Abstract

Dielectric microstructures have generated much interest in recent years as a means of accelerating charged particles when powered by solid state lasers. The acceleration gradient (or particle energy gain per unit length) is an important figure of merit. To design structures with high acceleration gradients, we explore the adjoint variable method, a highly efficient technique used to compute the sensitivity of an objective with respect to a large number of parameters. With this formalism, the sensitivity of the acceleration gradient of a dielectric structure with respect to its entire spatial permittivity distribution is calculated by the use of only two full-field electromagnetic simulations, the original and ‘adjoint’. The adjoint simulation corresponds physically to the reciprocal situation of a point charge moving through the accelerator gap and radiating. Using this formalism, we perform numerical optimizations aimed at maximizing acceleration gradients, which generate fabricable structures of greatly improved performance in comparison to previously examined geometries.

© 2017 Optical Society of America

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2015 (2)

K. J. Leedle, A. Ceballos, H. Deng, O. Solgaard, R. F. Pease, R. L. Byer, and J. S. Harris, “Dielectric laser acceleration of sub-100 kev electrons with silicon dual-pillar grating structures,” Opt. Lett. 40, 4344–4347 (2015).
[Crossref] [PubMed]

A. Y. Piggott, J. Lu, K. G. Lagoudakis, J. Petykiewicz, T. M. Babinec, and J. Vučković, “Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer,” Nat. Photonics 9, 374–377 (2015).
[Crossref]

2014 (4)

C. M. Chang and O. Solgaard, “Silicon buried gratings for dielectric laser electron accelerators,” Appl. Phys. Lett. 104, 184102 (2014).
[Crossref]

J. Breuer, J. McNeur, and P. Hommelhoff, “Dielectric laser acceleration of electrons in the vicinity of single and double grating structures; Theory and simulations,” J. Phys. B: At. Mol. Opt. Phys. 47, 234004 (2014).
[Crossref]

J. Breuer, R. Graf, A. Apolonski, and P. Hommelhoff, “Dielectric laser acceleration of nonrelativistic electrons at a single fused silica grating structure: Experimental part,” Phys. Rev. Spec. Top. Accel Beams 17, 021301 (2014).
[Crossref]

R. J. England, R. J. Noble, K. Bane, D. H. Dowell, C.-K. Ng, J. E. Spencer, S. Tantawi, Z. Wu, R. L. Byer, E. Peralta, and K. Soong, “Dielectric laser accelerators,” Rev. Mod. Phys. 86, 1337 (2014).
[Crossref]

2013 (2)

E. Peralta, K. Soong, R. England, E. Colby, Z. Wu, B. Montazeri, C. McGuinness, J. McNeur, K. Leedle, D. Walz, E. Sozer, B. Cowan, G. Travish, and R. Byer, “Demonstration of electron acceleration in a laser-driven dielectric microstructure,” Nature 503, 91–94 (2013).
[Crossref] [PubMed]

C. M. Lalau-Keraly, S. Bhargava, O. D. Miller, and E. Yablonovitch, “Adjoint shape optimization applied to electromagnetic design,” Opt. Express 21, 21693–21701 (2013).
[Crossref] [PubMed]

2012 (3)

W. Shin and S. Fan, “Choice of the perfectly matched layer boundary condition for frequency-domain Maxwell‘s equations solvers,” J. Comput. Phys. 231, 3406–3431 (2012).
[Crossref]

C. Joshi, “The Los Alamos Laser Acceleration of Particles Workshop and beginning of the advanced accelerator concepts field,” AIP Conf. Proc. 1507, 61–66 (2012).

K. Soong, R. Byer, E. Colby, R. England, and E. Peralta, “Laser damage threshold measurements of optical materials for direct laser accelerators,” AIP Conf. Proc. 1507, 511–515 (2012).

2008 (3)

J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. Barty, “Analysis of the scalability of diffraction-limited fiber lasers and amplifiers to high average power,” Opt. Express 16, 13240–13266 (2008).
[Crossref] [PubMed]

T. Plettner and R. Byer, “Microstructure-based laser-driven free-electron laser,” Nucl. Instrum. Methods Phys. Res., Sect. A  593, 63–66 (2008).
[Crossref]

B. M. Cowan, “Three-dimensional dielectric photonic crystal structures for laser-driven acceleration,” Phys. Rev. Spec. Top. Accel Beams 11, 011301 (2008).
[Crossref]

2006 (2)

R.-E. Plessix, “A review of the adjoint-state method for computing the gradient of a functional with geophysical applications,” Geophys. J. Int. 167, 495–503 (2006).
[Crossref]

T. Plettner, P. Lu, and R. Byer, “Proposed few-optical cycle laser-driven particle accelerator structure,” Phys. Rev. Spec. Top. Accel Beams 9, 111301 (2006).
[Crossref]

2005 (1)

P. Musumeci, S. Y. Tochitsky, S. Boucher, C. Clayton, A. Doyuran, R. England, C. Joshi, C. Pellegrini, J. Ralph, J. Rosenzweig, G. Sung, S. Tolmachev, A. Varfolomeev, A. J. Varfolomeev, T. Yarovoi, and R. Yoder, “High energy gain of trapped electrons in a tapered, diffraction-dominated inverse-free-electron laser,” Phys. Rev. Lett. 94, 154801 (2005).
[Crossref] [PubMed]

2004 (2)

2003 (1)

M. Bakr and N. Nikolova, “An adjoint variable method for frequency domain TLM problems with conducting boundaries,” IEEE Microwave Wireless Compon. Lett. 13, 408–410 (2003).
[Crossref]

2002 (1)

N. K. Georgieva, S. Glavic, M. H. Bakr, and J. W. Bandler, “Feasible adjoint sensitivity technique for EM design optimization,” IEEE Trans. Microwave Theory Tech. 50, 2751–2758 (2002).
[Crossref]

2000 (1)

M. B. Giles and N. A. Pierce, “An introduction to the adjoint approach to design,” Flow, Turbulence and Combustion 65, 393–415 (2000).
[Crossref]

1997 (1)

T. Zhang, J. Hirshfield, T. Marshall, and B. Hafizi, “Stimulated dielectric wake-field accelerator,” Phys. Rev. E: Stat. Nonlinear Soft Matter Phys. 56, 4647 (1997).
[Crossref]

1996 (1)

J. Hebling, “Derivation of the pulse front tilt caused by angular dispersion,” Opt. Quantum Electron. 28, 1759–1763 (1996).
[Crossref]

1995 (2)

R. B. Palmer, “Acceleration theorems,” AIP Conf. Proc. 335, 90–100 (1995).
[Crossref]

W. Kimura, G. Kim, R. Romea, L. Steinhauer, I. Pogorelsky, K. Kusche, R. Fernow, X. Wang, and Y. Liu, “Laser acceleration of relativistic electrons using the inverse Cherenkov effect,” Phys. Rev. Lett. 74, 546 (1995).
[Crossref] [PubMed]

1992 (1)

J. Bae, H. Shirai, T. Nishida, T. Nozokido, K. Furuya, and K. Mizuno, “Experimental verification of the theory on the inverse Smith–Purcell effect at a submillimeter wavelength,” Appl. Phys. Lett. 61, 870–872 (1992).
[Crossref]

1987 (1)

K. Mizuno, J. Pae, T. Nozokido, and K. Furuya, “Experimental evidence of the inverse Smith–Purcell effect,” Nature 328, 45–47 (1987).
[Crossref]

1985 (1)

E. Courant, C. Pellegrini, W. Zakowicz, M. Month, P. Dahl, and M. Dienes, “High-energy inverse free-electron laser accelerator,” AIP Conf. Proc. 127, 849–874 (1985).
[Crossref]

1983 (1)

J. Fontana and R. Pantell, “A high-energy, laser accelerator for electrons using the inverse Cherenkov effect,” J. Appl. Phys. 54, 4285–4288 (1983).
[Crossref]

1980 (1)

J. Nocedal, “Updating quasi-newton matrices with limited storage,” Math. Comput. 35, 773–782 (1980).
[Crossref]

Akturk, S.

Apolonski, A.

J. Breuer, R. Graf, A. Apolonski, and P. Hommelhoff, “Dielectric laser acceleration of nonrelativistic electrons at a single fused silica grating structure: Experimental part,” Phys. Rev. Spec. Top. Accel Beams 17, 021301 (2014).
[Crossref]

Avriel, M.

M. Avriel, Nonlinear Programming: Analysis and Methods (Courier Corporation, 2003).

Babinec, T. M.

A. Y. Piggott, J. Lu, K. G. Lagoudakis, J. Petykiewicz, T. M. Babinec, and J. Vučković, “Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer,” Nat. Photonics 9, 374–377 (2015).
[Crossref]

Bae, J.

J. Bae, H. Shirai, T. Nishida, T. Nozokido, K. Furuya, and K. Mizuno, “Experimental verification of the theory on the inverse Smith–Purcell effect at a submillimeter wavelength,” Appl. Phys. Lett. 61, 870–872 (1992).
[Crossref]

Bakr, M.

M. Bakr and N. Nikolova, “An adjoint variable method for frequency domain TLM problems with conducting boundaries,” IEEE Microwave Wireless Compon. Lett. 13, 408–410 (2003).
[Crossref]

Bakr, M. H.

N. K. Georgieva, S. Glavic, M. H. Bakr, and J. W. Bandler, “Feasible adjoint sensitivity technique for EM design optimization,” IEEE Trans. Microwave Theory Tech. 50, 2751–2758 (2002).
[Crossref]

Bandler, J. W.

N. K. Georgieva, S. Glavic, M. H. Bakr, and J. W. Bandler, “Feasible adjoint sensitivity technique for EM design optimization,” IEEE Trans. Microwave Theory Tech. 50, 2751–2758 (2002).
[Crossref]

Bane, K.

R. J. England, R. J. Noble, K. Bane, D. H. Dowell, C.-K. Ng, J. E. Spencer, S. Tantawi, Z. Wu, R. L. Byer, E. Peralta, and K. Soong, “Dielectric laser accelerators,” Rev. Mod. Phys. 86, 1337 (2014).
[Crossref]

Barty, C.

Beach, R. J.

Bhargava, S.

Boucher, S.

P. Musumeci, S. Y. Tochitsky, S. Boucher, C. Clayton, A. Doyuran, R. England, C. Joshi, C. Pellegrini, J. Ralph, J. Rosenzweig, G. Sung, S. Tolmachev, A. Varfolomeev, A. J. Varfolomeev, T. Yarovoi, and R. Yoder, “High energy gain of trapped electrons in a tapered, diffraction-dominated inverse-free-electron laser,” Phys. Rev. Lett. 94, 154801 (2005).
[Crossref] [PubMed]

Breuer, J.

J. Breuer, R. Graf, A. Apolonski, and P. Hommelhoff, “Dielectric laser acceleration of nonrelativistic electrons at a single fused silica grating structure: Experimental part,” Phys. Rev. Spec. Top. Accel Beams 17, 021301 (2014).
[Crossref]

J. Breuer, J. McNeur, and P. Hommelhoff, “Dielectric laser acceleration of electrons in the vicinity of single and double grating structures; Theory and simulations,” J. Phys. B: At. Mol. Opt. Phys. 47, 234004 (2014).
[Crossref]

Byer, R.

E. Peralta, K. Soong, R. England, E. Colby, Z. Wu, B. Montazeri, C. McGuinness, J. McNeur, K. Leedle, D. Walz, E. Sozer, B. Cowan, G. Travish, and R. Byer, “Demonstration of electron acceleration in a laser-driven dielectric microstructure,” Nature 503, 91–94 (2013).
[Crossref] [PubMed]

K. Soong, R. Byer, E. Colby, R. England, and E. Peralta, “Laser damage threshold measurements of optical materials for direct laser accelerators,” AIP Conf. Proc. 1507, 511–515 (2012).

T. Plettner and R. Byer, “Microstructure-based laser-driven free-electron laser,” Nucl. Instrum. Methods Phys. Res., Sect. A  593, 63–66 (2008).
[Crossref]

T. Plettner, P. Lu, and R. Byer, “Proposed few-optical cycle laser-driven particle accelerator structure,” Phys. Rev. Spec. Top. Accel Beams 9, 111301 (2006).
[Crossref]

M. Kozák, M. Förster, J. McNeur, N. Schönenberger, K. Leedle, H. Deng, J. Harris, R. Byer, and P. Hommelhoff, “Dielectric laser acceleration of sub-relativistic electrons by few-cycle laser pulses,” Nucl. Instrum. Methods Phys. Res., Sect. A (2016).
[Crossref]

Byer, R. L.

K. J. Leedle, A. Ceballos, H. Deng, O. Solgaard, R. F. Pease, R. L. Byer, and J. S. Harris, “Dielectric laser acceleration of sub-100 kev electrons with silicon dual-pillar grating structures,” Opt. Lett. 40, 4344–4347 (2015).
[Crossref] [PubMed]

R. J. England, R. J. Noble, K. Bane, D. H. Dowell, C.-K. Ng, J. E. Spencer, S. Tantawi, Z. Wu, R. L. Byer, E. Peralta, and K. Soong, “Dielectric laser accelerators,” Rev. Mod. Phys. 86, 1337 (2014).
[Crossref]

J. McNeur, M. Kozák, N. Schönenberger, K. J. Leedle, H. Deng, A. Ceballos, H. Hoogland, A. Ruehl, I. Hartl, O. Solgaard, J. S. Harris, R. L. Byer, and P. Hommelhof, “Elements of a dielectric laser accelerator,” https://arxiv.org/abs/1604.07684 (2016).

K. Soong, R. L. Byer, C. McGuinness, E. Peralta, and E. Colby, “Experimental determination of damage threshold characteristics of IR compatible optical materials,” 2011 Particle Accelerator Conference Proceedings277, (2011).

Ceballos, A.

K. J. Leedle, A. Ceballos, H. Deng, O. Solgaard, R. F. Pease, R. L. Byer, and J. S. Harris, “Dielectric laser acceleration of sub-100 kev electrons with silicon dual-pillar grating structures,” Opt. Lett. 40, 4344–4347 (2015).
[Crossref] [PubMed]

J. McNeur, M. Kozák, N. Schönenberger, K. J. Leedle, H. Deng, A. Ceballos, H. Hoogland, A. Ruehl, I. Hartl, O. Solgaard, J. S. Harris, R. L. Byer, and P. Hommelhof, “Elements of a dielectric laser accelerator,” https://arxiv.org/abs/1604.07684 (2016).

Chang, C. M.

C. M. Chang and O. Solgaard, “Silicon buried gratings for dielectric laser electron accelerators,” Appl. Phys. Lett. 104, 184102 (2014).
[Crossref]

Clayton, C.

P. Musumeci, S. Y. Tochitsky, S. Boucher, C. Clayton, A. Doyuran, R. England, C. Joshi, C. Pellegrini, J. Ralph, J. Rosenzweig, G. Sung, S. Tolmachev, A. Varfolomeev, A. J. Varfolomeev, T. Yarovoi, and R. Yoder, “High energy gain of trapped electrons in a tapered, diffraction-dominated inverse-free-electron laser,” Phys. Rev. Lett. 94, 154801 (2005).
[Crossref] [PubMed]

Colby, E.

E. Peralta, K. Soong, R. England, E. Colby, Z. Wu, B. Montazeri, C. McGuinness, J. McNeur, K. Leedle, D. Walz, E. Sozer, B. Cowan, G. Travish, and R. Byer, “Demonstration of electron acceleration in a laser-driven dielectric microstructure,” Nature 503, 91–94 (2013).
[Crossref] [PubMed]

K. Soong, R. Byer, E. Colby, R. England, and E. Peralta, “Laser damage threshold measurements of optical materials for direct laser accelerators,” AIP Conf. Proc. 1507, 511–515 (2012).

K. Soong, R. L. Byer, C. McGuinness, E. Peralta, and E. Colby, “Experimental determination of damage threshold characteristics of IR compatible optical materials,” 2011 Particle Accelerator Conference Proceedings277, (2011).

Courant, E.

E. Courant, C. Pellegrini, W. Zakowicz, M. Month, P. Dahl, and M. Dienes, “High-energy inverse free-electron laser accelerator,” AIP Conf. Proc. 127, 849–874 (1985).
[Crossref]

Cowan, B.

E. Peralta, K. Soong, R. England, E. Colby, Z. Wu, B. Montazeri, C. McGuinness, J. McNeur, K. Leedle, D. Walz, E. Sozer, B. Cowan, G. Travish, and R. Byer, “Demonstration of electron acceleration in a laser-driven dielectric microstructure,” Nature 503, 91–94 (2013).
[Crossref] [PubMed]

Cowan, B. M.

B. M. Cowan, “Three-dimensional dielectric photonic crystal structures for laser-driven acceleration,” Phys. Rev. Spec. Top. Accel Beams 11, 011301 (2008).
[Crossref]

Dahl, P.

E. Courant, C. Pellegrini, W. Zakowicz, M. Month, P. Dahl, and M. Dienes, “High-energy inverse free-electron laser accelerator,” AIP Conf. Proc. 127, 849–874 (1985).
[Crossref]

Dawson, J. W.

Deng, H.

K. J. Leedle, A. Ceballos, H. Deng, O. Solgaard, R. F. Pease, R. L. Byer, and J. S. Harris, “Dielectric laser acceleration of sub-100 kev electrons with silicon dual-pillar grating structures,” Opt. Lett. 40, 4344–4347 (2015).
[Crossref] [PubMed]

J. McNeur, M. Kozák, N. Schönenberger, K. J. Leedle, H. Deng, A. Ceballos, H. Hoogland, A. Ruehl, I. Hartl, O. Solgaard, J. S. Harris, R. L. Byer, and P. Hommelhof, “Elements of a dielectric laser accelerator,” https://arxiv.org/abs/1604.07684 (2016).

M. Kozák, M. Förster, J. McNeur, N. Schönenberger, K. Leedle, H. Deng, J. Harris, R. Byer, and P. Hommelhoff, “Dielectric laser acceleration of sub-relativistic electrons by few-cycle laser pulses,” Nucl. Instrum. Methods Phys. Res., Sect. A (2016).
[Crossref]

Dienes, M.

E. Courant, C. Pellegrini, W. Zakowicz, M. Month, P. Dahl, and M. Dienes, “High-energy inverse free-electron laser accelerator,” AIP Conf. Proc. 127, 849–874 (1985).
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R. J. England, R. J. Noble, K. Bane, D. H. Dowell, C.-K. Ng, J. E. Spencer, S. Tantawi, Z. Wu, R. L. Byer, E. Peralta, and K. Soong, “Dielectric laser accelerators,” Rev. Mod. Phys. 86, 1337 (2014).
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P. Musumeci, S. Y. Tochitsky, S. Boucher, C. Clayton, A. Doyuran, R. England, C. Joshi, C. Pellegrini, J. Ralph, J. Rosenzweig, G. Sung, S. Tolmachev, A. Varfolomeev, A. J. Varfolomeev, T. Yarovoi, and R. Yoder, “High energy gain of trapped electrons in a tapered, diffraction-dominated inverse-free-electron laser,” Phys. Rev. Lett. 94, 154801 (2005).
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K. Soong, R. Byer, E. Colby, R. England, and E. Peralta, “Laser damage threshold measurements of optical materials for direct laser accelerators,” AIP Conf. Proc. 1507, 511–515 (2012).

P. Musumeci, S. Y. Tochitsky, S. Boucher, C. Clayton, A. Doyuran, R. England, C. Joshi, C. Pellegrini, J. Ralph, J. Rosenzweig, G. Sung, S. Tolmachev, A. Varfolomeev, A. J. Varfolomeev, T. Yarovoi, and R. Yoder, “High energy gain of trapped electrons in a tapered, diffraction-dominated inverse-free-electron laser,” Phys. Rev. Lett. 94, 154801 (2005).
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R. J. England, R. J. Noble, K. Bane, D. H. Dowell, C.-K. Ng, J. E. Spencer, S. Tantawi, Z. Wu, R. L. Byer, E. Peralta, and K. Soong, “Dielectric laser accelerators,” Rev. Mod. Phys. 86, 1337 (2014).
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W. Shin and S. Fan, “Choice of the perfectly matched layer boundary condition for frequency-domain Maxwell‘s equations solvers,” J. Comput. Phys. 231, 3406–3431 (2012).
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W. Kimura, G. Kim, R. Romea, L. Steinhauer, I. Pogorelsky, K. Kusche, R. Fernow, X. Wang, and Y. Liu, “Laser acceleration of relativistic electrons using the inverse Cherenkov effect,” Phys. Rev. Lett. 74, 546 (1995).
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J. Fontana and R. Pantell, “A high-energy, laser accelerator for electrons using the inverse Cherenkov effect,” J. Appl. Phys. 54, 4285–4288 (1983).
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M. Kozák, M. Förster, J. McNeur, N. Schönenberger, K. Leedle, H. Deng, J. Harris, R. Byer, and P. Hommelhoff, “Dielectric laser acceleration of sub-relativistic electrons by few-cycle laser pulses,” Nucl. Instrum. Methods Phys. Res., Sect. A (2016).
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J. Bae, H. Shirai, T. Nishida, T. Nozokido, K. Furuya, and K. Mizuno, “Experimental verification of the theory on the inverse Smith–Purcell effect at a submillimeter wavelength,” Appl. Phys. Lett. 61, 870–872 (1992).
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N. K. Georgieva, S. Glavic, M. H. Bakr, and J. W. Bandler, “Feasible adjoint sensitivity technique for EM design optimization,” IEEE Trans. Microwave Theory Tech. 50, 2751–2758 (2002).
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J. Breuer, R. Graf, A. Apolonski, and P. Hommelhoff, “Dielectric laser acceleration of nonrelativistic electrons at a single fused silica grating structure: Experimental part,” Phys. Rev. Spec. Top. Accel Beams 17, 021301 (2014).
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Hafizi, B.

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K. J. Leedle, A. Ceballos, H. Deng, O. Solgaard, R. F. Pease, R. L. Byer, and J. S. Harris, “Dielectric laser acceleration of sub-100 kev electrons with silicon dual-pillar grating structures,” Opt. Lett. 40, 4344–4347 (2015).
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J. McNeur, M. Kozák, N. Schönenberger, K. J. Leedle, H. Deng, A. Ceballos, H. Hoogland, A. Ruehl, I. Hartl, O. Solgaard, J. S. Harris, R. L. Byer, and P. Hommelhof, “Elements of a dielectric laser accelerator,” https://arxiv.org/abs/1604.07684 (2016).

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J. McNeur, M. Kozák, N. Schönenberger, K. J. Leedle, H. Deng, A. Ceballos, H. Hoogland, A. Ruehl, I. Hartl, O. Solgaard, J. S. Harris, R. L. Byer, and P. Hommelhof, “Elements of a dielectric laser accelerator,” https://arxiv.org/abs/1604.07684 (2016).

Hommelhoff, P.

J. Breuer, R. Graf, A. Apolonski, and P. Hommelhoff, “Dielectric laser acceleration of nonrelativistic electrons at a single fused silica grating structure: Experimental part,” Phys. Rev. Spec. Top. Accel Beams 17, 021301 (2014).
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J. Breuer, J. McNeur, and P. Hommelhoff, “Dielectric laser acceleration of electrons in the vicinity of single and double grating structures; Theory and simulations,” J. Phys. B: At. Mol. Opt. Phys. 47, 234004 (2014).
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J. McNeur, M. Kozák, N. Schönenberger, K. J. Leedle, H. Deng, A. Ceballos, H. Hoogland, A. Ruehl, I. Hartl, O. Solgaard, J. S. Harris, R. L. Byer, and P. Hommelhof, “Elements of a dielectric laser accelerator,” https://arxiv.org/abs/1604.07684 (2016).

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W. Kimura, G. Kim, R. Romea, L. Steinhauer, I. Pogorelsky, K. Kusche, R. Fernow, X. Wang, and Y. Liu, “Laser acceleration of relativistic electrons using the inverse Cherenkov effect,” Phys. Rev. Lett. 74, 546 (1995).
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W. Kimura, G. Kim, R. Romea, L. Steinhauer, I. Pogorelsky, K. Kusche, R. Fernow, X. Wang, and Y. Liu, “Laser acceleration of relativistic electrons using the inverse Cherenkov effect,” Phys. Rev. Lett. 74, 546 (1995).
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J. McNeur, M. Kozák, N. Schönenberger, K. J. Leedle, H. Deng, A. Ceballos, H. Hoogland, A. Ruehl, I. Hartl, O. Solgaard, J. S. Harris, R. L. Byer, and P. Hommelhof, “Elements of a dielectric laser accelerator,” https://arxiv.org/abs/1604.07684 (2016).

Kusche, K.

W. Kimura, G. Kim, R. Romea, L. Steinhauer, I. Pogorelsky, K. Kusche, R. Fernow, X. Wang, and Y. Liu, “Laser acceleration of relativistic electrons using the inverse Cherenkov effect,” Phys. Rev. Lett. 74, 546 (1995).
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Leedle, K.

E. Peralta, K. Soong, R. England, E. Colby, Z. Wu, B. Montazeri, C. McGuinness, J. McNeur, K. Leedle, D. Walz, E. Sozer, B. Cowan, G. Travish, and R. Byer, “Demonstration of electron acceleration in a laser-driven dielectric microstructure,” Nature 503, 91–94 (2013).
[Crossref] [PubMed]

M. Kozák, M. Förster, J. McNeur, N. Schönenberger, K. Leedle, H. Deng, J. Harris, R. Byer, and P. Hommelhoff, “Dielectric laser acceleration of sub-relativistic electrons by few-cycle laser pulses,” Nucl. Instrum. Methods Phys. Res., Sect. A (2016).
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K. J. Leedle, A. Ceballos, H. Deng, O. Solgaard, R. F. Pease, R. L. Byer, and J. S. Harris, “Dielectric laser acceleration of sub-100 kev electrons with silicon dual-pillar grating structures,” Opt. Lett. 40, 4344–4347 (2015).
[Crossref] [PubMed]

J. McNeur, M. Kozák, N. Schönenberger, K. J. Leedle, H. Deng, A. Ceballos, H. Hoogland, A. Ruehl, I. Hartl, O. Solgaard, J. S. Harris, R. L. Byer, and P. Hommelhof, “Elements of a dielectric laser accelerator,” https://arxiv.org/abs/1604.07684 (2016).

Liu, Y.

W. Kimura, G. Kim, R. Romea, L. Steinhauer, I. Pogorelsky, K. Kusche, R. Fernow, X. Wang, and Y. Liu, “Laser acceleration of relativistic electrons using the inverse Cherenkov effect,” Phys. Rev. Lett. 74, 546 (1995).
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A. Y. Piggott, J. Lu, K. G. Lagoudakis, J. Petykiewicz, T. M. Babinec, and J. Vučković, “Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer,” Nat. Photonics 9, 374–377 (2015).
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T. Plettner, P. Lu, and R. Byer, “Proposed few-optical cycle laser-driven particle accelerator structure,” Phys. Rev. Spec. Top. Accel Beams 9, 111301 (2006).
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T. Zhang, J. Hirshfield, T. Marshall, and B. Hafizi, “Stimulated dielectric wake-field accelerator,” Phys. Rev. E: Stat. Nonlinear Soft Matter Phys. 56, 4647 (1997).
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E. Peralta, K. Soong, R. England, E. Colby, Z. Wu, B. Montazeri, C. McGuinness, J. McNeur, K. Leedle, D. Walz, E. Sozer, B. Cowan, G. Travish, and R. Byer, “Demonstration of electron acceleration in a laser-driven dielectric microstructure,” Nature 503, 91–94 (2013).
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K. Soong, R. L. Byer, C. McGuinness, E. Peralta, and E. Colby, “Experimental determination of damage threshold characteristics of IR compatible optical materials,” 2011 Particle Accelerator Conference Proceedings277, (2011).

McNeur, J.

J. Breuer, J. McNeur, and P. Hommelhoff, “Dielectric laser acceleration of electrons in the vicinity of single and double grating structures; Theory and simulations,” J. Phys. B: At. Mol. Opt. Phys. 47, 234004 (2014).
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E. Peralta, K. Soong, R. England, E. Colby, Z. Wu, B. Montazeri, C. McGuinness, J. McNeur, K. Leedle, D. Walz, E. Sozer, B. Cowan, G. Travish, and R. Byer, “Demonstration of electron acceleration in a laser-driven dielectric microstructure,” Nature 503, 91–94 (2013).
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J. McNeur, M. Kozák, N. Schönenberger, K. J. Leedle, H. Deng, A. Ceballos, H. Hoogland, A. Ruehl, I. Hartl, O. Solgaard, J. S. Harris, R. L. Byer, and P. Hommelhof, “Elements of a dielectric laser accelerator,” https://arxiv.org/abs/1604.07684 (2016).

M. Kozák, M. Förster, J. McNeur, N. Schönenberger, K. Leedle, H. Deng, J. Harris, R. Byer, and P. Hommelhoff, “Dielectric laser acceleration of sub-relativistic electrons by few-cycle laser pulses,” Nucl. Instrum. Methods Phys. Res., Sect. A (2016).
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Miller, O. D.

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J. Bae, H. Shirai, T. Nishida, T. Nozokido, K. Furuya, and K. Mizuno, “Experimental verification of the theory on the inverse Smith–Purcell effect at a submillimeter wavelength,” Appl. Phys. Lett. 61, 870–872 (1992).
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K. Mizuno, J. Pae, T. Nozokido, and K. Furuya, “Experimental evidence of the inverse Smith–Purcell effect,” Nature 328, 45–47 (1987).
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E. Peralta, K. Soong, R. England, E. Colby, Z. Wu, B. Montazeri, C. McGuinness, J. McNeur, K. Leedle, D. Walz, E. Sozer, B. Cowan, G. Travish, and R. Byer, “Demonstration of electron acceleration in a laser-driven dielectric microstructure,” Nature 503, 91–94 (2013).
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E. Courant, C. Pellegrini, W. Zakowicz, M. Month, P. Dahl, and M. Dienes, “High-energy inverse free-electron laser accelerator,” AIP Conf. Proc. 127, 849–874 (1985).
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P. Musumeci, S. Y. Tochitsky, S. Boucher, C. Clayton, A. Doyuran, R. England, C. Joshi, C. Pellegrini, J. Ralph, J. Rosenzweig, G. Sung, S. Tolmachev, A. Varfolomeev, A. J. Varfolomeev, T. Yarovoi, and R. Yoder, “High energy gain of trapped electrons in a tapered, diffraction-dominated inverse-free-electron laser,” Phys. Rev. Lett. 94, 154801 (2005).
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R. J. England, R. J. Noble, K. Bane, D. H. Dowell, C.-K. Ng, J. E. Spencer, S. Tantawi, Z. Wu, R. L. Byer, E. Peralta, and K. Soong, “Dielectric laser accelerators,” Rev. Mod. Phys. 86, 1337 (2014).
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R. J. England, R. J. Noble, K. Bane, D. H. Dowell, C.-K. Ng, J. E. Spencer, S. Tantawi, Z. Wu, R. L. Byer, E. Peralta, and K. Soong, “Dielectric laser accelerators,” Rev. Mod. Phys. 86, 1337 (2014).
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K. Mizuno, J. Pae, T. Nozokido, and K. Furuya, “Experimental evidence of the inverse Smith–Purcell effect,” Nature 328, 45–47 (1987).
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K. Mizuno, J. Pae, T. Nozokido, and K. Furuya, “Experimental evidence of the inverse Smith–Purcell effect,” Nature 328, 45–47 (1987).
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Pease, R. F.

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P. Musumeci, S. Y. Tochitsky, S. Boucher, C. Clayton, A. Doyuran, R. England, C. Joshi, C. Pellegrini, J. Ralph, J. Rosenzweig, G. Sung, S. Tolmachev, A. Varfolomeev, A. J. Varfolomeev, T. Yarovoi, and R. Yoder, “High energy gain of trapped electrons in a tapered, diffraction-dominated inverse-free-electron laser,” Phys. Rev. Lett. 94, 154801 (2005).
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Peralta, E.

R. J. England, R. J. Noble, K. Bane, D. H. Dowell, C.-K. Ng, J. E. Spencer, S. Tantawi, Z. Wu, R. L. Byer, E. Peralta, and K. Soong, “Dielectric laser accelerators,” Rev. Mod. Phys. 86, 1337 (2014).
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E. Peralta, K. Soong, R. England, E. Colby, Z. Wu, B. Montazeri, C. McGuinness, J. McNeur, K. Leedle, D. Walz, E. Sozer, B. Cowan, G. Travish, and R. Byer, “Demonstration of electron acceleration in a laser-driven dielectric microstructure,” Nature 503, 91–94 (2013).
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K. Soong, R. Byer, E. Colby, R. England, and E. Peralta, “Laser damage threshold measurements of optical materials for direct laser accelerators,” AIP Conf. Proc. 1507, 511–515 (2012).

K. Soong, R. L. Byer, C. McGuinness, E. Peralta, and E. Colby, “Experimental determination of damage threshold characteristics of IR compatible optical materials,” 2011 Particle Accelerator Conference Proceedings277, (2011).

Petykiewicz, J.

A. Y. Piggott, J. Lu, K. G. Lagoudakis, J. Petykiewicz, T. M. Babinec, and J. Vučković, “Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer,” Nat. Photonics 9, 374–377 (2015).
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M. B. Giles and N. A. Pierce, “An introduction to the adjoint approach to design,” Flow, Turbulence and Combustion 65, 393–415 (2000).
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A. Y. Piggott, J. Lu, K. G. Lagoudakis, J. Petykiewicz, T. M. Babinec, and J. Vučković, “Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer,” Nat. Photonics 9, 374–377 (2015).
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T. Plettner, P. Lu, and R. Byer, “Proposed few-optical cycle laser-driven particle accelerator structure,” Phys. Rev. Spec. Top. Accel Beams 9, 111301 (2006).
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Pogorelsky, I.

W. Kimura, G. Kim, R. Romea, L. Steinhauer, I. Pogorelsky, K. Kusche, R. Fernow, X. Wang, and Y. Liu, “Laser acceleration of relativistic electrons using the inverse Cherenkov effect,” Phys. Rev. Lett. 74, 546 (1995).
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P. Musumeci, S. Y. Tochitsky, S. Boucher, C. Clayton, A. Doyuran, R. England, C. Joshi, C. Pellegrini, J. Ralph, J. Rosenzweig, G. Sung, S. Tolmachev, A. Varfolomeev, A. J. Varfolomeev, T. Yarovoi, and R. Yoder, “High energy gain of trapped electrons in a tapered, diffraction-dominated inverse-free-electron laser,” Phys. Rev. Lett. 94, 154801 (2005).
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Romea, R.

W. Kimura, G. Kim, R. Romea, L. Steinhauer, I. Pogorelsky, K. Kusche, R. Fernow, X. Wang, and Y. Liu, “Laser acceleration of relativistic electrons using the inverse Cherenkov effect,” Phys. Rev. Lett. 74, 546 (1995).
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P. Musumeci, S. Y. Tochitsky, S. Boucher, C. Clayton, A. Doyuran, R. England, C. Joshi, C. Pellegrini, J. Ralph, J. Rosenzweig, G. Sung, S. Tolmachev, A. Varfolomeev, A. J. Varfolomeev, T. Yarovoi, and R. Yoder, “High energy gain of trapped electrons in a tapered, diffraction-dominated inverse-free-electron laser,” Phys. Rev. Lett. 94, 154801 (2005).
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Ruehl, A.

J. McNeur, M. Kozák, N. Schönenberger, K. J. Leedle, H. Deng, A. Ceballos, H. Hoogland, A. Ruehl, I. Hartl, O. Solgaard, J. S. Harris, R. L. Byer, and P. Hommelhof, “Elements of a dielectric laser accelerator,” https://arxiv.org/abs/1604.07684 (2016).

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J. McNeur, M. Kozák, N. Schönenberger, K. J. Leedle, H. Deng, A. Ceballos, H. Hoogland, A. Ruehl, I. Hartl, O. Solgaard, J. S. Harris, R. L. Byer, and P. Hommelhof, “Elements of a dielectric laser accelerator,” https://arxiv.org/abs/1604.07684 (2016).

M. Kozák, M. Förster, J. McNeur, N. Schönenberger, K. Leedle, H. Deng, J. Harris, R. Byer, and P. Hommelhoff, “Dielectric laser acceleration of sub-relativistic electrons by few-cycle laser pulses,” Nucl. Instrum. Methods Phys. Res., Sect. A (2016).
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W. Shin and S. Fan, “Choice of the perfectly matched layer boundary condition for frequency-domain Maxwell‘s equations solvers,” J. Comput. Phys. 231, 3406–3431 (2012).
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J. Bae, H. Shirai, T. Nishida, T. Nozokido, K. Furuya, and K. Mizuno, “Experimental verification of the theory on the inverse Smith–Purcell effect at a submillimeter wavelength,” Appl. Phys. Lett. 61, 870–872 (1992).
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Siders, C. W.

Solgaard, O.

K. J. Leedle, A. Ceballos, H. Deng, O. Solgaard, R. F. Pease, R. L. Byer, and J. S. Harris, “Dielectric laser acceleration of sub-100 kev electrons with silicon dual-pillar grating structures,” Opt. Lett. 40, 4344–4347 (2015).
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C. M. Chang and O. Solgaard, “Silicon buried gratings for dielectric laser electron accelerators,” Appl. Phys. Lett. 104, 184102 (2014).
[Crossref]

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Soong, K.

R. J. England, R. J. Noble, K. Bane, D. H. Dowell, C.-K. Ng, J. E. Spencer, S. Tantawi, Z. Wu, R. L. Byer, E. Peralta, and K. Soong, “Dielectric laser accelerators,” Rev. Mod. Phys. 86, 1337 (2014).
[Crossref]

E. Peralta, K. Soong, R. England, E. Colby, Z. Wu, B. Montazeri, C. McGuinness, J. McNeur, K. Leedle, D. Walz, E. Sozer, B. Cowan, G. Travish, and R. Byer, “Demonstration of electron acceleration in a laser-driven dielectric microstructure,” Nature 503, 91–94 (2013).
[Crossref] [PubMed]

K. Soong, R. Byer, E. Colby, R. England, and E. Peralta, “Laser damage threshold measurements of optical materials for direct laser accelerators,” AIP Conf. Proc. 1507, 511–515 (2012).

K. Soong, R. L. Byer, C. McGuinness, E. Peralta, and E. Colby, “Experimental determination of damage threshold characteristics of IR compatible optical materials,” 2011 Particle Accelerator Conference Proceedings277, (2011).

Sozer, E.

E. Peralta, K. Soong, R. England, E. Colby, Z. Wu, B. Montazeri, C. McGuinness, J. McNeur, K. Leedle, D. Walz, E. Sozer, B. Cowan, G. Travish, and R. Byer, “Demonstration of electron acceleration in a laser-driven dielectric microstructure,” Nature 503, 91–94 (2013).
[Crossref] [PubMed]

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R. J. England, R. J. Noble, K. Bane, D. H. Dowell, C.-K. Ng, J. E. Spencer, S. Tantawi, Z. Wu, R. L. Byer, E. Peralta, and K. Soong, “Dielectric laser accelerators,” Rev. Mod. Phys. 86, 1337 (2014).
[Crossref]

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Steinhauer, L.

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R. J. England, R. J. Noble, K. Bane, D. H. Dowell, C.-K. Ng, J. E. Spencer, S. Tantawi, Z. Wu, R. L. Byer, E. Peralta, and K. Soong, “Dielectric laser accelerators,” Rev. Mod. Phys. 86, 1337 (2014).
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P. Musumeci, S. Y. Tochitsky, S. Boucher, C. Clayton, A. Doyuran, R. England, C. Joshi, C. Pellegrini, J. Ralph, J. Rosenzweig, G. Sung, S. Tolmachev, A. Varfolomeev, A. J. Varfolomeev, T. Yarovoi, and R. Yoder, “High energy gain of trapped electrons in a tapered, diffraction-dominated inverse-free-electron laser,” Phys. Rev. Lett. 94, 154801 (2005).
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P. Musumeci, S. Y. Tochitsky, S. Boucher, C. Clayton, A. Doyuran, R. England, C. Joshi, C. Pellegrini, J. Ralph, J. Rosenzweig, G. Sung, S. Tolmachev, A. Varfolomeev, A. J. Varfolomeev, T. Yarovoi, and R. Yoder, “High energy gain of trapped electrons in a tapered, diffraction-dominated inverse-free-electron laser,” Phys. Rev. Lett. 94, 154801 (2005).
[Crossref] [PubMed]

Travish, G.

E. Peralta, K. Soong, R. England, E. Colby, Z. Wu, B. Montazeri, C. McGuinness, J. McNeur, K. Leedle, D. Walz, E. Sozer, B. Cowan, G. Travish, and R. Byer, “Demonstration of electron acceleration in a laser-driven dielectric microstructure,” Nature 503, 91–94 (2013).
[Crossref] [PubMed]

Trebino, R.

Varfolomeev, A.

P. Musumeci, S. Y. Tochitsky, S. Boucher, C. Clayton, A. Doyuran, R. England, C. Joshi, C. Pellegrini, J. Ralph, J. Rosenzweig, G. Sung, S. Tolmachev, A. Varfolomeev, A. J. Varfolomeev, T. Yarovoi, and R. Yoder, “High energy gain of trapped electrons in a tapered, diffraction-dominated inverse-free-electron laser,” Phys. Rev. Lett. 94, 154801 (2005).
[Crossref] [PubMed]

Varfolomeev, A. J.

P. Musumeci, S. Y. Tochitsky, S. Boucher, C. Clayton, A. Doyuran, R. England, C. Joshi, C. Pellegrini, J. Ralph, J. Rosenzweig, G. Sung, S. Tolmachev, A. Varfolomeev, A. J. Varfolomeev, T. Yarovoi, and R. Yoder, “High energy gain of trapped electrons in a tapered, diffraction-dominated inverse-free-electron laser,” Phys. Rev. Lett. 94, 154801 (2005).
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Walz, D.

E. Peralta, K. Soong, R. England, E. Colby, Z. Wu, B. Montazeri, C. McGuinness, J. McNeur, K. Leedle, D. Walz, E. Sozer, B. Cowan, G. Travish, and R. Byer, “Demonstration of electron acceleration in a laser-driven dielectric microstructure,” Nature 503, 91–94 (2013).
[Crossref] [PubMed]

Wang, X.

W. Kimura, G. Kim, R. Romea, L. Steinhauer, I. Pogorelsky, K. Kusche, R. Fernow, X. Wang, and Y. Liu, “Laser acceleration of relativistic electrons using the inverse Cherenkov effect,” Phys. Rev. Lett. 74, 546 (1995).
[Crossref] [PubMed]

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R. J. England, R. J. Noble, K. Bane, D. H. Dowell, C.-K. Ng, J. E. Spencer, S. Tantawi, Z. Wu, R. L. Byer, E. Peralta, and K. Soong, “Dielectric laser accelerators,” Rev. Mod. Phys. 86, 1337 (2014).
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E. Peralta, K. Soong, R. England, E. Colby, Z. Wu, B. Montazeri, C. McGuinness, J. McNeur, K. Leedle, D. Walz, E. Sozer, B. Cowan, G. Travish, and R. Byer, “Demonstration of electron acceleration in a laser-driven dielectric microstructure,” Nature 503, 91–94 (2013).
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Yarovoi, T.

P. Musumeci, S. Y. Tochitsky, S. Boucher, C. Clayton, A. Doyuran, R. England, C. Joshi, C. Pellegrini, J. Ralph, J. Rosenzweig, G. Sung, S. Tolmachev, A. Varfolomeev, A. J. Varfolomeev, T. Yarovoi, and R. Yoder, “High energy gain of trapped electrons in a tapered, diffraction-dominated inverse-free-electron laser,” Phys. Rev. Lett. 94, 154801 (2005).
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P. Musumeci, S. Y. Tochitsky, S. Boucher, C. Clayton, A. Doyuran, R. England, C. Joshi, C. Pellegrini, J. Ralph, J. Rosenzweig, G. Sung, S. Tolmachev, A. Varfolomeev, A. J. Varfolomeev, T. Yarovoi, and R. Yoder, “High energy gain of trapped electrons in a tapered, diffraction-dominated inverse-free-electron laser,” Phys. Rev. Lett. 94, 154801 (2005).
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E. Peralta, K. Soong, R. England, E. Colby, Z. Wu, B. Montazeri, C. McGuinness, J. McNeur, K. Leedle, D. Walz, E. Sozer, B. Cowan, G. Travish, and R. Byer, “Demonstration of electron acceleration in a laser-driven dielectric microstructure,” Nature 503, 91–94 (2013).
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P. Musumeci, S. Y. Tochitsky, S. Boucher, C. Clayton, A. Doyuran, R. England, C. Joshi, C. Pellegrini, J. Ralph, J. Rosenzweig, G. Sung, S. Tolmachev, A. Varfolomeev, A. J. Varfolomeev, T. Yarovoi, and R. Yoder, “High energy gain of trapped electrons in a tapered, diffraction-dominated inverse-free-electron laser,” Phys. Rev. Lett. 94, 154801 (2005).
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J. McNeur, M. Kozák, N. Schönenberger, K. J. Leedle, H. Deng, A. Ceballos, H. Hoogland, A. Ruehl, I. Hartl, O. Solgaard, J. S. Harris, R. L. Byer, and P. Hommelhof, “Elements of a dielectric laser accelerator,” https://arxiv.org/abs/1604.07684 (2016).

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K. Soong, R. L. Byer, C. McGuinness, E. Peralta, and E. Colby, “Experimental determination of damage threshold characteristics of IR compatible optical materials,” 2011 Particle Accelerator Conference Proceedings277, (2011).

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Figures (4)

Fig. 1
Fig. 1 Diagram outlining the system setup for side-coupled DLA with an arbitrary dielectric structure (x, y) (green). A charged particle moves through the vacuum gap with speed βc0. The periodicity is set at βλ where λ is the central wavelength of the laser pulse.
Fig. 2
Fig. 2 Demonstration of AVM in calculating sensitivities. (a) The acceleration gradient (G) of a square accelerator structure (inset) as a function of the square’s relative permittivity. We express the acceleration gradient in its dimensionless form, normalized by the electric field amplitude of the incident plane wave (E0). The particle traverses along the dotted line with a velocity of c0 (β = 1) and a plane wave is incident from the bottom of the structure. (b) The sensitivity d G d of the gradient with respect to changing the square relative permittivity for direct central difference (solid line) d G d = G ( + Δ ) G ( Δ ) 2 Δ and using AVM (circles). The two calculations agree with excellent precision. The dotted line at d G d = 0, corresponds to local minima and maxima of G() above.
Fig. 3
Fig. 3 Demonstration of the structure optimization for β = 0.5, laser wavelength λ = 2 μm, and a gap size of 400 nm. A plane wave is incident from the bottom in all cases. (a) Acceleration gradient as a function of iteration number for different maximum relative permittivity values, corresponding to those of Si, Si3N4, and SiO2 at the laser wavelength. The acceleration gradient is normalized by the electric field amplitude of the incident plane wave (E0). The optimizations converge after about five-hundred iterations. (b–d) Final structure permittivity distributions (white = vacuum, black = m) corresponding to the three curves in (a). Eight periods are shown, corresponding to four laser wavelengths. For each (b–d), design region widths on each side of the particle gap were given by 1, 2, and 4 μm for Si, Si3N4, and SiO2, respectively.
Fig. 4
Fig. 4 Demonstration of the final structures after optimization for (a) maximizing gradient only, (b) maximizing the acceleration factor. β = 0.5, laser wavelength λ = 2 μm, gap size of 400 nm. m = 2.1, corresponding to SiO2. In (a), the high gradients are achieved using reflective dielectric mirrors to confine and enhance the fields in the center region. In (b), these dielectric mirrors are removed and the pillar structures are augmented. The structure in (b) shows a 23% increase in the acceleration factor in the material region when compared to (a).

Tables (1)

Tables Icon

Table 1 Acceleration factor (fA) before and after maximization.

Equations (29)

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G = 1 T 0 T E ( r ( t ) , t ) d t ,
E ( r , t ) = Re { E ( r ) exp ( i ω t ) } ,
G = 1 β λ Re { exp ( i ϕ 0 ) 0 β λ d x E x ( x , 0 ) exp ( i 2 π β λ x ) } .
a , b = b , a = V d v ( a b ) = 0 β λ d x d y ( a b ) .
G = Re { E , η } ,
η = η ( x , y ) = 1 β λ exp ( i 2 π β λ x ) δ ( y ) x ^ .
d G d γ = Re { d E d γ , η } .
× × E ( r ) k 0 2 r ( r ) E ( r ) A ^ E ( r ) = i μ 0 ω J ( r ) .
d E d γ = A ^ 1 d A ^ d γ E .
d G d γ = Re { A ^ 1 d A ^ d γ E , η } = Re { E , d A ^ d γ A ^ 1 η } .
A ^ E a j = i μ 0 ω J a j = η ,
d G d γ = Re { E , d A ^ d γ E a j } .
d A ^ d ( r ) = { k 0 2 if r i n square 0 otherwsie .
d G d sq = k 0 2 Re { sq d 2 r E ( r ) E a j ( r ) } .
d G d ( r ) = k 0 2 Re { d 2 r E ( r ) E a j ( r ) δ ( r r ) }
= k 0 2 Re { E ( r ) E a j ( r ) } .
J rad ( x , y ; t ) = q β c 0 δ ( x x 0 c 0 β t ) δ ( y ) x ^ .
J rad ( x , y ; ω ) = q β c 0 δ ( y ) x ^ d t exp ( i ω t ) δ ( x x 0 c 0 β t )
= q exp ( i ω ( x x 0 ) c 0 β ) δ ( y ) x ^
= q exp ( i 2 π β λ x ) exp ( i ϕ 0 ) δ ( y ) x ^ .
J a j = i exp ( i ϕ 0 ) 2 π q β c 0 μ 0 J rad .
A e = b .
G = Re { e T η } ,
d G d i = k 0 2 Re { e i e ¯ i } ,
A e ¯ = η .
i : = i + α d G d i .
i ( j + 1 ) : = i ( j ) + α [ d G ( j ) d i + α d G ( j 1 ) d i ] .
f A = G max { | E | } .
max { | E | } i | E i | exp ( a | E i | ) i exp ( a | E i | ) .

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