Cascade-Like Nonlinearity Caused by Local-Field Effects: Extending Bloembergen's Result
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1 Cascade-Like Nonlinearity Caused by Local-Field Effects: Extending Bloembergen's Result Ksenia Dolgaleva 1, Robert W. Boyd 1, and John E. Sipe 2 1 Institute of Optics, University of Rochester 2 Department of Physics, University of Toronto
2 Motivation Cascading is a nonlinear process in which a lower-order nonlinearity contributes to a higher-order nonlinear response in a multi-step fashion: 3 = const 2 : 2. Cascading in a usual sense requires propagation and phase-matching. Local-field effects can cause cascading at a microscopic level. Such cascading does not require propagation and phase matching. Although macroscopic cascading is a well-known phenomenon, microscopic cascading is less well known.
3 Lorentz Local Field Consider a homogeneous medium exposed to an external optical field: E ext
4 Lorentz Local Field Consider a homogeneous medium exposed to an external optical field: E ext E E loc E loc E ext E
5 Lorentz Local Field E P b E loc E E loc R Imaginary sphere (boundary of virtual cavity) Contributions from inside dipoles are accounted exactly The dipoles outside the cavity are considered as a homogeneous medium b R
6 Lorentz Local Field E P b E loc E E loc R Imaginary sphere (boundary of virtual cavity) Contributions from inside dipoles are accounted exactly The dipoles outside the cavity are considered as a homogeneous medium E E loc P b R is average (macroscopic) field in the medium is the local field acting on a typical emitter is average (macroscopic) polarization E loc =E 4 3 P
7 Lorentz Local Field E loc =E 4 3 P or E loc =L E
8 Lorentz Local Field E loc =E 4 3 P or E loc =L E where L= is Lorentz local-field correction factor 1 is dielectric permittivity
9 Local-Field Effects in Linear Optics 1 1 =L N at 1 N 1 at is linear susceptibility is molecular or atomic density is linear microscopic polarizability
10 Local-Field Effects in Linear Optics 1 1 =L N at 1 N 1 at is linear susceptibility is molecular or atomic density is linear microscopic polarizability Linear optical properties depend on factor L.
11 Local-Field Effects in Nonlinear Optics = 1 2 =L 1 L 2 L 3 N at where L i = 1 i at is the second-order nonlinear susceptibility is the second-order nonlinear hyperpolarizability N. Bloembergen, Nonlinear Optics, 4 th ed. (Scientific, Singapore, 1996).
12 Extending Bloembergen's Result: Naïve Way 2 Since scales as the 3 rd power of factor L, n scales as the (n+1) th power of factor L Open your favorite textbook on Nonlinear Optics
13 Extending Bloembergen's Result: Naïve Way 2 Since scales as the 3 rd power of factor L, n scales as the (n+1) th power of factor L Consider degenerate n th -order nonlinearity: n n =... =N at L n 1 L 2 Open your favorite textbook on Nonlinear Optics
14 Third-Order Nonlinearity in Non-Centrosymmetric Media However, 3 3 =3 1 = d 3 c 3, 3 d =N 3 at L 3 1 L 3 where and c 3 [ 2 ] 2 L 2 1, 2 2 = 1 1 =N 2 at L 2 1 L 2. D. Bedeaux and N. Bloembergen, Physica 69, (1973).
15 Third-Order Nonlinearity in Non-Centrosymmetric Media where d 3 comes from and scales as the 4 th power of factor L; c =3 1 = d 3 c 3, 3 at 2 comes from at and scales as the 5 th power of factor L. D. Bedeaux and N. Bloembergen, Physica 69, (1973).
16 Therefore... It appears that in non-centrosymmetric media. 3 3 = N at L 2 L 2 n n =N at L n 1 L 2 Straight-forward generalization of Bloembergen's result for second-order susceptibility to a higher-order susceptibility is, generally, not correct. One cannot predict the expression for local-field-corrected n based on the result for a lower-order nonlinearity.
17 Our Contribution Why the straight-forward generalization does not work? To answer the question, we undertake calculation of local-field-corrected fifth-order susceptibility in a centrosymmetric medium.
18 LF-Corrected Degenerate 5 : Bloembergen's Prescription Starting from Lorentz local field E loc =E 4 3 P, P=P L P NL consider P=P 1 P 3 P 5, where P 1 =N 1 at E loc ; P 3 3 =N at E loc 2 E loc ; P 5 =N 5 E at loc 4 E loc. centrosymmetric medium
19 LF-Corrected Degenerate 5 : Bloembergen's Prescription With the use of the nonlinear source polarization P NLS =L P 3 P 5, and also L= and 1 = 1 1 4, P=P 1 P 3 P 5 becomes P= 1 E P NLS. centrosymmetric medium
20 LF-Corrected Degenerate 5 : Bloembergen's Prescription P= 1 E P NLS Alternatively, as a power-series expansion w. r. t. E : P= 1 E 3 3 E 2 E 10 5 E 4 E. centrosymmetric medium
21 LF-Corrected Degenerate 5 : Bloembergen's Prescription P= 1 E P NLS P NLS =L P 3 P 5 P 3 3 =N at E loc 2 E loc P 5 5 =N at E loc 4 E loc Alternatively, as a power-series expansion w. r. t. E : P= 1 E 3 3 E 2 E 10 5 E 4 E. If one can deduce P NLS, one can find n. centrosymmetric medium
22 LF-Corrected Degenerate 5 : Bloembergen's Prescription P= 1 E P NLS P NLS =L P 3 P 5 P 3 3 =N at E loc 2 E loc P 5 5 =N at E loc 4 E loc Alternatively, as a power-series expansion w. r. t. E : P= 1 E 3 3 E 2 E 10 5 E 4 E. If one can deduce P NLS, one can find n. centrosymmetric medium
23 LF-Corrected Degenerate 5 : Wrong Way P= 1 E P NLS P NLS =L P 3 P 5 P 3 3 =N at E loc 2 E loc P 5 5 =N at E loc 4 E loc Substitute linear E loc =L E to get 1 =N 1 at L ; 3 3 =N at L 2 L 2 ; 5 5 =N at L 4 L 2. Then n n =N at L n 1 L 2. centrosymmetric medium
24 LF-Corrected Degenerate 5 : Correct Way P= 1 E P NLS P NLS =L P 3 P 5 P 3 3 =N at E loc 2 E loc P 5 5 =N at E loc 4 E loc Substitute nonlinear E loc =E 4 3 P then compare to P= 1 E 3 3 E 2 E 10 5 E 4 E to deduce n. centrosymmetric medium
25 LF-Corrected Degenerate 5 : Correct Result The result: 1 =N 1 at L ; 3 3 =N at L 2 L 2 ; 5 5 =N at L 4 L N 2 3 at 2 L 4 L N at L 6 L. centrosymmetric medium
26 LF-Corrected Degenerate 5 : Correct Result The result: well-known 1 =N 1 at L ; 3 3 =N at L 2 L 2 ; 5 5 =N at L 4 L N 2 3 at 2 L 4 L N at L 6 L. centrosymmetric medium
27 LF-Corrected Degenerate 5 : Correct Result The result: well-known nothing peculiar 1 =N 1 at L ; 3 3 =N at L 2 L 2 ; 5 5 =N at L 4 L N 2 3 at 2 L 4 L N at L 6 L. centrosymmetric medium
28 LF-Corrected Degenerate 5 : Correct Result well-known nothing peculiar The result: 1 =N 1 at L ; 3 3 =N at L 2 L 2 ; 5 5 =N at L 4 L 2 deserves attention N 2 3 at 2 L 4 L N at L 6 L. centrosymmetric medium
29 LF-Corrected Degenerate 5 : Direct and Cascaded Contributions 5 5 =N at L 4 L N 2 3 at 2 L 4 L N at L 6 L. centrosymmetric medium
30 LF-Corrected Degenerate 5 : Direct and Cascaded Contributions direct contribution from fifth-order 5 hyperpolarizability at 5 5 =N at L 4 L N 2 3 at 2 L 4 L N at L 6 L. centrosymmetric medium
31 LF-Corrected Degenerate 5 : Direct and Cascaded Contributions direct contribution from fifth-order 5 hyperpolarizability at 5 5 =N at L 4 L N 2 3 at 2 L 4 L N at L 6 L. cascaded contributions from thirdorder hyperpolarizability 3 at centrosymmetric medium
32 LF-Corrected Degenerate 5 : Direct and Cascaded Contributions 5 5 direct =N at L 4 L 2 scales as 6 th power of factor L. 5 cascaded = N 2 3 at 2 L 4 L N at L 6 L scales as 7 th power of factor L. centrosymmetric medium
33 LF-Corrected Degenerate 5 : Direct and Cascaded Contributions 5 5 direct =N at L 4 L 2 scales as 6 th power of factor L. How significant? 5 cascaded = N 2 3 at 2 L 4 L N at L 6 L scales as 7 th power of factor L. centrosymmetric medium
34 Direct and Cascaded Contributions: Comparison Consider sodium 3 s 3 p transition: The dipole moment = esu Population relaxation time T 1 =16 ns Atomic density range N = cm -3
35 Direct and Cascaded Contributions: Comparison Ratio of absolute values of the contributions as a function of the normalized detuning and atomic density R= 5 cascaded 5 direct
36 Direct and Cascaded Contributions: Comparison Ratio of absolute values of the contributions as a function of the normalized detuning and atomic density R= 5 cascaded 5 direct Under certain conditions, the cascaded contribution can be as large as the direct contribution.
37 Conclusions Microscopic cascading is possible due to local-fieldinduced contributions of lower-order nonlinearities to higher-order nonlinearities. We demonstrated it based on 5 th -order degenerate nonlinearity, following Bloembergen's procedure and Maxwell-Bloch approach. We demonstrated that the cascaded contribution to can be as large as the direct contribution. 5 Experiment is in progress to verify the theory.
38 Acknowledgments Dr. Sergei Volkov for valuable discussion Prof. Boyd's research group
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