Abstract

We demonstrate a reverse design method for realizing a broad range of optical filters based on integrated optical waveguides and experimentally verify example designs on a CMOS-compatible silicon-on-insulator (SOI) platform. The reflectance-based filters allow for control of both phase and amplitude of the optical response. Among this device’s many potential applications we highlight and numerically demonstrate its use for ultrafast on-chip pulse shaping.

© 2014 Optical Society of America

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References

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  1. E. Yablonovitch and T. J. Gmitter, “Photonic band structure: the face-centered-cubic case,” Phys. Rev. Lett. 63(18), 1950–1953 (1989).
    [Crossref] [PubMed]
  2. K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a Photonic Gap in Periodic Dielectric Structures,” Phys. Rev. Lett. 65(25), 3152–3155 (1990).
    [Crossref] [PubMed]
  3. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, (Princeton University Press, 2008).
  4. J. C. Knight, T. A. Birks, P. S. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21(19), 1547–1549 (1996).
    [Crossref] [PubMed]
  5. K. J. Vahala, “Optical Microcavities,” Nature 424(6950), 839–846 (2003).
    [Crossref] [PubMed]
  6. E. Ozbay, “Plasmonics: Merging Photonics and Electronics at Nanoscale Dimensions,” Science 311(5758), 189–193 (2006).
    [Crossref] [PubMed]
  7. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
    [Crossref] [PubMed]
  8. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and Negative Refractive Index,” Science 305(5685), 788–792 (2004).
    [Crossref] [PubMed]
  9. V. M. Shalaev, “‘Optical negative-index metamaterials,” Nat. Photonics 1(1), 41–48 (2007).
    [Crossref]
  10. G. Steinmeyer, “A review of ultrafast optics and optoelectronics,” J. Opt. A, Pure Appl. Opt. 5(1), R1–R5 (2003).
    [Crossref]
  11. C. Brif, R. Chakrabarti, and H. Rabitz, “Control of quantum phenomena: past, present, and future,” New J. Phys. 12(7), 075008 (2010).
    [Crossref]
  12. Y. Zhang, C. Li, and M. Lončar, “Optimal Broadband Antireflective Taper,” Opt. Lett. 38(5), 646–648 (2013).
    [Crossref] [PubMed]
  13. G. B. Arkfen and H. J. Webber, Mathematical Methods for Physicists (Academic Press, 2005).
  14. M. Verbist, D. Van Thourhout, and W. Bogaerts, “Weak gratings in silicon-on-insulator for spectral filters based on volume holography,” Opt. Lett. 38(3), 386–388 (2013).
    [Crossref] [PubMed]
  15. T. Vallius, “Tailored bandgaps: iterative algorithms of diffractive optics,” Opt. Express 21(11), 13810–13817 (2013).
    [Crossref] [PubMed]
  16. D. T. H. Tan, K. Ikeda, R. E. Saperstein, B. Slutsky, and Y. Fainman, “Chip-scale dispersion engineering using chirped vertical gratings,” Opt. Lett. 33(24), 3013–3015 (2008).
    [Crossref] [PubMed]
  17. D. T. H. Tan, K. Ikeda, and Y. Fainman, “Coupled chirped vertical gratings for on-chip group velocity dispersion engineering,” APL 95, 141109 (2009).
  18. D. T. H. Tan, P. C. Sun, and Y. Fainman, “Monolithic nonlinear pulse compressor on a silicon chip,” Nat Commun 1(8), 116 (2010).
    [Crossref] [PubMed]
  19. A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71(5), 1929–1960 (2000).
    [Crossref]
  20. A. Präkelt, M. Wollenhaupt, A. Assion, Ch. Horn, C. Sarpe-Tudoran, M. Winter, and T. Baumert, “Compact, robust, and flexible setup for femtosecond pulse shaping,” Rev. Sci. Instrum. 74(11), 4950–4953 (2003).
    [Crossref]
  21. D. Meshulach and Y. Silberberg, “Coherent quantum control of multiphoton transitions by shaped ultrashort optical pulses,” PRA 60(2), 1287–1292 (1999).
    [Crossref]
  22. P. Nuernberger, G. Vogt, T. Brixner, and G. Gerber, “Femtosecond quantum control of molecular dynamics in the condensed phase,” Phys. Chem. Chem. Phys. 9(20), 2470–2497 (2007).
    [Crossref] [PubMed]
  23. A. Monmayrant, S. Weber, and B. Chatel, “A newcomer’s guide to ultrashort pulse shaping and characterization,” J. Phys. At. Mol. Opt. Phys. 43(10), 103001 (2010).
    [Crossref]
  24. Hamming pulse shape: w(n)=A[0.54−0.46cos(2πnN)],0≤n≤N.

2013 (3)

2010 (3)

C. Brif, R. Chakrabarti, and H. Rabitz, “Control of quantum phenomena: past, present, and future,” New J. Phys. 12(7), 075008 (2010).
[Crossref]

D. T. H. Tan, P. C. Sun, and Y. Fainman, “Monolithic nonlinear pulse compressor on a silicon chip,” Nat Commun 1(8), 116 (2010).
[Crossref] [PubMed]

A. Monmayrant, S. Weber, and B. Chatel, “A newcomer’s guide to ultrashort pulse shaping and characterization,” J. Phys. At. Mol. Opt. Phys. 43(10), 103001 (2010).
[Crossref]

2009 (1)

D. T. H. Tan, K. Ikeda, and Y. Fainman, “Coupled chirped vertical gratings for on-chip group velocity dispersion engineering,” APL 95, 141109 (2009).

2008 (1)

2007 (2)

V. M. Shalaev, “‘Optical negative-index metamaterials,” Nat. Photonics 1(1), 41–48 (2007).
[Crossref]

P. Nuernberger, G. Vogt, T. Brixner, and G. Gerber, “Femtosecond quantum control of molecular dynamics in the condensed phase,” Phys. Chem. Chem. Phys. 9(20), 2470–2497 (2007).
[Crossref] [PubMed]

2006 (1)

E. Ozbay, “Plasmonics: Merging Photonics and Electronics at Nanoscale Dimensions,” Science 311(5758), 189–193 (2006).
[Crossref] [PubMed]

2004 (1)

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and Negative Refractive Index,” Science 305(5685), 788–792 (2004).
[Crossref] [PubMed]

2003 (4)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref] [PubMed]

G. Steinmeyer, “A review of ultrafast optics and optoelectronics,” J. Opt. A, Pure Appl. Opt. 5(1), R1–R5 (2003).
[Crossref]

K. J. Vahala, “Optical Microcavities,” Nature 424(6950), 839–846 (2003).
[Crossref] [PubMed]

A. Präkelt, M. Wollenhaupt, A. Assion, Ch. Horn, C. Sarpe-Tudoran, M. Winter, and T. Baumert, “Compact, robust, and flexible setup for femtosecond pulse shaping,” Rev. Sci. Instrum. 74(11), 4950–4953 (2003).
[Crossref]

2000 (1)

A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71(5), 1929–1960 (2000).
[Crossref]

1999 (1)

D. Meshulach and Y. Silberberg, “Coherent quantum control of multiphoton transitions by shaped ultrashort optical pulses,” PRA 60(2), 1287–1292 (1999).
[Crossref]

1996 (1)

1990 (1)

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a Photonic Gap in Periodic Dielectric Structures,” Phys. Rev. Lett. 65(25), 3152–3155 (1990).
[Crossref] [PubMed]

1989 (1)

E. Yablonovitch and T. J. Gmitter, “Photonic band structure: the face-centered-cubic case,” Phys. Rev. Lett. 63(18), 1950–1953 (1989).
[Crossref] [PubMed]

Assion, A.

A. Präkelt, M. Wollenhaupt, A. Assion, Ch. Horn, C. Sarpe-Tudoran, M. Winter, and T. Baumert, “Compact, robust, and flexible setup for femtosecond pulse shaping,” Rev. Sci. Instrum. 74(11), 4950–4953 (2003).
[Crossref]

Atkin, D. M.

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref] [PubMed]

Baumert, T.

A. Präkelt, M. Wollenhaupt, A. Assion, Ch. Horn, C. Sarpe-Tudoran, M. Winter, and T. Baumert, “Compact, robust, and flexible setup for femtosecond pulse shaping,” Rev. Sci. Instrum. 74(11), 4950–4953 (2003).
[Crossref]

Birks, T. A.

Bogaerts, W.

Brif, C.

C. Brif, R. Chakrabarti, and H. Rabitz, “Control of quantum phenomena: past, present, and future,” New J. Phys. 12(7), 075008 (2010).
[Crossref]

Brixner, T.

P. Nuernberger, G. Vogt, T. Brixner, and G. Gerber, “Femtosecond quantum control of molecular dynamics in the condensed phase,” Phys. Chem. Chem. Phys. 9(20), 2470–2497 (2007).
[Crossref] [PubMed]

Chakrabarti, R.

C. Brif, R. Chakrabarti, and H. Rabitz, “Control of quantum phenomena: past, present, and future,” New J. Phys. 12(7), 075008 (2010).
[Crossref]

Chan, C. T.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a Photonic Gap in Periodic Dielectric Structures,” Phys. Rev. Lett. 65(25), 3152–3155 (1990).
[Crossref] [PubMed]

Chatel, B.

A. Monmayrant, S. Weber, and B. Chatel, “A newcomer’s guide to ultrashort pulse shaping and characterization,” J. Phys. At. Mol. Opt. Phys. 43(10), 103001 (2010).
[Crossref]

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref] [PubMed]

Ebbesen, T. W.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref] [PubMed]

Fainman, Y.

D. T. H. Tan, P. C. Sun, and Y. Fainman, “Monolithic nonlinear pulse compressor on a silicon chip,” Nat Commun 1(8), 116 (2010).
[Crossref] [PubMed]

D. T. H. Tan, K. Ikeda, and Y. Fainman, “Coupled chirped vertical gratings for on-chip group velocity dispersion engineering,” APL 95, 141109 (2009).

D. T. H. Tan, K. Ikeda, R. E. Saperstein, B. Slutsky, and Y. Fainman, “Chip-scale dispersion engineering using chirped vertical gratings,” Opt. Lett. 33(24), 3013–3015 (2008).
[Crossref] [PubMed]

Gerber, G.

P. Nuernberger, G. Vogt, T. Brixner, and G. Gerber, “Femtosecond quantum control of molecular dynamics in the condensed phase,” Phys. Chem. Chem. Phys. 9(20), 2470–2497 (2007).
[Crossref] [PubMed]

Gmitter, T. J.

E. Yablonovitch and T. J. Gmitter, “Photonic band structure: the face-centered-cubic case,” Phys. Rev. Lett. 63(18), 1950–1953 (1989).
[Crossref] [PubMed]

Ho, K. M.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a Photonic Gap in Periodic Dielectric Structures,” Phys. Rev. Lett. 65(25), 3152–3155 (1990).
[Crossref] [PubMed]

Horn, Ch.

A. Präkelt, M. Wollenhaupt, A. Assion, Ch. Horn, C. Sarpe-Tudoran, M. Winter, and T. Baumert, “Compact, robust, and flexible setup for femtosecond pulse shaping,” Rev. Sci. Instrum. 74(11), 4950–4953 (2003).
[Crossref]

Ikeda, K.

D. T. H. Tan, K. Ikeda, and Y. Fainman, “Coupled chirped vertical gratings for on-chip group velocity dispersion engineering,” APL 95, 141109 (2009).

D. T. H. Tan, K. Ikeda, R. E. Saperstein, B. Slutsky, and Y. Fainman, “Chip-scale dispersion engineering using chirped vertical gratings,” Opt. Lett. 33(24), 3013–3015 (2008).
[Crossref] [PubMed]

Knight, J. C.

Li, C.

Loncar, M.

Meshulach, D.

D. Meshulach and Y. Silberberg, “Coherent quantum control of multiphoton transitions by shaped ultrashort optical pulses,” PRA 60(2), 1287–1292 (1999).
[Crossref]

Monmayrant, A.

A. Monmayrant, S. Weber, and B. Chatel, “A newcomer’s guide to ultrashort pulse shaping and characterization,” J. Phys. At. Mol. Opt. Phys. 43(10), 103001 (2010).
[Crossref]

Nuernberger, P.

P. Nuernberger, G. Vogt, T. Brixner, and G. Gerber, “Femtosecond quantum control of molecular dynamics in the condensed phase,” Phys. Chem. Chem. Phys. 9(20), 2470–2497 (2007).
[Crossref] [PubMed]

Ozbay, E.

E. Ozbay, “Plasmonics: Merging Photonics and Electronics at Nanoscale Dimensions,” Science 311(5758), 189–193 (2006).
[Crossref] [PubMed]

Pendry, J. B.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and Negative Refractive Index,” Science 305(5685), 788–792 (2004).
[Crossref] [PubMed]

Präkelt, A.

A. Präkelt, M. Wollenhaupt, A. Assion, Ch. Horn, C. Sarpe-Tudoran, M. Winter, and T. Baumert, “Compact, robust, and flexible setup for femtosecond pulse shaping,” Rev. Sci. Instrum. 74(11), 4950–4953 (2003).
[Crossref]

Rabitz, H.

C. Brif, R. Chakrabarti, and H. Rabitz, “Control of quantum phenomena: past, present, and future,” New J. Phys. 12(7), 075008 (2010).
[Crossref]

Russell, P. S.

Saperstein, R. E.

Sarpe-Tudoran, C.

A. Präkelt, M. Wollenhaupt, A. Assion, Ch. Horn, C. Sarpe-Tudoran, M. Winter, and T. Baumert, “Compact, robust, and flexible setup for femtosecond pulse shaping,” Rev. Sci. Instrum. 74(11), 4950–4953 (2003).
[Crossref]

Shalaev, V. M.

V. M. Shalaev, “‘Optical negative-index metamaterials,” Nat. Photonics 1(1), 41–48 (2007).
[Crossref]

Silberberg, Y.

D. Meshulach and Y. Silberberg, “Coherent quantum control of multiphoton transitions by shaped ultrashort optical pulses,” PRA 60(2), 1287–1292 (1999).
[Crossref]

Slutsky, B.

Smith, D. R.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and Negative Refractive Index,” Science 305(5685), 788–792 (2004).
[Crossref] [PubMed]

Soukoulis, C. M.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a Photonic Gap in Periodic Dielectric Structures,” Phys. Rev. Lett. 65(25), 3152–3155 (1990).
[Crossref] [PubMed]

Steinmeyer, G.

G. Steinmeyer, “A review of ultrafast optics and optoelectronics,” J. Opt. A, Pure Appl. Opt. 5(1), R1–R5 (2003).
[Crossref]

Sun, P. C.

D. T. H. Tan, P. C. Sun, and Y. Fainman, “Monolithic nonlinear pulse compressor on a silicon chip,” Nat Commun 1(8), 116 (2010).
[Crossref] [PubMed]

Tan, D. T. H.

D. T. H. Tan, P. C. Sun, and Y. Fainman, “Monolithic nonlinear pulse compressor on a silicon chip,” Nat Commun 1(8), 116 (2010).
[Crossref] [PubMed]

D. T. H. Tan, K. Ikeda, and Y. Fainman, “Coupled chirped vertical gratings for on-chip group velocity dispersion engineering,” APL 95, 141109 (2009).

D. T. H. Tan, K. Ikeda, R. E. Saperstein, B. Slutsky, and Y. Fainman, “Chip-scale dispersion engineering using chirped vertical gratings,” Opt. Lett. 33(24), 3013–3015 (2008).
[Crossref] [PubMed]

Vahala, K. J.

K. J. Vahala, “Optical Microcavities,” Nature 424(6950), 839–846 (2003).
[Crossref] [PubMed]

Vallius, T.

Van Thourhout, D.

Verbist, M.

Vogt, G.

P. Nuernberger, G. Vogt, T. Brixner, and G. Gerber, “Femtosecond quantum control of molecular dynamics in the condensed phase,” Phys. Chem. Chem. Phys. 9(20), 2470–2497 (2007).
[Crossref] [PubMed]

Weber, S.

A. Monmayrant, S. Weber, and B. Chatel, “A newcomer’s guide to ultrashort pulse shaping and characterization,” J. Phys. At. Mol. Opt. Phys. 43(10), 103001 (2010).
[Crossref]

Weiner, A. M.

A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71(5), 1929–1960 (2000).
[Crossref]

Wiltshire, M. C. K.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and Negative Refractive Index,” Science 305(5685), 788–792 (2004).
[Crossref] [PubMed]

Winter, M.

A. Präkelt, M. Wollenhaupt, A. Assion, Ch. Horn, C. Sarpe-Tudoran, M. Winter, and T. Baumert, “Compact, robust, and flexible setup for femtosecond pulse shaping,” Rev. Sci. Instrum. 74(11), 4950–4953 (2003).
[Crossref]

Wollenhaupt, M.

A. Präkelt, M. Wollenhaupt, A. Assion, Ch. Horn, C. Sarpe-Tudoran, M. Winter, and T. Baumert, “Compact, robust, and flexible setup for femtosecond pulse shaping,” Rev. Sci. Instrum. 74(11), 4950–4953 (2003).
[Crossref]

Yablonovitch, E.

E. Yablonovitch and T. J. Gmitter, “Photonic band structure: the face-centered-cubic case,” Phys. Rev. Lett. 63(18), 1950–1953 (1989).
[Crossref] [PubMed]

Zhang, Y.

APL (1)

D. T. H. Tan, K. Ikeda, and Y. Fainman, “Coupled chirped vertical gratings for on-chip group velocity dispersion engineering,” APL 95, 141109 (2009).

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

G. Steinmeyer, “A review of ultrafast optics and optoelectronics,” J. Opt. A, Pure Appl. Opt. 5(1), R1–R5 (2003).
[Crossref]

J. Phys. At. Mol. Opt. Phys. (1)

A. Monmayrant, S. Weber, and B. Chatel, “A newcomer’s guide to ultrashort pulse shaping and characterization,” J. Phys. At. Mol. Opt. Phys. 43(10), 103001 (2010).
[Crossref]

Nat Commun (1)

D. T. H. Tan, P. C. Sun, and Y. Fainman, “Monolithic nonlinear pulse compressor on a silicon chip,” Nat Commun 1(8), 116 (2010).
[Crossref] [PubMed]

Nat. Photonics (1)

V. M. Shalaev, “‘Optical negative-index metamaterials,” Nat. Photonics 1(1), 41–48 (2007).
[Crossref]

Nature (2)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref] [PubMed]

K. J. Vahala, “Optical Microcavities,” Nature 424(6950), 839–846 (2003).
[Crossref] [PubMed]

New J. Phys. (1)

C. Brif, R. Chakrabarti, and H. Rabitz, “Control of quantum phenomena: past, present, and future,” New J. Phys. 12(7), 075008 (2010).
[Crossref]

Opt. Express (1)

Opt. Lett. (4)

Phys. Chem. Chem. Phys. (1)

P. Nuernberger, G. Vogt, T. Brixner, and G. Gerber, “Femtosecond quantum control of molecular dynamics in the condensed phase,” Phys. Chem. Chem. Phys. 9(20), 2470–2497 (2007).
[Crossref] [PubMed]

Phys. Rev. Lett. (2)

E. Yablonovitch and T. J. Gmitter, “Photonic band structure: the face-centered-cubic case,” Phys. Rev. Lett. 63(18), 1950–1953 (1989).
[Crossref] [PubMed]

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a Photonic Gap in Periodic Dielectric Structures,” Phys. Rev. Lett. 65(25), 3152–3155 (1990).
[Crossref] [PubMed]

PRA (1)

D. Meshulach and Y. Silberberg, “Coherent quantum control of multiphoton transitions by shaped ultrashort optical pulses,” PRA 60(2), 1287–1292 (1999).
[Crossref]

Rev. Sci. Instrum. (2)

A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71(5), 1929–1960 (2000).
[Crossref]

A. Präkelt, M. Wollenhaupt, A. Assion, Ch. Horn, C. Sarpe-Tudoran, M. Winter, and T. Baumert, “Compact, robust, and flexible setup for femtosecond pulse shaping,” Rev. Sci. Instrum. 74(11), 4950–4953 (2003).
[Crossref]

Science (2)

E. Ozbay, “Plasmonics: Merging Photonics and Electronics at Nanoscale Dimensions,” Science 311(5758), 189–193 (2006).
[Crossref] [PubMed]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and Negative Refractive Index,” Science 305(5685), 788–792 (2004).
[Crossref] [PubMed]

Other (3)

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, (Princeton University Press, 2008).

G. B. Arkfen and H. J. Webber, Mathematical Methods for Physicists (Academic Press, 2005).

Hamming pulse shape: w(n)=A[0.54−0.46cos(2πnN)],0≤n≤N.

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Figures (4)

Fig. 1
Fig. 1 (a) An SEM micrograph of a fabricated waveguide showing the W(x) profile. (b) Cross-sections of the electric field intensity of the modes supported by our ridge waveguides. The TE-like mode features an E-field discontinuity at sides of the waveguide, while the TM mode’s discontinuity is at the top and bottom of the waveguide. (c) An example target R(λ). (d) The width profile, W(x), that is obtained by applying the inverse Fourier transform obtained from Eq. (2) to the spectrum from b. (e) Simulated reflections, using FDTD analysis, from four iterations of our design method. The final step (thick black line) has come very close to replicating the spectrum from (b). Further optimizations are possible, though each subsequent iteration yields diminishing returns.
Fig. 2
Fig. 2 (a) Iterative design process to compensate for inaccuracies due to assumptions in solution to Eq. (1). The result of the design process is checked via simulation, and any discrepancies in the resulting spectrum are used to feed back a wavelength dependent correction factor into the design process. (b) A comparison of the efficacy of our design method. The target reflection spectrum, shown in blue, was used to calculate the width profile using the Eq. (6). The reflection spectrum resulting from the obtained width profile was then found by solving Eq. (1) numerically, using an ordinary differential equation (ODE) solver, and results are plotted in red. The reflection spectra are also calculated using 3D FDTD modeling in the case of both TE and TM modes. Excellent agreement between ODE and FDTD solutions is found for TM mode, and is attributed to the lack of E-field discontinuity in the direction of interest (along waveguide width).
Fig. 3
Fig. 3 Numerical analysis, using full 3D FDTD simulation, of ultra-fast pulse shaping. The sech2 input pulse is incident on two different width modulated filters designed to reflect (a) “triple Hamming” pulse (shown in light blue), and (b) “square” pulse (shown in purple) with constant amplitude over a wide range. In both cases input sech2 pulse is shown to the left of reflected pulses.
Fig. 4
Fig. 4 (a) SEM micrograph of example device; the inset shows a magnification of the width modulated region. Cartoons show the flow of the experiment: light is launched from a tunable telecom laser through a lensed fiber, the light is then coupled through a polymer waveguide to the silicon waveguides through an inverse taper, the light reflected from the modulated waveguides is extracted using a directional coupler to another polymer waveguide, and finally collected through a second lensed fiber into a photodetector. The light on-chip is propagating in the TM mode (b) The spectral shape of the “triple Hamming” filter, overlayed with the target design (c) The spectral shape of the square-pulse filter as probed by the tunable laser. The dashed overlay represents our target design. The measured spectra in both (b) and (c) are normalized by the magnitude of the light transmitted through the filter. We note that in both cases the signal is distorted by Fabry-Perot resonances in the system (e.g. from the SU-8 waveguide facets) as well as truncation effects due to the finite size and resolution of the width-modulated filters.

Equations (6)

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r ( x ^ , λ ) + i 4 π r ( x ^ , λ ) = 1 2 [ 1 r ( x ^ , λ ) 2 ] ln [ n ( x ^ , λ ) ]
r ( λ ) = 1 2 J ( λ ) 0 l d W d x exp [ i 2 π 2 n e f f ( λ ) λ x ] d x
d W ( x ) d x = 2 r ( λ ) J ( λ ) e x p [ i 2 π 2 n e f f ( λ ) λ x ] d { 2 n e f f ( λ ) λ }
r ( x ^ , λ ) [ 1 r ( x ^ , λ ) 2 ] + i 4 π L λ r ( x ^ , λ ) [ 1 r ( x ^ , λ ) 2 ] = 1 2 ln [ n ( x ^ , λ ) ]
tan h 1 [ r ( x ^ , λ ) ] ' + i 4 π L λ tan h 1 [ r ( x ^ , λ ) ] = 1 2 ln [ n ( x ^ , λ ) ]
W ( x ) = 0 x d x d { 2 n e f f ( λ ) λ } 2 tan h 1 [ r ( λ ) ] J ( λ ) e x p [ i 2 π 2 n e f f ( λ ) λ x ]

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