Abstract

A noncentrosymmetric nanocrystal can produce second-harmonic generation (SHG) and sum-frequency generation (SFG) upon interaction with a laser field. The SHG or SFG radiation depends on the orientation of the nanocrystal as well as the field polarization, which allows for modulating the second-order emission of an arbitrarily oriented nanocrystal by specially tailoring the field polarization. We theoretically study SHG and SFG signals produced by nanocrystals driven with broad-bandwidth laser pulses. Several simulations explore the influence of the field polarization and temporal pulse profile. The latter two factors are decoupled in their influence upon the SHG and SFG signals, and thus polarization and temporal shaping can be independently performed to modulate a nanocrystals’ second-order emission. We consider the possibility of enhancing (suppressing) the signal from one nanocrystal among others by choosing the appropriate polarization, thereby opening up the prospect of selectively addressing optically nonlinear nanocrystals.

© 2014 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2012 (1)

O. Shir, J. Roslund, Z. Leghtas, and H. Rabitz, “Quantum control experiments as a testbed for evolutionary multi-objective algorithms,” Genetic Program. Evolvable Mach. 13, 445–491 (2012).

2011 (2)

2010 (2)

2009 (5)

J. Extermann, L. Bonacina, E. Cua, C. Kasparian, Y. Mugnier, T. Feurer, and J.-P. Wolf, “Nanodoublers as deep imaging markers for multi-photon microscopy,” Opt. Express 17, 15342 (2009).
[CrossRef]

C.-L. Hsieh, R. Grange, Y. Pu, and D. Psaltis, “Three-dimensional harmonic holographic microcopy using nanoparticles as probes for cell imaging,” Opt. Express 17, 2880–2891 (2009).
[CrossRef]

J. Roslund, O. Shir, T. Back, and H. Rabitz, “Accelerated optimization and automated discovery with covariance matrix adaptation for experimental quantum control,” Phys. Rev. A 80, 043415 (2009).
[CrossRef]

M. Roth, L. Guyon, J. Roslund, V. Boutou, F. Courvoisier, J. Wolf, and H. Rabitz, “Quantum control of tightly competitive product channels,” Phys. Rev. Lett. 102, 253001 (2009).
[CrossRef]

J. Roslund and H. Rabitz, “Experimental quantum control landscapes: inherent monotonicity and artificial structure,” Phys. Rev. A 80, 013408 (2009).
[CrossRef]

2008 (1)

A. V. Kachynski, A. N. Kuzmin, M. Nyk, I. Roy, and P. N. Prasad, “Zinc oxide nanocrystals for nonresonant nonlinear optical microscopy in biology and medicine,” J. Phys. Chem. C 112, 10721–10724 (2008).
[CrossRef]

2007 (1)

L. Bonacina, Y. Mugnier, F. Courvoisier, R. Le Dantec, J. Extermann, Y. Lambert, V. Boutou, C. Galez, and J.-P. Wolf, “Polar Fe(IO3)3: nanocrystals as local probes for nonlinear microscopy,” Appl. Phys. B 87, 399–403 (2007).
[CrossRef]

2006 (1)

S. Yoon, S. Baik, M. G. Kim, and N. Shin, “Formation mechanisms of tetragonal barium titanate nanoparticles in AlkoxideHydroxide sol-precipitation synthesis,” J. Am. Ceram. Soc. 89, 1816–1821 (2006).
[CrossRef]

2005 (1)

B. Li, H. Rabitz, and J. P. Wolf, “Optimal dynamic discrimination of similar quantum systems with time series data,” J. Chem. Phys. 122, 154103 (2005).
[CrossRef]

2004 (2)

S. Brasselet, V. Le Floch, F. Treussart, J.-F. Roch, J. Zyss, E. Botzung-Appert, and A. Ibanez, “In situ diagnostics of the crystalline nature of single organic nanocrystals by nonlinear microscopy,” Phys. Rev. Lett. 92, 207401 (2004).
[CrossRef]

J. Teyssier, R. L. Dantec, C. Galez, Y. Mugnier, J. Bouillot, and J. Plenet, “Lithium iodate nanocrystals in laponite matrix for nonlinear optical applications,” Appl. Phys. Lett. 85, 710–711 (2004).
[CrossRef]

2002 (3)

Y. Jiang, L. Sun, and M. C. Downer, “Second-harmonic spectroscopy of two-dimensional Si nanocrystal layers embedded in SiO2 films,” Appl. Phys. Lett. 81, 3034–3036 (2002).
[CrossRef]

T. Brixner, G. Krampert, P. Niklaus, and G. Gerber, “Generation and characterization of polarization-shaped femtosecond laser pulses,” Appl. Phys. B 74, s133–s144 (2002).
[CrossRef]

B. Li, G. Turinici, V. Ramakrishna, and H. Rabitz, “Optimal dynamic discrimination of similar molecules through quantum learning control,” J. Phys. Chem. B 106, 8125–8131 (2002).
[CrossRef]

2001 (1)

2000 (1)

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

1976 (1)

F. C. Zumsteg, J. D. Bierlein, and T. E. Gier, “KxRb1−xTiOPO4: a new nonlinear optical material,” J. Appl. Phys. 47, 4980–4985 (1976).
[CrossRef]

1949 (1)

S. Bhagavantam and D. Suryanarayana, “Crystal symmetry and physical properties: application of group theory,” Acta Crystallogr. 2, 21–26 (1949).
[CrossRef]

Back, T.

J. Roslund, O. Shir, T. Back, and H. Rabitz, “Accelerated optimization and automated discovery with covariance matrix adaptation for experimental quantum control,” Phys. Rev. A 80, 043415 (2009).
[CrossRef]

Baik, S.

S. Yoon, S. Baik, M. G. Kim, and N. Shin, “Formation mechanisms of tetragonal barium titanate nanoparticles in AlkoxideHydroxide sol-precipitation synthesis,” J. Am. Ceram. Soc. 89, 1816–1821 (2006).
[CrossRef]

Bäumner, R.

Bhagavantam, S.

S. Bhagavantam and D. Suryanarayana, “Crystal symmetry and physical properties: application of group theory,” Acta Crystallogr. 2, 21–26 (1949).
[CrossRef]

Bierlein, J. D.

F. C. Zumsteg, J. D. Bierlein, and T. E. Gier, “KxRb1−xTiOPO4: a new nonlinear optical material,” J. Appl. Phys. 47, 4980–4985 (1976).
[CrossRef]

Bonacina, L.

Botzung-Appert, E.

S. Brasselet, V. Le Floch, F. Treussart, J.-F. Roch, J. Zyss, E. Botzung-Appert, and A. Ibanez, “In situ diagnostics of the crystalline nature of single organic nanocrystals by nonlinear microscopy,” Phys. Rev. Lett. 92, 207401 (2004).
[CrossRef]

Bouillot, J.

J. Teyssier, R. L. Dantec, C. Galez, Y. Mugnier, J. Bouillot, and J. Plenet, “Lithium iodate nanocrystals in laponite matrix for nonlinear optical applications,” Appl. Phys. Lett. 85, 710–711 (2004).
[CrossRef]

Boutou, V.

M. Roth, L. Guyon, J. Roslund, V. Boutou, F. Courvoisier, J. Wolf, and H. Rabitz, “Quantum control of tightly competitive product channels,” Phys. Rev. Lett. 102, 253001 (2009).
[CrossRef]

L. Bonacina, Y. Mugnier, F. Courvoisier, R. Le Dantec, J. Extermann, Y. Lambert, V. Boutou, C. Galez, and J.-P. Wolf, “Polar Fe(IO3)3: nanocrystals as local probes for nonlinear microscopy,” Appl. Phys. B 87, 399–403 (2007).
[CrossRef]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, 2008).

Brasselet, S.

S. Brasselet, V. Le Floch, F. Treussart, J.-F. Roch, J. Zyss, E. Botzung-Appert, and A. Ibanez, “In situ diagnostics of the crystalline nature of single organic nanocrystals by nonlinear microscopy,” Phys. Rev. Lett. 92, 207401 (2004).
[CrossRef]

Brixner, T.

T. Brixner, G. Krampert, P. Niklaus, and G. Gerber, “Generation and characterization of polarization-shaped femtosecond laser pulses,” Appl. Phys. B 74, s133–s144 (2002).
[CrossRef]

T. Brixner and G. Gerber, “Femtosecond polarization pulse shaping,” Opt. Lett. 26, 557–559 (2001).
[CrossRef]

Chen, C.-H.

Chen, J.-T.

Courvoisier, F.

M. Roth, L. Guyon, J. Roslund, V. Boutou, F. Courvoisier, J. Wolf, and H. Rabitz, “Quantum control of tightly competitive product channels,” Phys. Rev. Lett. 102, 253001 (2009).
[CrossRef]

L. Bonacina, Y. Mugnier, F. Courvoisier, R. Le Dantec, J. Extermann, Y. Lambert, V. Boutou, C. Galez, and J.-P. Wolf, “Polar Fe(IO3)3: nanocrystals as local probes for nonlinear microscopy,” Appl. Phys. B 87, 399–403 (2007).
[CrossRef]

Cua, E.

Dantec, R. L.

J. Teyssier, R. L. Dantec, C. Galez, Y. Mugnier, J. Bouillot, and J. Plenet, “Lithium iodate nanocrystals in laponite matrix for nonlinear optical applications,” Appl. Phys. Lett. 85, 710–711 (2004).
[CrossRef]

Downer, M. C.

Y. Jiang, L. Sun, and M. C. Downer, “Second-harmonic spectroscopy of two-dimensional Si nanocrystal layers embedded in SiO2 films,” Appl. Phys. Lett. 81, 3034–3036 (2002).
[CrossRef]

Enderlein, J.

Extermann, J.

Feurer, T.

Fricke-Begemann, T.

Galez, C.

L. Bonacina, Y. Mugnier, F. Courvoisier, R. Le Dantec, J. Extermann, Y. Lambert, V. Boutou, C. Galez, and J.-P. Wolf, “Polar Fe(IO3)3: nanocrystals as local probes for nonlinear microscopy,” Appl. Phys. B 87, 399–403 (2007).
[CrossRef]

J. Teyssier, R. L. Dantec, C. Galez, Y. Mugnier, J. Bouillot, and J. Plenet, “Lithium iodate nanocrystals in laponite matrix for nonlinear optical applications,” Appl. Phys. Lett. 85, 710–711 (2004).
[CrossRef]

Gerber, G.

T. Brixner, G. Krampert, P. Niklaus, and G. Gerber, “Generation and characterization of polarization-shaped femtosecond laser pulses,” Appl. Phys. B 74, s133–s144 (2002).
[CrossRef]

T. Brixner and G. Gerber, “Femtosecond polarization pulse shaping,” Opt. Lett. 26, 557–559 (2001).
[CrossRef]

Gier, T. E.

F. C. Zumsteg, J. D. Bierlein, and T. E. Gier, “KxRb1−xTiOPO4: a new nonlinear optical material,” J. Appl. Phys. 47, 4980–4985 (1976).
[CrossRef]

Grange, R.

Guyon, L.

M. Roth, L. Guyon, J. Roslund, V. Boutou, F. Courvoisier, J. Wolf, and H. Rabitz, “Quantum control of tightly competitive product channels,” Phys. Rev. Lett. 102, 253001 (2009).
[CrossRef]

Hsieh, C.-L.

Ibanez, A.

S. Brasselet, V. Le Floch, F. Treussart, J.-F. Roch, J. Zyss, E. Botzung-Appert, and A. Ibanez, “In situ diagnostics of the crystalline nature of single organic nanocrystals by nonlinear microscopy,” Phys. Rev. Lett. 92, 207401 (2004).
[CrossRef]

Jiang, Y.

Y. Jiang, L. Sun, and M. C. Downer, “Second-harmonic spectroscopy of two-dimensional Si nanocrystal layers embedded in SiO2 films,” Appl. Phys. Lett. 81, 3034–3036 (2002).
[CrossRef]

Kachynski, A. V.

A. V. Kachynski, A. N. Kuzmin, M. Nyk, I. Roy, and P. N. Prasad, “Zinc oxide nanocrystals for nonresonant nonlinear optical microscopy in biology and medicine,” J. Phys. Chem. C 112, 10721–10724 (2008).
[CrossRef]

Kasparian, C.

Kim, M. G.

S. Yoon, S. Baik, M. G. Kim, and N. Shin, “Formation mechanisms of tetragonal barium titanate nanoparticles in AlkoxideHydroxide sol-precipitation synthesis,” J. Am. Ceram. Soc. 89, 1816–1821 (2006).
[CrossRef]

Krampert, G.

T. Brixner, G. Krampert, P. Niklaus, and G. Gerber, “Generation and characterization of polarization-shaped femtosecond laser pulses,” Appl. Phys. B 74, s133–s144 (2002).
[CrossRef]

Kuzmin, A. N.

A. V. Kachynski, A. N. Kuzmin, M. Nyk, I. Roy, and P. N. Prasad, “Zinc oxide nanocrystals for nonresonant nonlinear optical microscopy in biology and medicine,” J. Phys. Chem. C 112, 10721–10724 (2008).
[CrossRef]

Lai, L.-W.

Lai, W.-C.

Lambert, Y.

L. Bonacina, Y. Mugnier, F. Courvoisier, R. Le Dantec, J. Extermann, Y. Lambert, V. Boutou, C. Galez, and J.-P. Wolf, “Polar Fe(IO3)3: nanocrystals as local probes for nonlinear microscopy,” Appl. Phys. B 87, 399–403 (2007).
[CrossRef]

Lanvin, T.

Le Dantec, R.

L. Bonacina, Y. Mugnier, F. Courvoisier, R. Le Dantec, J. Extermann, Y. Lambert, V. Boutou, C. Galez, and J.-P. Wolf, “Polar Fe(IO3)3: nanocrystals as local probes for nonlinear microscopy,” Appl. Phys. B 87, 399–403 (2007).
[CrossRef]

Le Floch, V.

S. Brasselet, V. Le Floch, F. Treussart, J.-F. Roch, J. Zyss, E. Botzung-Appert, and A. Ibanez, “In situ diagnostics of the crystalline nature of single organic nanocrystals by nonlinear microscopy,” Phys. Rev. Lett. 92, 207401 (2004).
[CrossRef]

Leghtas, Z.

O. Shir, J. Roslund, Z. Leghtas, and H. Rabitz, “Quantum control experiments as a testbed for evolutionary multi-objective algorithms,” Genetic Program. Evolvable Mach. 13, 445–491 (2012).

Li, B.

B. Li, H. Rabitz, and J. P. Wolf, “Optimal dynamic discrimination of similar quantum systems with time series data,” J. Chem. Phys. 122, 154103 (2005).
[CrossRef]

B. Li, G. Turinici, V. Ramakrishna, and H. Rabitz, “Optimal dynamic discrimination of similar molecules through quantum learning control,” J. Phys. Chem. B 106, 8125–8131 (2002).
[CrossRef]

Marowsky, G.

Mugnier, Y.

J. Extermann, L. Bonacina, E. Cua, C. Kasparian, Y. Mugnier, T. Feurer, and J.-P. Wolf, “Nanodoublers as deep imaging markers for multi-photon microscopy,” Opt. Express 17, 15342 (2009).
[CrossRef]

L. Bonacina, Y. Mugnier, F. Courvoisier, R. Le Dantec, J. Extermann, Y. Lambert, V. Boutou, C. Galez, and J.-P. Wolf, “Polar Fe(IO3)3: nanocrystals as local probes for nonlinear microscopy,” Appl. Phys. B 87, 399–403 (2007).
[CrossRef]

J. Teyssier, R. L. Dantec, C. Galez, Y. Mugnier, J. Bouillot, and J. Plenet, “Lithium iodate nanocrystals in laponite matrix for nonlinear optical applications,” Appl. Phys. Lett. 85, 710–711 (2004).
[CrossRef]

Niklaus, P.

T. Brixner, G. Krampert, P. Niklaus, and G. Gerber, “Generation and characterization of polarization-shaped femtosecond laser pulses,” Appl. Phys. B 74, s133–s144 (2002).
[CrossRef]

Nyk, M.

A. V. Kachynski, A. N. Kuzmin, M. Nyk, I. Roy, and P. N. Prasad, “Zinc oxide nanocrystals for nonresonant nonlinear optical microscopy in biology and medicine,” J. Phys. Chem. C 112, 10721–10724 (2008).
[CrossRef]

Plenet, J.

J. Teyssier, R. L. Dantec, C. Galez, Y. Mugnier, J. Bouillot, and J. Plenet, “Lithium iodate nanocrystals in laponite matrix for nonlinear optical applications,” Appl. Phys. Lett. 85, 710–711 (2004).
[CrossRef]

Prasad, P. N.

A. V. Kachynski, A. N. Kuzmin, M. Nyk, I. Roy, and P. N. Prasad, “Zinc oxide nanocrystals for nonresonant nonlinear optical microscopy in biology and medicine,” J. Phys. Chem. C 112, 10721–10724 (2008).
[CrossRef]

Psaltis, D.

Pu, Y.

Rabitz, H.

O. Shir, J. Roslund, Z. Leghtas, and H. Rabitz, “Quantum control experiments as a testbed for evolutionary multi-objective algorithms,” Genetic Program. Evolvable Mach. 13, 445–491 (2012).

J. Roslund, O. Shir, T. Back, and H. Rabitz, “Accelerated optimization and automated discovery with covariance matrix adaptation for experimental quantum control,” Phys. Rev. A 80, 043415 (2009).
[CrossRef]

M. Roth, L. Guyon, J. Roslund, V. Boutou, F. Courvoisier, J. Wolf, and H. Rabitz, “Quantum control of tightly competitive product channels,” Phys. Rev. Lett. 102, 253001 (2009).
[CrossRef]

J. Roslund and H. Rabitz, “Experimental quantum control landscapes: inherent monotonicity and artificial structure,” Phys. Rev. A 80, 013408 (2009).
[CrossRef]

B. Li, H. Rabitz, and J. P. Wolf, “Optimal dynamic discrimination of similar quantum systems with time series data,” J. Chem. Phys. 122, 154103 (2005).
[CrossRef]

B. Li, G. Turinici, V. Ramakrishna, and H. Rabitz, “Optimal dynamic discrimination of similar molecules through quantum learning control,” J. Phys. Chem. B 106, 8125–8131 (2002).
[CrossRef]

Ramakrishna, V.

B. Li, G. Turinici, V. Ramakrishna, and H. Rabitz, “Optimal dynamic discrimination of similar molecules through quantum learning control,” J. Phys. Chem. B 106, 8125–8131 (2002).
[CrossRef]

Roch, J.-F.

S. Brasselet, V. Le Floch, F. Treussart, J.-F. Roch, J. Zyss, E. Botzung-Appert, and A. Ibanez, “In situ diagnostics of the crystalline nature of single organic nanocrystals by nonlinear microscopy,” Phys. Rev. Lett. 92, 207401 (2004).
[CrossRef]

Roslund, J.

O. Shir, J. Roslund, Z. Leghtas, and H. Rabitz, “Quantum control experiments as a testbed for evolutionary multi-objective algorithms,” Genetic Program. Evolvable Mach. 13, 445–491 (2012).

J. Roslund and H. Rabitz, “Experimental quantum control landscapes: inherent monotonicity and artificial structure,” Phys. Rev. A 80, 013408 (2009).
[CrossRef]

J. Roslund, O. Shir, T. Back, and H. Rabitz, “Accelerated optimization and automated discovery with covariance matrix adaptation for experimental quantum control,” Phys. Rev. A 80, 043415 (2009).
[CrossRef]

M. Roth, L. Guyon, J. Roslund, V. Boutou, F. Courvoisier, J. Wolf, and H. Rabitz, “Quantum control of tightly competitive product channels,” Phys. Rev. Lett. 102, 253001 (2009).
[CrossRef]

Roth, M.

M. Roth, L. Guyon, J. Roslund, V. Boutou, F. Courvoisier, J. Wolf, and H. Rabitz, “Quantum control of tightly competitive product channels,” Phys. Rev. Lett. 102, 253001 (2009).
[CrossRef]

Roy, I.

A. V. Kachynski, A. N. Kuzmin, M. Nyk, I. Roy, and P. N. Prasad, “Zinc oxide nanocrystals for nonresonant nonlinear optical microscopy in biology and medicine,” J. Phys. Chem. C 112, 10721–10724 (2008).
[CrossRef]

Sheu, J.-K.

Shin, N.

S. Yoon, S. Baik, M. G. Kim, and N. Shin, “Formation mechanisms of tetragonal barium titanate nanoparticles in AlkoxideHydroxide sol-precipitation synthesis,” J. Am. Ceram. Soc. 89, 1816–1821 (2006).
[CrossRef]

Shir, O.

O. Shir, J. Roslund, Z. Leghtas, and H. Rabitz, “Quantum control experiments as a testbed for evolutionary multi-objective algorithms,” Genetic Program. Evolvable Mach. 13, 445–491 (2012).

J. Roslund, O. Shir, T. Back, and H. Rabitz, “Accelerated optimization and automated discovery with covariance matrix adaptation for experimental quantum control,” Phys. Rev. A 80, 043415 (2009).
[CrossRef]

Sun, L.

Y. Jiang, L. Sun, and M. C. Downer, “Second-harmonic spectroscopy of two-dimensional Si nanocrystal layers embedded in SiO2 films,” Appl. Phys. Lett. 81, 3034–3036 (2002).
[CrossRef]

Suryanarayana, D.

S. Bhagavantam and D. Suryanarayana, “Crystal symmetry and physical properties: application of group theory,” Acta Crystallogr. 2, 21–26 (1949).
[CrossRef]

Teyssier, J.

J. Teyssier, R. L. Dantec, C. Galez, Y. Mugnier, J. Bouillot, and J. Plenet, “Lithium iodate nanocrystals in laponite matrix for nonlinear optical applications,” Appl. Phys. Lett. 85, 710–711 (2004).
[CrossRef]

Treussart, F.

S. Brasselet, V. Le Floch, F. Treussart, J.-F. Roch, J. Zyss, E. Botzung-Appert, and A. Ibanez, “In situ diagnostics of the crystalline nature of single organic nanocrystals by nonlinear microscopy,” Phys. Rev. Lett. 92, 207401 (2004).
[CrossRef]

Turinici, G.

B. Li, G. Turinici, V. Ramakrishna, and H. Rabitz, “Optimal dynamic discrimination of similar molecules through quantum learning control,” J. Phys. Chem. B 106, 8125–8131 (2002).
[CrossRef]

Weiner, A. M.

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

Wolf, J.

M. Roth, L. Guyon, J. Roslund, V. Boutou, F. Courvoisier, J. Wolf, and H. Rabitz, “Quantum control of tightly competitive product channels,” Phys. Rev. Lett. 102, 253001 (2009).
[CrossRef]

Wolf, J. P.

B. Li, H. Rabitz, and J. P. Wolf, “Optimal dynamic discrimination of similar quantum systems with time series data,” J. Chem. Phys. 122, 154103 (2005).
[CrossRef]

Wolf, J.-P.

Yang, Y.-Y.

Yoon, S.

S. Yoon, S. Baik, M. G. Kim, and N. Shin, “Formation mechanisms of tetragonal barium titanate nanoparticles in AlkoxideHydroxide sol-precipitation synthesis,” J. Am. Ceram. Soc. 89, 1816–1821 (2006).
[CrossRef]

Zumsteg, F. C.

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Figures (6)

Fig. 1.
Fig. 1.

Specifications of crystal orientation and E-field polarization. (a) Euler angle (θ,φ,γ) rotation from the (x, y, z) laboratory frame to the (a, b, c) crystal frame. (b) Representation of the E-field of an optical pulse propagating in the z direction. When the x and y components of the E-field are in phase, the pulse is linearly polarized and the field oscillates along a constant direction (cosβ˜,sinβ˜) in the xy plane (solid and dotted blue line). When the x and y components of the E-field are in quadrature, the pulse is right (left) elliptically polarized and the field vector rotates counterclockwise (clockwise) within the ellipse inscribed in the dashed rectangle (red arrow).

Fig. 2.
Fig. 2.

BaTiO3 nanocrystal SHG signal landscapes with respect to the crystal orientation for particular field polarizations. (a), (b), and (c) SHG landscapes generated with a linearly polarized E-field of increasing β˜ angle. (d), (e), and (f) Progression with elliptical polarization and increasing β˜ phase [note that (a) and (c) also represent the extremes of the latter progression since when β˜=0° or 90° elliptical and linear polarizations are equivalent]. (e) The case for circular polarization.

Fig. 3.
Fig. 3.

Generic SFG “virtual” landscapes. The surfaces L11(2), L22(2), L33(2), and L12(2) all possess reflection symmetry with respect to the φ=90° vertical plane. Only the L11(2) landscape significantly contributes to the second-order polarization at small θ.

Fig. 4.
Fig. 4.

SFG intensity landscapes with respect to crystal orientation. (a), (b), and (c) Landscapes obtained with linearly polarized fields. (d)–(i) Landscapes using elliptically polarized fields.

Fig. 5.
Fig. 5.

SFG intensity landscapes for a collection of nanocrystals with respect to the sum and difference of the E-fields’ polarization angles. (a) Linear polarization scheme. (b) Elliptical polarization scheme.

Fig. 6.
Fig. 6.

Maximum contrast attainable between two nanocrystals’ nonlinear responses with linearly polarized E-fields as Jφ1=0°,θ1max(min) landscapes with respect to nanocrystal 2 orientation. In (a), (b), and (c) nanocrystal 1 is standing up vertically, titled 45°, and 80°, respectively.

Equations (55)

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P˜(2)(t)=χ(2)·E˜1(t)·E˜2(t),
P˜(2)(t)=deffF˜(t),
deff=(d11d12d13d14d15d16d21d22d23d24d25d26d31d32d33d34d35d36),
F˜(t)=(E˜a1(t)E˜a2(t)E˜b1(t)E˜b2(t)E˜c1(t)E˜c2(t)E˜b1(t)E˜c2(t)+E˜b2(t)E˜c1(t)E˜c1(t)E˜a2(t)+E˜c2(t)E˜a1(t)E˜a1(t)E˜b2(t)+E˜a2(t)E˜b1(t)),
F˜(t)=(Ex1(t)Ex2(t)Ex1(t)Ey2(t)+Ex2(t)Ey1(t)Ey1(t)Ey2(t)).
P˜(2)(t)=deffΩF˜(t),
I˜(2)(t)=nλNLϵ0cV22|P˜(2)(t)|2,
|P˜(2)(t)|2=F˜(t)ΩdeffdeffΩF˜(t)=F˜(t)QF˜(t),
Q=UΛU.
E˜(t)=E˜0(t)(100)ei(ϕ˜0(t)ω0t)+c.c.,
E˜(t)=E˜(t)(cosβ˜(t)eiξ˜(t)sinβ˜(t)eiξ˜(t)0)ei(ϕ˜(t)ω0t)+c.c.,
F˜(t)=E˜2(t)f˜(t)e2i(ϕ˜(t)ω0t)+c.c.,
f˜(t)=(cos2β˜(t)e2iξ˜(t)2cosβ˜(t)sinβ˜(t)sin2β˜(t)e2iξ˜(t)).
I˜(2)(t)F˜(t)QF˜(t)=2E˜4(t)f˜(t)Qf˜(t).
deff=(0000d150000d1500d31d31d33000),
Ω=(cos2φ1/2sin2φsin2φsin2φcos2θ1/2cos2θsin2φcos2θcos2φsin2φsin2θ1/2sin2θsin2φsin2θcos2φsin2φsin2θ1/2sin2φsin2θcos2φsin2θsinθsin2φsinθcos2φsinθsin2φcosθsin2φcosθcos2φcosθsin2φ).
φφ+180°,θ180°θ,P˜lab(2)(t)P˜lab(2)(t).
E˜A=E˜A(cosβ˜Aeiξ˜Asinβ˜Aeiξ˜A0)ei(ϕ˜AωAt)+c.c.,
E˜B=E˜B(cosβ˜Beiξ˜Bsinβ˜Beiξ˜B0)ei(ϕ˜BωBt)+c.c.,
F˜=E˜AE˜B(f˜SFGei(ϕ˜A+ϕ˜B)i(ωA+ωB)t+c.c.),
f˜SFG=(1/2(cosβ˜s+cosβ˜d)eiξ˜ssinβ˜scosξ˜disinβ˜dsinξ˜d1/2(cosβ˜dcosβ˜s)eiξ˜s),
I˜(2)2E˜A2E˜B2(f˜SFG)Qf˜SFG.
f˜SFG=a˜v1+b˜v2+c˜v3,
a˜=1/2(cosβ˜dcosξ˜s+icosβ˜ssinξ˜s),b˜=1/2(cosβ˜scosξ˜s+icosβ˜dsinξ˜s),c˜=(sinβ˜scosξ˜disinβ˜dsinξ˜d),
v1=(1/201/2),v2=(1/201/2),andv3=(010).
I˜(2)2E˜A2E˜B2×[|a˜|2L11(2)+|b˜|2L22(2)+|c˜|2L33(2)+(a˜*b˜+b˜*a˜)L12(2)+(a˜*c˜+c˜*a˜)L13(2)+(c˜*b˜+b˜*c˜)L23(2)],
I˜(2)2E˜4[12cos22β˜L11(2)+12L22(2)+(sin22β˜)L33(2)+cos2β˜L12(2)].
I˜(2)cos22β˜+cos2β˜2ILPx(2)+(sin22β˜)ICP(2)+cos22β˜cos2β˜2ILPy(2),
f˜AB=12cosβ˜dv1+12cosβ˜sv2+sinβ˜sv3.
f˜AB=i2cosβ˜sv1+i2cosβ˜dv2+sinβ˜sv3.
f˜AB=12cosβ˜dv1+12cosβ˜sv2isinβ˜dv3.
I˜tot(2)=kI˜k(2)kVk2F˜QkF˜,
I˜tot(2)nVavg2F˜[1nkQk]F˜,
Q¯(21.0509.8805.609.88021.05).
I˜tot(2)|a˜|2λ¯1+|b˜|2λ¯2+|c˜|2λ¯3,
I˜tot(2)12λ¯1(cos2β˜dcos2ξ˜s+cos2β˜ssin2ξ˜s)+12λ¯2(cos2β˜scos2ξ˜s+cos2β˜dsin2ξ˜s)+λ¯3(sin2β˜scos2ξ˜d+sin2β˜dsin2ξ˜d).
J˜=I˜2(2)I˜1(2)=F˜Q2F˜F˜Q1F˜.
Q1=U1Λ1U1.
Q=Λ112U1Q2U1Λ112.
I˜1(2)=Y˜Y˜andI˜2(2)=Y˜QY˜.
J˜=Y˜QY˜Y˜Y˜=Y˜Y˜Y˜QY˜Y˜Y˜=y˜Qy˜.
λ3J˜λ1.
F˜=E˜AE˜Bf˜AB,
f˜AB=f˜SFGei(ϕ˜A+ϕ˜B)i(ωA+ωB)t+c.c.+f˜DFGei(ϕ˜Aϕ˜B)i(ωAωB)t+c.c.,
f˜SFG=(cosβ˜Acosβ˜Bei(ξ˜A+ξ˜B)cosβ˜Asinβ˜Bei(ξ˜Aξ˜B)+sinβ˜Acosβ˜Bei(ξ˜Bξ˜A)sinβ˜Asinβ˜Bei(ξ˜A+ξ˜B)),
f˜DFG=(cosβ˜Acosβ˜Bei(ξ˜Aξ˜B)cosβ˜Asinβ˜Bei(ξ˜A+ξ˜B)+sinβ˜Acosβ˜Bei(ξ˜A+ξ˜B)sinβ˜Asinβ˜Bei(ξ˜Bξ˜A)).
f˜SFG=(1/2(cosβ˜s+cosβ˜d)eiξ˜ssinβ˜scosξ˜disinβ˜dsinξ˜d1/2(cosβ˜dcosβ˜s)eiξ˜s),
f˜DFG=(1/2(cosβ˜s+cosβ˜d)eiξ˜dsinβ˜scosξ˜sisinβ˜dsinξ˜s1/2(cosβ˜dcosβ˜s)eiξ˜d),
I˜(2)(t)F˜(t)QF˜(t)=E˜A2(t)E˜B2(t)f˜ABQf˜AB.
I˜(DFG)(t)2E˜A2(t)E˜B2(t)(f˜DFG)Qf˜DFG.
f˜DFG=av1+bv2+cv3=12(cosβdcosξd+icosβssinξd)v1+12(cosβscosξd+icosβdsinξd)v2+(sinβ˜scosξ˜sisinβ˜dsinξ˜s)v3.
(a˜*c˜+c˜*a˜)=12[sin2β˜Acos2ξ˜A+sin2β˜Bcos2ξ˜B](c˜*b˜+b˜*c˜)=12[sin2β˜Acos2β˜Bcos2ξ˜A+cos2β˜Asin2β˜Bcos2ξ˜B].
ξ˜A=ξ˜B=45°,
β˜A=0°,90°andβ˜B=0°,90°,
β˜A=0°,ξ˜B=45°orβ˜B=0°,ξ˜A=45°.

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