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Large Hyperpolarizabilities of the Second Harmonic Generation Induced by Nonuniform Optical Near Fields Maiku Yamaguchi, and Katsuyuki Nobusada J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b08507 • Publication Date (Web): 27 Sep 2016 Downloaded from http://pubs.acs.org on October 3, 2016

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The Journal of Physical Chemistry

Large Hyperpolarizabilities of the Second Harmonic Generation Induced by Nonuniform Optical Near Fields Maiku Yamaguchi† and Katsuyuki Nobusada∗,‡ Department of Electrical Engineering and Information Systems, Graduate School of Engineering, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan, and Department of Theoretical and Computational Molecular Science, Institute for Molecular Science, Myodaiji, Okazaki 444-8585, Japan, Phone: +81-564-55-7311 E-mail: [email protected]

∗

To whom correspondence should be addressed Department of Electrical Engineering and Information Systems, Graduate School of Engineering, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan ‡ Department of Theoretical and Computational Molecular Science, Institute for Molecular Science, Myodaiji, Okazaki 444-8585, Japan, Phone: +81-564-55-7311 †

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Abstract We studied the optical selection rules and hyperpolarizabilities of a second harmonic generation (SHG) induced by a nonuniform optical near field (ONF) on the basis of first-principles calculations. The excitations of the symmetric and asymmetric molecules para-dinitrobenzene (pDNB) and para-nitroaniline (pNA), respectively, were investigated according to the time-dependent density functional theory. By calculating the ONF excitation dynamics of symmetric pDNB, we demonstrated that the ONF causes the SHG even in symmetric materials due to nonuniformity of the field in space. To quantitatively evaluate the intensity of the SHG induced by the ONF, we estimated the first hyperpolarizability β(2ω) of asymmetric pNA for both of the usual far-field light and ONF excitations. Our results showed that β(2ω) is one to two orders of magnitude larger for the ONF excitation than for the far-field excitation when the distance between the source of the ONF and pNA is around 10 Å.

Introduction The interaction between light and a molecule is generally considered under the electric dipole approximation, 1,2 which postulates that the molecule feels the spatially uniform electric field of the light. The electric dipole approximation is valid when the variation of the light in space is small relative to the length scale of the molecule. In most cases of light–matter interactions, this condition is satisfied. However, one situation in which the electric dipole approximation obviously breaks down is the interaction between an optical near field (ONF) and molecule. An ONF is localized light generated around a surface or interface of materials irradiated by far-field light. 3 The ONF has quantitative and qualitative differences from far-field light. The quantitative difference is that the intensity of an ONF derived from plasmon resonance can be 102 to 103 times stronger than that of the incident light. This phenomenon, which is called electric field enhancement, has been applied to, for example, surface-enhanced 2 ACS Paragon Plus Environment

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Raman scattering, 4 enhanced solar cells, 5 electron emitters, 6 chemical reactions, 7 photothermal therapy, 8 and nonlinear optical effects. 9 The qualitative difference is that the ONF is nonuniform in space. Because the ONF is localized around its source, such as nanostructures or interfaces of materials, the intensity of the ONF decreases acutely with increasing distance from the source. This change of the intensity in space leads to the nonuniformity, or large gradient, of the field. Owing to the nonuniformity, the ONF–matter interaction is no longer described by the electric dipole approximation and causes phenomena absent in usual light–matter interaction. Recently, several studies have shown both theoretically 10–14 and experimentally 15,16 that the nonuniformity of the ONF induces electric quadrupole allowed transitions, which are forbidden under the selection rules based on the electric dipole approximation. The selection rules of Raman scattering are also changed by the ONF due to the nonuniformity. 17,18 Furthermore, we previously clarified that a second harmonic generation (SHG) is induced by the ONF–matter interaction in a different manner from that induced by an intense laser field. 19–21 The SHG induced by the ONF is remarkable in that it occurs even in symmetric materials, even though the usual SHG is only caused in asymmetric materials. In other words, the ONF breaks the selection rules of the SHG. However, we previously only gave qualitative explanations of the SHG mechanism and presented calculation results based on model systems. 20,21 Although our research group presented the results of first-principles calculations, 19 the intensity of the SHG has not been quantitatively clarified. In the present study, we investigated the selection rules and hyperpolarizabilities of the SHG induced by the ONF by using a first-principles calculation method based on the timedependent density functional theory (TDDFT). 22–24 First, we examined the SHG mechanism by using the perturbation theory of optical nonlinear effects. 25 In particular, we focused on the origin of the breakdown of the selection rules. Then, we calculated the ONF excitation dynamics for two molecules: para-dinitrobenzene (pDNB) and para-nitroaniline (pNA). Although these two molecules have similar molecular structures, the crucial difference is that pDNB is symmetric while pNA is asymmetric. Based on the calculation of the symmetric



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pDNB, we demonstrated that the ONF causes the SHG even in symmetric materials because of the spatial nonuniformity of the ONF. By the calculation of the asymmetric pNA, we quantitatively compared the first hyperpolarizabilities β, which are the coefficient of the amplitude of the SHG of both far-field and ONF excitations. Because pNA is asymmetric, the far-field light also causes the SHG; thus, we were able to compare the hyperpolarizabilities of both excitations on equal footing. The hyperpolarizabilities are usually determined only by the electronic structure of a molecule. However, for the SHG induced by the ONF, β also depends on the spatial arrangement of the ONF source and molecule because the ONF induces the SHG via the field nonuniformity, which varies more rapidly than the intensity. Therefore, we presented the dependence of β on the distance between the ONF source and molecule.

Theory In this section, we present the theory of an SHG induced by a nonuniform ONF resorting to the perturbation theory. Consider the light excitation dynamics of a discrete system with the eigenstates |n⟩ for n = 0, · · · N . The general form of the Hamiltonian of the system including light–matter interaction is, in atomic units, êxt )2 + Vˆ0 + Vêxt , ˆ = 1 (ˆ H p−A 2

(1)

where pˆ = −i∇ is the momentum operator and V0 is the static scalar potential of the system. êxt and Vêxt represent the vector and scalar potentials of the excitation light. The terms A In the following, we consider the excitations by far-field propagating light and the ONF. For the far-field excitation, the electric dipole approximation can be reasonably applied.


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Based on this approximation, Eq. (1) is rewritten in the length gauge as 2 ˆ = pˆ + Vˆ0 − µ ˆ · E(t) H 2 ˆ 0 + Vˆ (far) (t), ≡H

(2) (3)

ˆ 0 ≡ pˆ2 /2 + Vˆ0 is the static Hamiltonian of the system and where H ˆ · E(t) Vˆ (far) (t) ≡ −µ

(4)

is the light–matter interaction term for the far-field excitation. This is represented by the ˆ = −ˆ scalar product of the electric dipole operator µ r and electric field amplitude of the light E(t). For the ONF excitation, the electric dipole approximation is no longer valid because of the field nonuniformity. Therefore, an explicit form of the ONF potential should be considered. Here, we represent the source of the ONF with an oscillating electric dipole. The potential generated by the oscillating electric dipole placed at the origin r = 0 is written in the Lorentz gauge as r · P (t) r · P˙ (t) − , r3 cr2 r · P˙ (t) Aext (r, t) = , c2 r Vext (r, t) = −

(5) (6)

where P (t) is the amplitude of the electric dipole moment, r = |r|, and c is the speed of light. In close proximity to the electric dipole, the first term of Eq. (5) has a dominant effect among the three terms. 3 Thus, we approximate the Hamiltonian of the ONF excitation as ˆ =H ˆ 0 − r · P (t) H r3 ˆ 0 + Vˆ (ONF) (t), ≡H


(7) (8)


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where the interaction term for the ONF excitation is r · P (t) . Vˆ (ONF) (t) ≡ − r3

(9)

Because the Hamiltonians of the far-field and ONF excitations in Eqs. (3) and (8) are written ˆ =H ˆ 0 + Vˆ , we can analyze both excitations with the same framework. in the same form H To solve the time-dependent Schrödinger equation

i

[ ] ∂ ˆ 0 + Vˆ |Ψ⟩ , |Ψ⟩ = H ∂t

(10)

where |Ψ⟩ is the time-dependent wave function, we employ the perturbation theory. Vˆ and |Ψ⟩ are expressed by Vˆ → λVˆ ,

(11)

|Ψ⟩ → |Ψ(0) ⟩ + λ |Ψ(1) ⟩ + λ2 |Ψ(2) ⟩ + · · · ,

(12)

and Eq. (10) is solved for each order of λ separately as ∂ ˆ 0 |Ψ(0) ⟩ |Ψ(0) ⟩ = H ∂t ∂ ˆ 0 |Ψ(l) ⟩ + Vˆ |Ψ(l−1) ⟩ (l = 1, 2, · · · ). i |Ψ(l) ⟩ = H ∂t

i

(13) (14)

By postulating that the ground state |0⟩ is occupied in equilibrium, Eq. (13) is solved as |Ψ(0) ⟩ = e−iω0 t |0⟩ ,

(15)

where ω0 is the eigenfrequency of |0⟩. The solution of Eq. (14) is obtained by expanding


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ˆ 0 so that |Ψ(l) ⟩ by the eigenstates of the unperturbed Hamiltonian H |Ψ(l) ⟩ =

∑

ck (t)e−iωk t |k⟩ , (l)

(16)

k

where |k⟩ and ωk are the k-th eigenstate and its eigenfrequency, respectively. By inserting Eq. (16) into Eq. (14) in combination with a time harmonic representation of the potential with the frequency ω Vˆ = Vˆ0 e−iωt ,

(17) (l)

the expansion coefficient of |n⟩ in the l-th order solution cn (t) is gained as c(l) n (t)

= −i

∑∫ k

t

−∞

dt′ V0,kn ck

(l−1)

(t′ )ei(ωnk −ω)t ,

(18)

where ωnk = ωn − ωk and V0,nk = ⟨k| Vˆ0 |n⟩. By inserting Eq. (15) and assuming that the contribution from the lower limit of the integral in Eq. (18) vanishes, the first-order solution |Ψ(1) ⟩ is obtained as |Ψ(1) ⟩ = e−i(ω0 +ω)t

∑ V0,n1 |n⟩ . ω − ω n0 n

(19)

The second-order solution |Ψ(2) ⟩ is subsequently gained by inserting Eq. (19) into (18): |Ψ(2) ⟩ = e−i(ω0 +2ω)t

∑ n,m

V0,nm V0,m0 |n⟩ . (2ω − ωn0 )(ω − ωm0 )

(20)

ˆ = ⟨Ψ| µ ˆ |Ψ⟩. In The expectation value of the induced dipole moment is calculated by ⟨µ⟩ particular, the expectation value of the second harmonic component of the dipole moment



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is ˆ ˆ |Ψ(2) ⟩ + c.c. ⟨µ(2ω)⟩ e−i2ωt = ⟨Ψ(0) | µ ∑ µ0n V0,nm V0,m0 = e−i2ωt + c.c., (2ω − ω )(ω − ω ) n0 m0 n,m

(21) (22)

ˆ |n⟩. Equation (22) indicates that the SHG occurs when there is a state where µ0n = ⟨0| µ |n⟩ that simultaneously satisfies the conditions µ0n ̸= 0,

(23)

V0,nm V0,m0 ̸= 0 for arbitrary m.

(24)

As is widely known, these conditions cannot be fulfilled for far-field excitations of symmetric materials. When the potential of the far-field excitation is represented as Eq. (4) with E(t) = E0 e−iωt , it follows that far far V0,nm V0,m0 = µnm µm0 |E0 |2 .

(25)

In general, µab is nonzero when the states |a⟩ and |b⟩ have opposite parity. Therefore, V0,nm V0,m0 is nonzero under the condition that {|n⟩ and |m⟩} and {|m⟩ and |0⟩} have opposite parity. Under such a condition, |n⟩ and |0⟩ should have the same parity; thus, µ0n = 0. That is why Eqs. (23) and (24) are not simultaneously satisfied; consequently, the SHG is not caused by the far-field excitation. This selection rule of the SHG originates from the form ˆ · E(t) based on the electric dipole of the light–matter interaction potential Vˆ (far) (t) = −µ approximation. For the ONF excitation, the situation is quite different. Owing to the spatial (ONF)

nonuniformity of the ONF, V0,ab

can be nonzero even though the states |a⟩ and |b⟩ have (ONF)

(ONF)

the same parity. Therefore, it is possible that V0,nm V0,m0

is nonzero when the states |n⟩

and |0⟩ have the opposite parity; consequently, µ0n is nonzero. Therefore, Eqs. (23) and (24) can be satisfied simultaneously for the ONF excitation. This is the origin of the breakdown 8 ACS Paragon Plus Environment

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(a)

(b)

3

QW

2

1

V (ONF)

(c) V (far)

0

z

QW

x

Figure 1: (Color online) (a) Four eigenstates of a two-dimensional quantum well. Schematics of the (b) far-field and (c) ONF excitations. in the selection rules of the SHG induced by the ONF. To gain a physical insight into the mechanism of the SHG induced by the ONF explained above, consider the excitation dynamics of a four-level model system representing a twodimensional circular quantum well (QW), as shown in Fig. 1(a). In the schematics of the eigenstates, the blue and yellow colors represent positive and negative values of the wave functions. This system has the ground state |0⟩, the two-fold degenerated first excited states |1⟩ and |2⟩, and the second excited state |3⟩. First, the system is in the ground state |0⟩ and is excited by both a z-polarized far-field light (Fig. 1(b)) and an ONF generated by a z-polarized oscillating electric dipole placed in the −x direction with respect to the QW (Fig. 1(c)). The states |a⟩ and |b⟩ connected by the blue and red arrows have nonzero values (far)

(ONF)

of V0,ab and V0,ab

, respectively. The blue arrows indicate that the state |n⟩ which satisfies

Eq. (24) for the far-field excitation is only |3⟩ with m = 2. At the same time, because |3⟩ and |0⟩ have the same parity, µ03 = 0. Thus, n = 3 does not fulfill Eq. (23), and the SHG (ONF) does not occur. A crucial difference between Vˆ (ONF) and Vˆ (far) is that Vˆ0,12 is nonzero even

though |1⟩ and |2⟩ have the same parity. This is because the ONF excitation is nonuniform in space and breaks the electric dipole approximation. The explicit form of the second-order Taylor expansion of the ONF potential Vˆ (ONF) in the xz-plane with the origin at the center



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of the QW is ∂V (ONF) (0) x2 ∂ 2 V (ONF) (0) ∂V (ONF) (0) +z + ∂x ∂z 2 ∂x2 2 2 (ONF) 2 (ONF) z ∂ V (0) ∂ V (0) + + xz 2 2 ∂z ∂x∂z (ONF) (0) x2 ∂Ex = xEx(ONF) (0) + zEz(ONF) (0) + 2 ∂x (ONF) (ONF) 2 z ∂Ez (0) ∂Ez (0) + + xz 2 ∂z ∂x (ONF) ∂Ez (0) = zEz(ONF) (0) + xz . ∂x

V (ONF) (r) = x

(26)

(27) (28)

We omit the terms with zero value for the present system in the deformation from Eq. (27) (ONF)

to Eq. (28). Thus, V0,12 ∫ (ONF) V0,12

=

is calculated to be [

dr ϕ∗1 (r) zEz(ONF) (0) + xz ∫

=

(ONF)

Ez(ONF) (0)

∂Ez

(0)

∂x (ONF)

dr

ϕ∗1 (r)zϕ2 (r)

+

]

∂Ez

(0)

∂x

ϕ2 (r) ∫

dr ϕ∗1 (r)xzϕ2 (r).

(29)

As can be deduced from the geometry of the wave functions of |1⟩ and |2⟩, the first integral of Eq. (29) is zero, but the second integral is nonzero. For the far-field excitation, the second term vanishes because the field gradient ∂Ez (0)/∂x = 0. In contrast, for the ONF (ONF)

excitation, ∂Ez (0)/∂x ̸= 0 due to the field nonuniformity. As a result, V0,12

(ONF)

V0,20

has a

nonzero value. Furthermore, the x-component of µ01 has a nonzero value, as can be deduced from the wave functions of |0⟩ and |1⟩. Therefore, for Vˆ = Vˆ (ONF) , n = 1 satisfies both Eqs. (23) and (24) with m = 2. The amplitude of the SHG induced by the ONF in the system drawn in Fig. 1(c) is represented as ˆ ⟨µ(2ω)⟩ e−i2ωt = e−i2ωt

µ01 V0,12 V0,20 + c.c.. (2ω − ω10 )(ω − ω20 )

(30)

Because the second term of Eq. (29) is proportional to the electric field gradient, the origin


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of the SHG induced by the ONF is the nonuniformity of the electric field.

Computational Details To demonstrate the SHG induced by a nonuniform ONF in real molecular systems, we performed first-principles calculations for the ONF excitation dynamics of two molecules by using TDDFT. The molecules were symmetric pDNB and asymmetric pNA. For the excitation of pNA, the hyperpolarizability β was estimated by using the finite field method. 26–29 In this section, we briefly explain the theoretical framework of TDDFT and the finite field method and describe the optical properties of pDNB and pNA.

Real Space TDDFT To simulate the electron excitation dynamics of the real molecules, we solved the timedependent Kohn–Sham equation 23,24 [ ] ∇2 ∂ + Veff [n](r, t) ϕj (r, t), i ϕj (r, t) = − ∂t 2

(31)

where ϕj is a j-th Kohn–Sham orbital. The one-particle electron density n(r, t) is calculated as

n(r, t) = 2

N/2 ∑

|ϕj (r, t)|2 ,

(32)

j=1

where N is the number of the electrons. The effective potential Veff [n](r, t) is the sum of four potentials:

Veff [n](r, t) = vion (r) + vH [n](r, t) + vxc [n](r, t) + vext (r, t).


(33)


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The ion-electron interaction potential vion (r) is calculated by using norm-conserving pseudopotentials. The Hartree potential vH [n](r, t) is calculated as ∫ vH [n](r, t) =

dr ′

n(r ′ , t) . |r − r ′ |

(34)

For the exchange-correlation potential vxc [n](r, t), the adiabatic local density approximation is employed. A external potential vext (r, t) represents an optical excitation. In the same manner as described in the previous section, we considered excitations by far-field propagating light and an ONF. The far-field excitation is described by the electric dipole approximation as (far)

vext (r, t) = −r · E0 cos(ωt)f (t),

(35)

where E0 is the electric field amplitude, ω is the frequency of the light, and f (t) is an envelope function. The ONF excitation is represented by the Coulomb potential of an electric dipole oscillation as (ONF)

vext

(r, t) = −

r · P0 cos(ωt)f (t), |r − rp |

(36)

where rp is the position of the electric dipole. All of the calculations presented in this paper were performed with a highly efficient TDDFT program called GCEED (Grid-based Coupled Electron and Electric field Dynamics) 30 that was developed by our group. GCEED solves the Kohn–Sham equation with real space basis and implements the time evolution by a fourth-order Taylor expansion after a self-consistent ground state calculation. A simulation box with dimensions of 26 Å ×26 Å ×26 Å was employed, and the spacing of the real space grid was set to 0.23 Å. The time evolution was performed with a time step of 1.114 fs.


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Finite Field Method When a molecule is irradiated by an external light represented by the electric field Ej cos(ωt), the electric dipole moment of the molecule Pi (t) is induced, where the indices i and j represent the Cartesian coordinates x, y, and z. The induced dipole Pi (t) includes all orders of the optical responses: (1)

(2)

(3)

Pi (t) = µij (t)Ej + µijj (t)Ej 2 + µijjj (t)Ej 3 + · · · , (1)

(2)

(37)

(3)

where µij (t), µijj (t), and µijjj (t) represent the first-, second-, and third-order dipole responses, respectively. The first- and second-order responses are represented as (1)

µij (t) = αij (ω) cos(ωt)Ej (2)

µijj (t) =

1 {βijj (2ω) cos(2ωt) + βijj (0)} Ej 2 , 4

(38) (39)

where αij and βijj are the linear polarizability and first hyperpolarizability, respectively. The intensity of the SHG is determined by the first hyperpolarizability βijj (2ω). The finite field method 26–29 extracts the optical response of a specific order by calculating the induced dipole moment Pi (t) for several intensities of the incident light and adding or subtracting them. In particular, we adopted the finite field method by using continuous-wave incident fields 29 to estimate the value of βijj (2ω). The first- and second-order responses are calculated as (1)

µij (t) =

1 [8 {Pi (t, Ej ) − Pi (t, −Ej )} 12Ej − {Pi (t, 2Ej ) − Pi (t, −2Ej )}] + O(Ej 4 )

(2)

µijj (t) =

(40)

1 [16 {Pi (t, Ej ) + Pi (t, −Ej )} 24Ej 2 − {Pi (t, 2Ej ) + Pi (t, −2Ej )}] + O(Ej 4 ),


(41)


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(a)

pDNB

(b)

y

pNA O

z x

(c)

dfx /dE dfy /dE dfz /dE

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H C

N

(d)

Figure 2: (Color online) Molecular structures of (a) pDNB and (b) pNA. Oscillator strengths of (c) pDNB and (d) pNA. x-, y-, and z-components are shown by the red solid, green dotted, and blue dashed lines, respectively. The y-component is nearly zero for this photon energy range. where Pi (t, ±Ej ) and Pi (t, ±2Ej ) are the induced dipole moments for each amplitude of the incident electric field (see Ref. 29 for the derivation of Eqs. (40) and (41)). We estimate the (hyper)polarizabilities of molecules with the following procedure. First, the induced dipole moments Pi (t) for several intensities of the electric field are calculated. (1)

(2)

Second, the first- and second-order responses µij (t) and µijj (t) are extracted from the induced dipoles by using Eqs. (40) and (41). Finally, the extracted first- and secondorder responses are fitted to the theoretical Eqs. (38) and (39), respectively, with the parameters αij (ω), βijj (2ω), and βijj (0). The resultant parameters are the estimated (hyper)polarizabilities.

Molecular Systems We investigated the SHG in symmetric pDNB and asymmetric pNA, whose molecular structures are shown in Figs. 2(a) and (b), respectively. The valence electrons of the atoms were treated in the TDDFT calculation: one electron (1s1 ) of H, four electrons (2s2 2p2 ) of C, five electrons (2s2 2p3 ) of N, and six electrons (2s2 2p4 ) of O. The other core electrons were frozen by employing the norm-conserving pseudopotentials. The structures were optimized by using Gaussian09 31 in the theory level of LSDA/CEP-31g.


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Figures 2(c) and (d) show the oscillator strengths df /dE of pDNB and pNA, respectively. The x-, y-, and z-components dfx /dE, dfy /dE, and dfz /dE are drawn with red solid, green dotted, and blue dashed lines, respectively. The photon energies of the peak oscillator strengths for the x-, y-, and z-polarizations correspond to the resonant excitation energies for the light of each polarization. To obtain the oscillator strength, the molecules were kicked by a pulse-like electric field at t = 0, and the time evolution was performed by solving the Kohn–Sham equation. The induced dipole moment Pi (t) is obtained from ∫ Pi (t) = −

dr ri [n(r, t) − n(r, 0)].

(42)

The oscillator strength dfi /dE (i = x, y, z) is calculated from Pi (t) as dfi 2ω = dE πF

∫

T

[ ] dt Pi (t) 1 − 3(t/T )2 + 2(t/T )3 eiωt ,

(43)

0

where F is the amplitude of the incident pulse and the third-order polynomial of (t/T ), for which T is the total propagation time, is employed as a damping function. 32

Results and Discussion SHG in pDNB Figure 3 shows both the far-field and ONF excitation dynamics of pDNB. The schematics of the excitations are drawn in Fig. 3(a). The photon energy of the excitations ω = 1.95 eV was set to half of the resonant energy in the x-direction indicated by the black arrow in Fig. 2(c) in order to clearly observe the dynamics of the SHG. The far-field light was z-polarized and had a power intensity of 1.05 × 1010 W/cm2 . The electric dipole source of the ONF was placed at a distance of 8 Å from the center of pDNB (4.68 Å from the nearest atom) in the −x direction. The dipole source oscillated along the z-direction with an amplitude of 1 Å. The ONF generated by the dipole with these parameters had the same intensity and 15 ACS Paragon Plus Environment


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Far field

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Near field

(a)

(b) Px (x20) Pz

(c)

2ω

ω

(d) ω

|Px (ω)|2 |Pz (ω)|2

ω 2ω

Figure 3: (Color online) Excitation dynamics of pDNB by far-field light (left column) and the ONF (right column). (a) Schematics of the excitations. (b) Induced dipole moments Px (t) and Pz (t). (c) Time evolution of the point (−Px (t), −Pz (t)) representing the averaged trajectory of the electron on the xz-plane during the excitations. (d) Power spectra of the dipole moments. polarization of the electric field as those of the far-field light at the center of pDNB. The envelope function of the external potential was

f (t) = sin2 (

πt ) (0 < t < τ ), τ

(44)

where the time duration of the pulse τ = 35 fs for both excitations. Figure 3(b) represents the x- and z-components of the induced dipole moment. The zcomponents Pz (t) drawn by the blue dashed lines show similar behavior for both the far-field and ONF excitations. The largest value is at t = 17.5 fs, which coincides with the peak of the incident pulse excitation, and the oscillation disappears after the end of pulse. This


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disappearance is because ω = 1.95 eV is nonresonant energy for the z-polarized excitation as can be seed from the fact that there is no peak around 1.95 eV in the oscillator strength of pDNB, as shown in Fig. 2(c). The ONF induced a larger Pz (t) than the far-field excitation because the intensity of the ONF is greater than that of the far field at positions closer to the dipole source than the center of pDNB. In contrast to Pz (t), the x-components Px (t), which are drawn with red solid lines, indicate completely different dynamics for the far-field and ONF excitations. Because the far-field light is z-polarized, Px (t) for the far-field excitation is nearly zero the entire time. However, Px (t) induced by the ONF oscillates at the frequency 2ω, which is twice that of the incident ONF. This means that the SHG is induced by the ONF in the x-direction even in the symmetric molecule pDNB. The origin of the breakdown of the selection rule is the nonuniformity of the ONF, as explained in the theory section. The oscillation of Px (t) remains even after the end of pulse because the oscillation frequency 2ω (i.e., twice the incident ONF frequency ω) of Px (t) is the resonant frequency 3.9 eV corresponding to the black arrow in Fig. 2(c). Figure 3(c) shows the movement of the point (−Px (t), −Pz (t)). Because the dipole moˆ = −ˆ ment P (t) is defined as the expectation value of the dipole operator µ r , as shown in Eq. (42), −P (t) represents the expectation value for the positions of electrons rˆ. Therefore, Fig. 3(c) indicates the averaged trajectory of the electrons on the xz-plane. For the far-field excitation, the electron oscillates along the z-axis. On the other hand, the electron excited by the ONF moves in both the x- and z-directions. More precisely, the electron is excited and repeats the following movement: the electron at the origin first moves in the z-direction, shifts in the x-direction, and then returns to the origin. This trajectory clearly indicates that the electron oscillates twice in the x-direction and once in the z-direction. This movement of the electron reflects the mechanism of the SHG explained by the model system shown in Fig. 1. Equation (30) indicates that, although the SHG is induced along the x-direction, the state |2⟩ with the transition dipole moment in the z-direction is involved in the SHG



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process. The movement of the electron shown in Fig. 3(c) indicates the involvement of the excited states in the z-direction. The power spectrum of the induced dipole moment is shown in Fig. 3(d). The peak corresponding to the frequency 2ω for the ONF excitation confirms the occurrence of the SHG in the x-direction.

Calculation of β in pNA Because pNA has asymmetry in the x direction, as shown in Fig. 2(b), not only the ONF but also the far-field excitation causes SHG, unlike the case of pDNB. Therefore, we calculated the first hyperpolarizability βijj (2ω) for both the far-field and ONF excitations and compared them to quantitatively evaluate the intensity of the SHG induced by the ONF. The hyperpolarizabilities were calculated with the finite field method. 29 The frequency of the incident light was set to ω = 1.165 eV, which is the same as that adopted in Ref. 29 . Because continuous-wave incident light is used in the framework, the envelope function was set to f (t) = 1. First, we calculated the hyperpolarizabilities for the far-field excitation. The amplitude of the electric field was set to Ej = 0.001 a.u.. Table 1 presents the calculated linear polarizability αij (ω) and first hyperpolarizabilities βijj (0) and βijj (2ω). These values approximately correspond to those calculated in Ref. 29 . This correspondence validates the present calculation. The difference between the values of the (hyper)polarizabilities presented in this study and Ref. 29 should mainly derive from the difference in the exchange-correlation potential employed in the TDDFT calculation. Then, we compared the first hyperpolarizability for the far-field and ONF excitations. Figure 4(a) shows the schematic of the excitations. The far-field light is z-polarized, and the ONF is generated by the z-polarized oscillating electric dipole placed in the −x direction with respect to pNA. Because the electric field of such an ONF has only the z-component at the center of pNA, it is reasonable to compare the hyperpolarizability βxzz (2ω) for both excitations. The electric field of the ONF was controlled by the amplitude of the electric 18 ACS Paragon Plus Environment

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Table 1: Calculated (hyper)polarizabilities of pNA in atomic units. α(ω)

β(0)

β(2ω)

201.35 60.07 113.43 2410.26 -200.34 -276.94 4516.88 -117.34 -283.35

αxx αyy αzz βxxx βxyy βxzz βxxx βxyy βxzz

Far field

Near field

(a)

(b)

µ(2)(t) fi!ed

Figure 4: Second-order dipole responses of pNA for both the far-field and ONF excitations. (2) (a) Schematics of the excitations. (b) The black solid lines are µxz (t) calculated by the finite field method of Eq. (41), and the red dashed lines are the fitted curves using Eq. (39). dipole source. The parameters were the distance between the dipole source and the center of pNA d = 8 Å, and the electric field amplitude at the center of pNA Ej = 0.00025 a.u.. Figure 4(b) shows the calculated second-order responses for both the far-field and ONF (2)

excitations. The black solid and red dashed lines represent µxz (t) and the fitted curve using Eq.

(39), respectively. (ONF)

−13489.05 a.u. and βxzz

(ONF)

The resultant first hyperpolarizabilities are βxzz

(0) =

(2ω) = −18991.52 a.u., where β (ONF) denotes the hyperpolar-

izabilities for the ONF excitation. Compared with the hyperpolarizabilities for the far-field (far)

(far)

excitation in Table 1 (βxzz (0) = −276.94 a.u. and βxzz (2ω) = −283.35 a.u., where β (far) denotes the hyperpolarizability for the far-field excitation), the ONF excitation enhances the first hyperpolarizabilities by two orders of magnitude. (ONF)

Note that βijj

depends on d, which is the distance between the ONF source and



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(a) (far)

|βxxx (2ω)|

(R)

(far)

|βxzz (2ω)| (L)

(b) far 20 20 38 70 70 38

14

d=14Å

(ONF)

Figure 5: d-dependence of |βxzz (2ω)|. (a) The red circles represent the d-dependence for the spatial arrangement where the ONF source is placed in the −x direction with respect to pNA (arrangement L), as drawn in the lower-left inset, and the blue triangles represent the same but for the inverse arrangement (R), as drawn in the upper-right inset. The (far) (far) black dashed and dotted lines are the values of |βxzz (2ω)| and |βxxx (2ω)|, respectively. (b) Averaged trajectories of the electron in the xz-plane excited by both the far-field light and ONF. The reddish and blueish lines show the trajectories for the ONF excitations (d = 14, 20, 38, 70 Å) for arrangements L and R, respectively. The black lines show the trajectory for the far-field excitation. (2)

pNA. This is because, while µijj (t) is normalized by the absolute value of the electric field amplitude as given in Eq. (41), the ONF induces the SHG via the gradient—in other words, the nonuniformity of the electric field. Because the amplitude and gradient of the electric field of the ONF scale as ∝ d−3 and ∝ d−4 , respectively, the ratio of the gradient to the (ONF)

amplitude changes with respect to d. That is why βijj

depends on d, unlike the usual

(far)

hyperpolarizability βijj . (ONF)

Figure 5(a) shows how the absolute value of βxzz

(2ω) depends on d. Both the red circles

(ONF)

and blue triangles represent the d-dependence of |βxzz

(2ω)| calculated for different spatial

arrangements of the system. The red circles indicate the d-dependence for the arrangement


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where the ONF source is placed in the −x direction with respect to pNA (arrangement L), as drawn in the lower-left inset of Fig. 5(a), and the blue triangles represent the same for the inverse arrangement (R), as drawn in the upper-right inset. For comparison, we show (far)

(far)

the values of |βxzz (2ω)| (black dashed line) and |βxxx (2ω)| (black dotted line). When d (ONF)

is below 10 Å, |βxzz

(far)

(2ω)| is one to two orders of magnitude larger than |βxzz (2ω)| for (far)

both arrangements and even larger than the diagonal element |βxxx (2ω)|. With increasing (ONF)

d, |βxzz

(ONF)

(2ω)| acutely decreases. For arrangement L, the sign of βxzz (ONF)

approximately d = 38 Å because the signs of βxzz

(2ω) changes at (far)

(2ω) for small d and βxzz (2ω) are

opposite. Figure 5(b) shows the averaged trajectories of the electrons excited by the far-field light (black line) and the ONF for arrangements L (reddish lines) and R (blueish lines), respectively. For d = 14 and 20 Å, the trajectories for arrangement L curves in the direction opposite to that for the far-field excitation; this difference in the direction of the curves is (ONF)

consistent with the opposite signs of βxzz

(far)

(2ω) and βxzz (2ω). With increasing d, the effect

of the ONF for SHG is weakened, and the trajectory of the electron converges to that of the far-field excitation. Around d = 38 Å, the contributions of the nonuniformity and intensity to the SHG are balanced. Thus, the electron trajectory for d = 38 Å of the arrangement L is almost straight along the z-axis, as shown in Fig. 5(b). For d = 70 Å, the effect of the intensity is dominant, and the nature of the SHG is like that for the far-field excitation. In (ONF)

contrast, for arrangement L, the signs of βxzz

(far)

(2ω) and βxzz (2ω) are the same for all d.

Thus, the directions of the curve of the electron trajectories are also the same. The acute (ONF)

dependence of |βxzz

(2ω)| on d and the change in the d-dependence with respect to the

spatial arrangement, as shown in Fig. 5, can be useful for an imaging method based on the SHG using a nonuniform ONF.



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Conclusion The selection rules and first hyperpolarizabilities of the SHG induced by a nonuniform ONF were investigated. First, we explained the mechanism of the SHG induced by the ONF in terms of the perturbation theory of nonlinear optical processes. We performed TDDFT calculations of the far-field and ONF excitation dynamics of two molecules: symmetric pDNB and asymmetric pNA. Based on the calculation of pDNB, we demonstrated that the ONF induces the SHG even in symmetric molecules. This breakdown of the selection rules of the SHG is because the SHG induced by the ONF is derived from the field nonuniformity. To evaluate the intensity of the SHG induced by the ONF, we calculated the first hyperpolarizability β for both the far-field and ONF excitations of pNA. The calculated hyperpolarizability for the (ONF)

ONF excitation βxzz

(2ω) depended on the distance d between the ONF source and pNA (far)

and was two orders of magnitude larger than that for the far-field excitation βxzz (2ω) when d was around 10 Å. An imaging method highly sensitive in space is expected to be developed on the basis of our theoretical approach to the ONF–SHG dynamics and the computed (ONF)

findings of the acute dependence of |βxzz

(2ω)| on d.

Acknowledgement This research was supported by JSPS KAKENHI Grant No. 25288012 and by MEXT as a social and scientific priority issue (Creation of new functional devices and high-performance materials to support next-generation industries) to be tackled by using the post-K computer (ID: hp160046, hp 160204). M. Y. acknowledges support from a Grant-in-Aid for JSPS Research Fellows (Grant No. 16J08037). This work was also partly supported by the JSPS Core-to-Core Program, A. Advanced Research Networks. Theoretical computations were performed in part at the Research Center for Computational Science, Okazaki, Japan.


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Dipole Moment

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Optical Near Field

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SHG

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