Abstract

A matrix approach is formulated to describe third-harmonic (TH) generation in stacked materials in the small signal limit, in both transmission and reflection geometries. The model takes into account the contribution from the substrate to the total generated TH, interference of fundamental and nonlinear fields inside the stack, the nonlinear signal generation in forward and backward direction, the beam profile of the focused incident beam in the substrate, and the finite spectrum associated with short laser pulses. The model is applied to design stacks of thin films for efficient TH generation.

© 2014 Optical Society of America

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Corrections

Cristina Rodríguez and Wolfgang Rudolph, "Modeling third harmonic generation from layered materials using nonlinear optical matrices: erratum," Opt. Express 23, 26670-26671 (2015)
http://proxy.osapublishing.org/oe/abstract.cfm?uri=oe-23-20-26670

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References

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  1. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. 127, 1918–1939 (1962).
    [Crossref]
  2. R.C. Miller, “Optical Harmonic Generation in Single Crystal BaTiO3,” Phys. Rev. 134, 1313–1319 (1964).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  7. S. Zhang, W. You, and Z. Huang, “Nonlinear Cascaded Femtosecond Third Harmonic Generation by Multi-grating Periodically Poled MgO-doped Lithium Niobate,” J. Opt. Photonics 3, 50–52 (2013).
    [Crossref]
  8. D. L. Williams, D. P. West, and T. A. King, “Quasi-phase matched third harmonic generation,” Opt. Commun. 148, 208–214 (1998).
    [Crossref]
  9. G. I. Petrov, V. Shcheslavskiy, V. V. Yakovlev, I. Ozerov, E. Chelnokov, and W. Marine, “Efficient third-harmonic generation in a thin nanocrystalline film of ZnO,” Appl. Phys. Lett. 83, 3993–3995 (2003).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  13. D. S. Bethune, “Optical harmonic generation and mixing in multilayer media: analysis using optical transfer matrix techniques,” J. Opt. Soc. Am. B 6, 910–916 (1989).
    [Crossref]
  14. C. Rodriguez and W. Rudolph, “Characterization and χ(3) measurements of thin films by third-harmonic microscopy,” Opt. Lett. (to be published).
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  16. E. O. Potma, W. P. de Boeij, and D. A. Wiersma, “Nonlinear coherent four-wave mixing in optical microscopy,” J. Opt. Soc. Am. B 17, 1678–1684 (2000).
    [Crossref]
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2013 (1)

S. Zhang, W. You, and Z. Huang, “Nonlinear Cascaded Femtosecond Third Harmonic Generation by Multi-grating Periodically Poled MgO-doped Lithium Niobate,” J. Opt. Photonics 3, 50–52 (2013).
[Crossref]

2011 (1)

2007 (1)

D. S. Hum and M. M. Fejer, “Quasi-phasematching,” Comptes Rendus Physique 8, 180–198 (2007).
[Crossref]

2005 (1)

S. M. Saltiel, A. A. Sukhorukov, and Yu. S. Kivshar, “Multistep Parametric Processes in Nonlinear Optics,” Progress in Optics 47, 1–73 (2005).
[Crossref]

2003 (1)

G. I. Petrov, V. Shcheslavskiy, V. V. Yakovlev, I. Ozerov, E. Chelnokov, and W. Marine, “Efficient third-harmonic generation in a thin nanocrystalline film of ZnO,” Appl. Phys. Lett. 83, 3993–3995 (2003).
[Crossref]

2002 (1)

2000 (1)

1998 (1)

D. L. Williams, D. P. West, and T. A. King, “Quasi-phase matched third harmonic generation,” Opt. Commun. 148, 208–214 (1998).
[Crossref]

1994 (1)

1989 (2)

F. Krausz, E. Wintner, and G. Leising, “Optical third-harmonic generation in polyacetylene,” Phys. Rev. B 39, 3701–3710 (1989).
[Crossref]

D. S. Bethune, “Optical harmonic generation and mixing in multilayer media: analysis using optical transfer matrix techniques,” J. Opt. Soc. Am. B 6, 910–916 (1989).
[Crossref]

1964 (1)

R.C. Miller, “Optical Harmonic Generation in Single Crystal BaTiO3,” Phys. Rev. 134, 1313–1319 (1964).
[Crossref]

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

Banks, P. S.

Bethune, D. S.

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

Chelnokov, E.

G. I. Petrov, V. Shcheslavskiy, V. V. Yakovlev, I. Ozerov, E. Chelnokov, and W. Marine, “Efficient third-harmonic generation in a thin nanocrystalline film of ZnO,” Appl. Phys. Lett. 83, 3993–3995 (2003).
[Crossref]

de Boeij, W. P.

Diels, J. C.

J. C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic Press, 2006), Chap. 3.

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

Feit, M. D.

Fejer, M. M.

D. S. Hum and M. M. Fejer, “Quasi-phasematching,” Comptes Rendus Physique 8, 180–198 (2007).
[Crossref]

Huang, Z.

S. Zhang, W. You, and Z. Huang, “Nonlinear Cascaded Femtosecond Third Harmonic Generation by Multi-grating Periodically Poled MgO-doped Lithium Niobate,” J. Opt. Photonics 3, 50–52 (2013).
[Crossref]

Hum, D. S.

D. S. Hum and M. M. Fejer, “Quasi-phasematching,” Comptes Rendus Physique 8, 180–198 (2007).
[Crossref]

King, T. A.

D. L. Williams, D. P. West, and T. A. King, “Quasi-phase matched third harmonic generation,” Opt. Commun. 148, 208–214 (1998).
[Crossref]

Kivshar, Yu. S.

S. M. Saltiel, A. A. Sukhorukov, and Yu. S. Kivshar, “Multistep Parametric Processes in Nonlinear Optics,” Progress in Optics 47, 1–73 (2005).
[Crossref]

Krausz, F.

F. Krausz, E. Wintner, and G. Leising, “Optical third-harmonic generation in polyacetylene,” Phys. Rev. B 39, 3701–3710 (1989).
[Crossref]

Leising, G.

F. Krausz, E. Wintner, and G. Leising, “Optical third-harmonic generation in polyacetylene,” Phys. Rev. B 39, 3701–3710 (1989).
[Crossref]

Marine, W.

G. I. Petrov, V. Shcheslavskiy, V. V. Yakovlev, I. Ozerov, E. Chelnokov, and W. Marine, “Efficient third-harmonic generation in a thin nanocrystalline film of ZnO,” Appl. Phys. Lett. 83, 3993–3995 (2003).
[Crossref]

Miller, R.C.

R.C. Miller, “Optical Harmonic Generation in Single Crystal BaTiO3,” Phys. Rev. 134, 1313–1319 (1964).
[Crossref]

Miyata, K.

Noack, F.

Ozerov, I.

G. I. Petrov, V. Shcheslavskiy, V. V. Yakovlev, I. Ozerov, E. Chelnokov, and W. Marine, “Efficient third-harmonic generation in a thin nanocrystalline film of ZnO,” Appl. Phys. Lett. 83, 3993–3995 (2003).
[Crossref]

Pedrotti, F. L.

F. L. Pedrotti, L. S. Pedrotti, and L. M. Pedrotti, Introduction to optics (Prentice hall, 2007), Chap. 22.

Pedrotti, L. M.

F. L. Pedrotti, L. S. Pedrotti, and L. M. Pedrotti, Introduction to optics (Prentice hall, 2007), Chap. 22.

Pedrotti, L. S.

F. L. Pedrotti, L. S. Pedrotti, and L. M. Pedrotti, Introduction to optics (Prentice hall, 2007), Chap. 22.

Perry, M. D.

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

Petrov, G. I.

G. I. Petrov, V. Shcheslavskiy, V. V. Yakovlev, I. Ozerov, E. Chelnokov, and W. Marine, “Efficient third-harmonic generation in a thin nanocrystalline film of ZnO,” Appl. Phys. Lett. 83, 3993–3995 (2003).
[Crossref]

Petrov, V.

Potma, E. O.

Rodriguez, C.

C. Rodriguez and W. Rudolph, “Characterization and χ(3) measurements of thin films by third-harmonic microscopy,” Opt. Lett. (to be published).

Rudolph, W.

C. Rodriguez and W. Rudolph, “Characterization and χ(3) measurements of thin films by third-harmonic microscopy,” Opt. Lett. (to be published).

J. C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic Press, 2006), Chap. 3.

Saltiel, S. M.

S. M. Saltiel, A. A. Sukhorukov, and Yu. S. Kivshar, “Multistep Parametric Processes in Nonlinear Optics,” Progress in Optics 47, 1–73 (2005).
[Crossref]

Shcheslavskiy, V.

G. I. Petrov, V. Shcheslavskiy, V. V. Yakovlev, I. Ozerov, E. Chelnokov, and W. Marine, “Efficient third-harmonic generation in a thin nanocrystalline film of ZnO,” Appl. Phys. Lett. 83, 3993–3995 (2003).
[Crossref]

Shen, Y. R.

Y. R. Shen, The Principles of Nonlinear Optics (John Wiley & Sons, 1984), Chap. 3.

Sukhorukov, A. A.

S. M. Saltiel, A. A. Sukhorukov, and Yu. S. Kivshar, “Multistep Parametric Processes in Nonlinear Optics,” Progress in Optics 47, 1–73 (2005).
[Crossref]

Tsang, T.

West, D. P.

D. L. Williams, D. P. West, and T. A. King, “Quasi-phase matched third harmonic generation,” Opt. Commun. 148, 208–214 (1998).
[Crossref]

Wiersma, D. A.

Williams, D. L.

D. L. Williams, D. P. West, and T. A. King, “Quasi-phase matched third harmonic generation,” Opt. Commun. 148, 208–214 (1998).
[Crossref]

Wintner, E.

F. Krausz, E. Wintner, and G. Leising, “Optical third-harmonic generation in polyacetylene,” Phys. Rev. B 39, 3701–3710 (1989).
[Crossref]

Yakovlev, V. V.

G. I. Petrov, V. Shcheslavskiy, V. V. Yakovlev, I. Ozerov, E. Chelnokov, and W. Marine, “Efficient third-harmonic generation in a thin nanocrystalline film of ZnO,” Appl. Phys. Lett. 83, 3993–3995 (2003).
[Crossref]

You, W.

S. Zhang, W. You, and Z. Huang, “Nonlinear Cascaded Femtosecond Third Harmonic Generation by Multi-grating Periodically Poled MgO-doped Lithium Niobate,” J. Opt. Photonics 3, 50–52 (2013).
[Crossref]

Zhang, S.

S. Zhang, W. You, and Z. Huang, “Nonlinear Cascaded Femtosecond Third Harmonic Generation by Multi-grating Periodically Poled MgO-doped Lithium Niobate,” J. Opt. Photonics 3, 50–52 (2013).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

G. I. Petrov, V. Shcheslavskiy, V. V. Yakovlev, I. Ozerov, E. Chelnokov, and W. Marine, “Efficient third-harmonic generation in a thin nanocrystalline film of ZnO,” Appl. Phys. Lett. 83, 3993–3995 (2003).
[Crossref]

Comptes Rendus Physique (1)

D. S. Hum and M. M. Fejer, “Quasi-phasematching,” Comptes Rendus Physique 8, 180–198 (2007).
[Crossref]

J. Opt. Photonics (1)

S. Zhang, W. You, and Z. Huang, “Nonlinear Cascaded Femtosecond Third Harmonic Generation by Multi-grating Periodically Poled MgO-doped Lithium Niobate,” J. Opt. Photonics 3, 50–52 (2013).
[Crossref]

J. Opt. Soc. Am. B (3)

Opt. Commun. (1)

D. L. Williams, D. P. West, and T. A. King, “Quasi-phase matched third harmonic generation,” Opt. Commun. 148, 208–214 (1998).
[Crossref]

Opt. Lett. (1)

Phys. Rev. (2)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

R.C. Miller, “Optical Harmonic Generation in Single Crystal BaTiO3,” Phys. Rev. 134, 1313–1319 (1964).
[Crossref]

Phys. Rev. B (1)

F. Krausz, E. Wintner, and G. Leising, “Optical third-harmonic generation in polyacetylene,” Phys. Rev. B 39, 3701–3710 (1989).
[Crossref]

Progress in Optics (1)

S. M. Saltiel, A. A. Sukhorukov, and Yu. S. Kivshar, “Multistep Parametric Processes in Nonlinear Optics,” Progress in Optics 47, 1–73 (2005).
[Crossref]

Other (4)

C. Rodriguez and W. Rudolph, “Characterization and χ(3) measurements of thin films by third-harmonic microscopy,” Opt. Lett. (to be published).

Y. R. Shen, The Principles of Nonlinear Optics (John Wiley & Sons, 1984), Chap. 3.

J. C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic Press, 2006), Chap. 3.

F. L. Pedrotti, L. S. Pedrotti, and L. M. Pedrotti, Introduction to optics (Prentice hall, 2007), Chap. 22.

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Figures (2)

Fig. 1
Fig. 1 TH generation in a stack of layers on a semi-infinite substrate. TH and fundamental fields are labeled E and F, respectively. Fields propagating to the right (r) and left () are distinguished in each layer. Fields on the right interfaces are labeled with a prime.
Fig. 2
Fig. 2 (a) TH generation from stacks of films in which silica layers are progressively added, keeping the total thickness of hafnia constant, equal to one coherence length, D = Lcoh ≈ 627 nm. TH signal in (b) transmission and (c) reflection with interference (crosses) and without interference (circles). Normalization is performed with respect to a single hafnia layer of D = Lcoh. Dotted line shows expected TH from a single hafnia layer, of D = Lcoh, assuming perfect phase matching. The nine-layer stack parameters (thicknesses) are substrate/[HfO2/SiO2]4HfO2 = substrate/98/131/98/122/93/696/103/48/235 (layer thicknesses in nm).

Equations (19)

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= n + 1 , n n n , n 1 2 , 1 1 1 , 0 ,
i , j = 1 t i j ( 1 r i j r i j 1 ) and i = ( e i k i d i 0 0 e + i k i d i )
v n + 1 = ( F r F ) = ( 11 12 21 22 ) ( F 0 , r F 0 , ) ,
F 0 , = 21 22 F 0 , r .
v i = ( F i , r F i , ) = i , i 1 i 1 2 , 1 1 1 , 0 ( F 0 , r F 0 , ) ,
M = H n + 1 , n L n H n , n 1 H 2 , 1 L 1 H 1 , 0 ,
H i , j = 1 τ i j ( 1 ρ i j ρ i j 1 ) and L i = ( e i κ i d i 0 0 e + i κ i d i ) .
E i , r = E i , r e i κ i d i + Δ E i , r ,
( E i , r E i , ) = L i ( E i , r E i , ) + Δ i .
w n + 1 = H n + 1 , n L n H n , n 1 L 1 H 1 , 0 w 0 + H n + 1 , n L n H n , n 1 L 3 H 3 , 2 L 2 H 2 , 1 Δ 1 + H n + 1 , n L n H n , n 1 L 3 H 3 , 2 Δ 2 + + H n + 1 , n Δ n .
( E r E 0 , ) = H 0 , 1 L 1 1 M 11 1 ( 1 0 M 21 1 M 11 1 ) [ Δ 1 + H 1 , 2 L 2 1 Δ 2 + + H 1 , 2 L 2 1 H n 1 , n L n 1 Δ n ] + 1 M 11 1 ( 1 M 12 1 M 21 1 M 11 1 M 22 1 M 12 1 M 21 1 ) ( E 0 , r E ) ,
2 i κ i Δ i , r z e i κ i ( z z i ) = 9 ω 2 χ i ( 3 ) c 2 [ F i , r e i k i ( z z i ) + F i , e i k i ( z z i ) ] 3 ,
Δ E i , r = 9 i ω 2 χ i ( 3 ) d i e i κ i d i 2 κ i c 2 × { F i , r 3 e i Δ k i d i / 2 sinc ( Δ k i d i / 2 ) + F i , 3 e + i Δ k i + d i / 2 sinc ( Δ k i + d i / 2 ) + 3 F i , r 2 F i , e i Δ ϕ i d i / 2 sinc ( Δ ϕ i d i / 2 ) + 3 F i , r F i , 2 e i Δ ϕ i + d i / 2 sinc ( Δ ϕ i + d i / 2 ) } ,
Δ E i , = 9 i ω 2 χ i ( 3 ) d i 2 κ i c 2 × { F i , r 3 e i Δ k i + d i / 2 sinc ( Δ k i + d i / 2 ) + F i , 3 e + i Δ k i d i / 2 sinc ( Δ k i d i / 2 ) + 3 F i , r 2 F i , e i Δ ϕ i + d i / 2 sinc ( Δ ϕ i + d i / 2 ) + 3 F i , r F i , 2 e i Δ ϕ i d i / 2 sinc ( Δ ϕ i d i / 2 ) } .
E T ( ρ ) = τ n + 1 , 0 [ π w 0 2 3 e π 2 w 0 2 ρ 2 / 3 e i a ρ 2 L E r b 0 L d z P n + 1 ( ρ , z ) e i Δ k n + 1 z e i a ρ 2 ( L z ) ] ,
E R ( ρ ) = π w 0 2 3 e π 2 w 0 2 ρ 2 / 3 E 0 , + b T n + 1 , 0 0 L d z P n + 1 ( ρ , z ) e i Δ k n + 1 + z e i a ρ 2 ( D z ) ,
2 i κ i ( z + 1 v E t ) Δ i , r e i κ i ( z z i ) = 9 ω 2 χ i ( 3 ) c 2 [ F i , r e i k i ( z z i ) + F i , e i k i ( z z i ) ] 3 ,
Δ i , r ξ e i κ i ξ = 9 i ω 2 χ i ( 3 ) 2 κ i c 2 [ F i , r e i k i ξ + F i , e i k i ξ ] 3 ,
Δ E i , r ( ν ) = 9 i ω 2 χ i ( 3 ) e i κ i d i 2 κ i c 2 × { 0 d i d ξ e i Δ K ξ d ν F i , r ( ν ν ) d ν F i , r ( ν ) F i , r ( ν ν ) + 3 0 d i d ξ e i Δ Φ ξ d ν F i , l ( ν ν ) e 2 i ξ ( ν ν ) v F d ν F i , r ( ν ) F i , r ( ν ν ) + 3 0 d i d ξ e i Δ Φ + ξ d ν F i , r ( ν ν ) e 2 i ξ ν v F d ν F i , l ( ν ) F i , l ( ν ν ) + 0 d i d ξ e i Δ K + ξ d ν F i , l ( ν ν ) e 2 i ξ ν v F d ν F i , l ( ν ) F i , l ( ν ν ) } ,

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