gu, “Wide-angle stop-gap chalcogenide photonic crystals generated by direct multiple-line laser writing,” Appl. Phys. B 105, 847–850 (2011).
[CrossRef]

D. Choi, S. Madden, D. Bulla, R. Wang, A. Rode, B. Luther-Davies, “Thermal annealing of arsenic trisulphide thin film and its influence on device performance,” J. Appl. Phys. 107,053106 (2010).
[CrossRef]

E. Nicoletti, G. Zhou, B. Jia, M. J. Ventura, D. Bulla, B. Luther-Davies, M. Gu, “Observation of multiple higher-order stopgaps from three-dimensional chalcogenide glass photonic crystals,” Opt. Lett. 33, 2311–2313 (2008).
[CrossRef] [PubMed]

Madden, S.

D. Choi, S. Madden, D. Bulla, R. Wang, A. Rode, B. Luther-Davies, “Thermal annealing of arsenic trisulphide thin film and its influence on device performance,” J. Appl. Phys. 107,053106 (2010).
[CrossRef]

Marshall, G. D.

Martin, G.

Mauclair, C.

Mecke, K.

M. Saba, M. Thiel, M. D. Turner, S. T. Hyde, M. Gu, K. Grosse-Brauckmann, D. N. Neshev, K. Mecke, G. E. Schröder-Turk, “Circular dichroism in biological photonic crystals and cubic chiral nets,” Phys. Rev. Lett. 106,103902 (2011).
[CrossRef] [PubMed]

Mermillod-Blondin, A.

Neshev, D. N.

M. Saba, M. Thiel, M. D. Turner, S. T. Hyde, M. Gu, K. Grosse-Brauckmann, D. N. Neshev, K. Mecke, G. E. Schröder-Turk, “Circular dichroism in biological photonic crystals and cubic chiral nets,” Phys. Rev. Lett. 106,103902 (2011).
[CrossRef] [PubMed]

Nicoletti, E.

E. Nicoletti, D. Bulla, B. Luther-Davies, M. Gu, “Wide-angle stop-gap chalcogenide photonic crystals generated by direct multiple-line laser writing,” Appl. Phys. B 105, 847–850 (2011).
[CrossRef]

E. Nicoletti, D. Bulla, B. Luther-Davies, M. Gu, “Generation of λ/12 nanowires in chalcogenide glasses,” Nano Lett., 11, 4218–4221 (2011).
[CrossRef] [PubMed]

E. Nicoletti, G. Zhou, B. Jia, M. J. Ventura, D. Bulla, B. Luther-Davies, M. Gu, “Observation of multiple higher-order stopgaps from three-dimensional chalcogenide glass photonic crystals,” Opt. Lett. 33, 2311–2313 (2008).
[CrossRef] [PubMed]

O’Keeffe, M.

S. T. Hyde, M. O’Keeffe, D. M. Proserpio, “A short history of an elusive yet ubiquitous structure in chemistry, materials, and mathematics,” Angew. Chem. Int. Ed. 47, 7996–8000 (2008).
[CrossRef]

Ozin, G.A.

S. Wong, M. Deubel, F. Pérez-Willard, S. John, G.A. Ozin, M. Wegener, G. vonFreymann, “Direct laser writing of three- dimensional photonic crystals with a complete photonic bandgap in chalcogenide glasses,” Adv. Mater. 18, 265–269 (2006).
[CrossRef]

Pérez-Willard, F.

S. Wong, M. Deubel, F. Pérez-Willard, S. John, G.A. Ozin, M. Wegener, G. vonFreymann, “Direct laser writing of three- dimensional photonic crystals with a complete photonic bandgap in chalcogenide glasses,” Adv. Mater. 18, 265–269 (2006).
[CrossRef]

Proserpio, D. M.

S. T. Hyde, M. O’Keeffe, D. M. Proserpio, “A short history of an elusive yet ubiquitous structure in chemistry, materials, and mathematics,” Angew. Chem. Int. Ed. 47, 7996–8000 (2008).
[CrossRef]

Psaila, N.

Richardson, K.

B. J. Eggleton, B. Luther-Davies, K. Richardson, “Chalcogenide photonics,” Nat. Photonics 5, 141–148 (2011).

Rode, A.

D. Choi, S. Madden, D. Bulla, R. Wang, A. Rode, B. Luther-Davies, “Thermal annealing of arsenic trisulphide thin film and its influence on device performance,” J. Appl. Phys. 107,053106 (2010).
[CrossRef]

Ródenas, A.

Saba, M.

M. D. Turner, M. Saba, Q. Zhang, B. P. Cumming, G. E. Schröder-Turk, M. Gu, “Miniature chiral beam-splitter based on Gyroid photonic crystals,” Nat. Photonics 7, 801–805 (2013).
[CrossRef]

M. Saba, M. Thiel, M. D. Turner, S. T. Hyde, M. Gu, K. Grosse-Brauckmann, D. N. Neshev, K. Mecke, G. E. Schröder-Turk, “Circular dichroism in biological photonic crystals and cubic chiral nets,” Phys. Rev. Lett. 106,103902 (2011).
[CrossRef] [PubMed]

Salter, P. S.

Schröder-Turk, G. E.

M. D. Turner, M. Saba, Q. Zhang, B. P. Cumming, G. E. Schröder-Turk, M. Gu, “Miniature chiral beam-splitter based on Gyroid photonic crystals,” Nat. Photonics 7, 801–805 (2013).
[CrossRef]

M. D. Turner, G. E. Schröder-Turk, M. Gu, “Fabrication and characterization of three-dimensional biomimetic chiral composites,” Opt. Express 19, 10001–10008 (2011).
[CrossRef] [PubMed]

M. Saba, M. Thiel, M. D. Turner, S. T. Hyde, M. Gu, K. Grosse-Brauckmann, D. N. Neshev, K. Mecke, G. E. Schröder-Turk, “Circular dichroism in biological photonic crystals and cubic chiral nets,” Phys. Rev. Lett. 106,103902 (2011).
[CrossRef] [PubMed]

Simmonds, R. D.

Stoian, R.

Thiel, M.

M. Saba, M. Thiel, M. D. Turner, S. T. Hyde, M. Gu, K. Grosse-Brauckmann, D. N. Neshev, K. Mecke, G. E. Schröder-Turk, “Circular dichroism in biological photonic crystals and cubic chiral nets,” Phys. Rev. Lett. 106,103902 (2011).
[CrossRef] [PubMed]

Thomson, R.

Török, P.

Turner, M. D.

M. D. Turner, M. Saba, Q. Zhang, B. P. Cumming, G. E. Schröder-Turk, M. Gu, “Miniature chiral beam-splitter based on Gyroid photonic crystals,” Nat. Photonics 7, 801–805 (2013).
[CrossRef]

M. D. Turner, G. E. Schröder-Turk, M. Gu, “Fabrication and characterization of three-dimensional biomimetic chiral composites,” Opt. Express 19, 10001–10008 (2011).
[CrossRef] [PubMed]

M. Saba, M. Thiel, M. D. Turner, S. T. Hyde, M. Gu, K. Grosse-Brauckmann, D. N. Neshev, K. Mecke, G. E. Schröder-Turk, “Circular dichroism in biological photonic crystals and cubic chiral nets,” Phys. Rev. Lett. 106,103902 (2011).
[CrossRef] [PubMed]

Varga, P.

Ventura, M. J.

vonFreymann, G.

S. Wong, M. Deubel, F. Pérez-Willard, S. John, G.A. Ozin, M. Wegener, G. vonFreymann, “Direct laser writing of three- dimensional photonic crystals with a complete photonic bandgap in chalcogenide glasses,” Adv. Mater. 18, 265–269 (2006).
[CrossRef]

Wang, R.

D. Choi, S. Madden, D. Bulla, R. Wang, A. Rode, B. Luther-Davies, “Thermal annealing of arsenic trisulphide thin film and its influence on device performance,” J. Appl. Phys. 107,053106 (2010).
[CrossRef]

Wegener, M.

S. Wong, M. Deubel, F. Pérez-Willard, S. John, G.A. Ozin, M. Wegener, G. vonFreymann, “Direct laser writing of three- dimensional photonic crystals with a complete photonic bandgap in chalcogenide glasses,” Adv. Mater. 18, 265–269 (2006).
[CrossRef]

Wehrspohn, R. B.

K. Busch, S. Lölkes, R. B. Wehrspohn, H. Föll, Photonic Crystals: Advances in Design, Fabrication, and Characterization (Wiley-VCH, Weinheim, 2004).
[CrossRef]

Williams, H. E.

Wilson, T.

Wong, S.

S. Wong, M. Deubel, F. Pérez-Willard, S. John, G.A. Ozin, M. Wegener, G. vonFreymann, “Direct laser writing of three- dimensional photonic crystals with a complete photonic bandgap in chalcogenide glasses,” Adv. Mater. 18, 265–269 (2006).
[CrossRef]

Zhang, Q.

M. D. Turner, M. Saba, Q. Zhang, B. P. Cumming, G. E. Schröder-Turk, M. Gu, “Miniature chiral beam-splitter based on Gyroid photonic crystals,” Nat. Photonics 7, 801–805 (2013).
[CrossRef]

Zhou, G.

Adv. Mater. (1)

S. Wong, M. Deubel, F. Pérez-Willard, S. John, G.A. Ozin, M. Wegener, G. vonFreymann, “Direct laser writing of three- dimensional photonic crystals with a complete photonic bandgap in chalcogenide glasses,” Adv. Mater. 18, 265–269 (2006).
[CrossRef]

Angew. Chem. Int. Ed. (1)

S. T. Hyde, M. O’Keeffe, D. M. Proserpio, “A short history of an elusive yet ubiquitous structure in chemistry, materials, and mathematics,” Angew. Chem. Int. Ed. 47, 7996–8000 (2008).
[CrossRef]

Appl. Phys. B (2)

E. Nicoletti, D. Bulla, B. Luther-Davies, M. Gu, “Wide-angle stop-gap chalcogenide photonic crystals generated by direct multiple-line laser writing,” Appl. Phys. B 105, 847–850 (2011).
[CrossRef]

B. P. Cumming, S. Debbarma, B. Luther-Davies, M. Gu, “Effect of refractive index mismatch aberration in arsenic trisulfide,” Appl. Phys. B 109, 227–232 (2012).
[CrossRef]

J. Appl. Phys. (2)

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Figures (7)

Fig. 1
Fig. 1

The gyroid PhC simple cubic unit cell.

Fig. 2
Fig. 2

(a) Defocus decoupled focusing geometry. (b) Plot of the peak to peak value of ψ′R as a function of s for d1 = 20 μm. The five images within the plot are phase wrapped images of ψ′R when s = 0, 0.25, 0.498, 0.75 and 1.

Fig. 3
Fig. 3

Five refractive-index mismatch aberration compensation patterns at fabrication depths of 0, 3, 6, 9 and 12 μm when (a) s = 1 (i.e. no defocus decoupling) and (b) s = 0.498 (i.e. optimised defocus decoupling).

Fig. 4
Fig. 4

(a) Side and (c) top view SEM images of gyroids for the case of no aberration compensation. (b) Side and (d) top view SEM images of gyroids for the case of defocus decoupled aberration compensation. (e) and (f) show plots of the measured lateral and axial feature sizes, respectively, at the marked depths in the SEM images for both uncompensated and compensated cases. The scale bars are 3 μm and the dashed line in (f) indicates the trend to the broken lateral line segment shown at a depth of 9 μm in (c).

Fig. 5
Fig. 5

(a) Measured RCP, (b) Measured LCP, (c) Simulated RCP and (d) Simulated LCP transmission spectra along the [001] direction of a gyroid with a size of 33×33×4 unit cells. (e) and (f) are SEM images of the measured structure which were used to determine the lateral and axial feature sizes for the simulations.

Fig. 6
Fig. 6

(a) Measured transmission spectra of uncompensated gyroids fabricated with increasing laser power. (b) Measured transmission spectra of aberration compensated gyroids fabricated with increasing laser power. The size of the gyroids was 33×33×4 unit cells.

Fig. 7
Fig. 7

Experimental setup for adaptive optics enhanced DLW.

Equations (6)

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ψ R ( ρ ) = k 0 d 1 [ s n 1 2 NA 2 ρ 2 n 2 2 NA 2 ρ 2 ] ,
d 0 = s d 1
ψ R ( ρ ) = k 0 d 0 [ n 2 2 NA 2 ρ 2 n 1 2 NA 2 ρ 2 ] ,
ψ D ( ρ ) = k 0 d 1 ( 1 s ) n 2 2 NA 2 ρ 2 ,
ψ R ( ρ ) = ψ R ( ρ ) ψ D ( ρ ) = k 0 d 1 [ s n 1 2 NA 2 ρ 2 n 2 2 NA 2 ρ 2 ] ,
s = n 2 n 2 2 NA 2 n 1 n 1 2 NA 2

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