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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 1 — Jan. 2, 2012
  • pp: 173–185
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Closed-form Maker fringe formulas for poled polymer thin films in multilayer structures

Dong Hun Park and Warren N. Herman  »View Author Affiliations


Optics Express, Vol. 20, Issue 1, pp. 173-185 (2012)
http://dx.doi.org/10.1364/OE.20.000173


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Abstract

We report new closed-form expressions for Maker fringes of anisotropic and absorbing poled polymer thin films in multilayer structures that include back reflections of both fundamental and second-harmonic waves. The expressions, based on boundary conditions at each interface, can be applied to multilayer structures containing a buffer and a transparent conducting oxide layer, which might enhance multiple reflections of fundamental and second-harmonic waves inside a nonlinear thin film layer. This formulation facilitates Maker fringe analysis for a sample containing additional multilayer structures on either side of a poled polymer thin film. Experimental data and numerical simulations are given to indicate the importance of inclusion of such a reflective layer in analyses for reliable characterization of second-harmonic tensor elements.

© 2011 OSA

1. Introduction

In this paper, we extend the transmission HH Maker fringe formulation 1) to account for simultaneous anisotropy and absorption of the NLO polymer, 2) a multilayer structure that includes dielectric or non-dielectric materials, and 3) back reflections of both SH and fundamental waves. Back reflections are considered because some absorbing layers, such as a TCO layer, may appreciably enhance multiple reflections. These closed-form mathematical expressions based on boundary conditions are intuitively applicable to a multilayer structure containing a single nonlinear layer with additional absorbing and/or anisotropic linear media.

We also provide a Maker fringe formulation for reflection. This is useful when a highly reflecting NLO sample is used or when a metal layer used as a poling electrode is not removed after poling. Such a formulation is expected to be useful for comparing SHG and EO measurements on the same film. We present experimental results and simulations in order to validate the proposed formulation and indicate the importance of inclusion of full layer structures with multiple reflections for reliable estimation of dμj coefficients.

2. Theory

As much as possible, we have attempted to follow the notation used in Ref. [10

10. W. N. Herman and L. M. Hayden, “Maker fringes revisited: second-harmonic generation from birefringent or absorbing materials,” J. Opt. Soc. Am. B 12(3), 416–427 (1995). [CrossRef]

]. A poled organic thin film prepared by spin coating belongs to the ∞mm point-group symmetry (space group Cv) and has complex ordinary and extraordinary indices of refraction, no and ne, respectively [19

19. I. P. Kaminow, An Introduction to Electrooptic Devices (Academic, 1974).

]. The SH d-coefficient tensor contains three independent complex elements and is given by [10

10. W. N. Herman and L. M. Hayden, “Maker fringes revisited: second-harmonic generation from birefringent or absorbing materials,” J. Opt. Soc. Am. B 12(3), 416–427 (1995). [CrossRef]

]

d=(0000d150000d1500d31d31d33000).
(1)

Because a contact-poled thin film usually requires a substrate for support and two electrodes such as gold and TCO for both contact poling and subsequent reflective EO measurement [4

4. C. C. Teng and H. T. Man, “Simple reflection technique for measuring the electro-optic coefficient of poled polymers,” Appl. Phys. Lett. 56(18), 1734–1736 (1990). [CrossRef]

6

6. D. H. Park, C. H. Lee, and W. N. Herman, “Analysis of multiple reflection effects in reflective measurements of electro-optic coefficients of poled polymers in multilayer structures,” Opt. Express 14(19), 8866–8884 (2006). [CrossRef] [PubMed]

], we consider a five-layer structure (air/buffer/NLO film/TCO/substrate) as shown in Fig. 1(a)
Fig. 1 (a) Multilayer structure geometry containing buffer and TCO layers. Black solid arrows represent fundamental waves and red arrows SH waves. First and second subscripts on n and θ are used to represent dependence of wavelength and layer symbols, respectively. For NLO film, o and e represent ordinary and extraordinary refractive indices, respectively. (b) Equivalent three layer structure using virtual layers (dashed lines). The numeral subscript/superscript i represents a wavelength (i = 1: fundamental and i = 2: second-harmonic). For example, taS(1p) means a p-polarization transmission coefficient from air to substrate at the fundamental (i = 1) wavelength.
. Later we discuss generalization to more than five layers. For transmission type Maker fringe experiments, the top electrode layer on a NLO film, typically gold for contact poling and reflective EO measurement, should be removed after the poling process using an appropriate etchant. The buffer layer might be a protection layer for efficient poling [20

20. J. P. Drummond, S. J. Clarson, J. S. Zetts, F. K. Hopkins, and S. J. Caracci, “Enhanced electro-optic poling in guest–host systems using conductive polymer-based cladding layers,” Appl. Phys. Lett. 74(3), 368–370 (1999). [CrossRef]

] or residual metal owing to imperfect etching. In the following analysis, SHG from interfaces or metal is assumed to be negligible compared with that from the poled polymer thin film. We assume that a p-polarized plane wave of angular frequency ω (fundamental wave) is incident from the left at an angle of incidence θ. Derivation of the incident s-polarized case is similar and straightforward. The SH electric (E) and magnetic (H) fields in each region are given as
RegionI:{E2a=e^2arraS(pp)eik2arrH2a=y^n2araS(pp)eik2arrRegionII:{E2B=e^2BAeik2Br+e^2BrBeik2BrrH2B=y^n2BAeik2Br+y^n2BBeik2BrrRegionIII:{E2f=e^2fCeik2fr+e^2frDeik2frr+ebe2ik1fr+ebre2ik1frrH2f=y^n2f(θ2f)Ceik2fr+y^n2f(θ2f)Deik2frr+hbe2ik1fr+hbre2ik1frrRegionIV:{E2T=e^2TEeik2Tr+e^2TrFeik2TrrH2T=y^n2TEeik2Tr+y^n2TFeik2TrrRegionV:{E2S=e^2StaS(pp)eik2SrH2S=y^n2StaS(pp)eik2Sr,
(2)
where raS(pp) and taS(pp)represent SH p-wave reflection and transmission coefficients, respectively (pp: fundamental p-wave input to SH p-wave output, sp: fundamental s-wave input to SH p-wave output) and A, B, C, D, E, and F are constants. In case of isotropic layers, e^2l=(cosθ2l,0,sinθ2l)and e^2lr=(cosθ2l,0,sinθ2l) are the polarization unit vectors for the electric fields of the transmitted and reflected SH waves, respectively, at layer l=a,B,f,T,S representing air, buffer, NLO film, TCO, and substrate, respectively, and y^=(0,1,0) is the unit vector for the magnetic field H. In the anisotropic layers, the unit vectors along the SH electric field directions are e^2l=(cos(θ2lγ2l),0,sin(θ2lγ2l)) and e^2lr=(cos(θ2lγ2l),0,sin(θ2lγ2l)), where the walk-off angles γil at the fundamental (i = 1) and SH (i = 2) wavelengths are included because of birefringence of the anisotropic layers [10

10. W. N. Herman and L. M. Hayden, “Maker fringes revisited: second-harmonic generation from birefringent or absorbing materials,” J. Opt. Soc. Am. B 12(3), 416–427 (1995). [CrossRef]

]. Similarly, k2l=2k0n2l(sinθ2l,0,cosθ2l) and k2lr=2k0n2l(sinθ2l,0,cosθ2l) are wave vectors for the forward and backward SH waves, respectively. Here, k0=2π/λ=ω/c is the wave vector of the fundamental in free space. n1l and n2l are indices of refraction of the layer l at the fundamental and SH wavelengths, respectively. For birefringent NLO films, we have for p-polarization
nif(θif)=(cos2θifnio2+sin2θifnie2)1/2,
(3)
where i=1,2 [21

21. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).

,22

22. P. Yeh, Optical Waves in Layered Media (Wiley, 1988)

]. We have omitted the subscript for layer information in the anisotropic indices, nio and nie, to prevent the notation from becoming unduly cumbersome, but Eq. (3) also can be applied to other anisotropic layers. The bound waves [8

8. J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41(4), 1667–1681 (1970). [CrossRef]

,9

9. N. Okamoto, Y. Hirano, and O. Sugihara, “Precise estimation of nonlinear-optical coefficients for anisotropic nonlinear films with C∞v symmetry,” J. Opt. Soc. Am. B 9(11), 2083–2087 (1970). [CrossRef]

]eb,ebr,hb, andhbrin region III are inhomogeneous solutions that satisfy Maxwell’s equation given in Eq. (A1). A detailed description for these bound waves is given in Appendix A.

Continuity of tangential E and H components at each interface gives the boundary conditions. The matrix method employing multiple interface boundary conditions [22

22. P. Yeh, Optical Waves in Layered Media (Wiley, 1988)

] is used to calculate and , which are the p-wave SH transmission and reflection coefficients, respectively for the case of a fundamental p-wave input. From Eq. (B3), we find that the SH transmission coefficient from the multilayer structure containing an absorptive and birefringent NLO film is given as
taS(ηp)=ei(ϕ2SL+ϕ2ST)4πik0Ln2fcγ2c2γ[taS(1η)E1atfTS(1η)]2taS(2p)taBf(2p)×{deff(ηp)[eiΦ(ηp)+rfBa(2p)(rfTS(1η))2eiΦ(ηp)]sinc(Ψ(ηp))+deffr(ηp)[(rfTS(1η))2eiΨ(ηp)+rfBa(2p)eiΨ(ηp)]sinc(Φ(ηp))},
(4)
where cγ2=cosγ2f, η=s,p, and E1a is the electric field of the incident fundamental wave. Ψ(ημ) and Φ(ημ) are defined as
Ψ(ημ)=ϕ1(η)ϕ2(μ)2,Φ(ημ)=ϕ1(η)+ϕ2(μ)2.
(5)
where μ=s,p and the phase terms are ϕ1(p)=k0n1f(θ1f)cos(θ1f)L, ϕ1(s)=k0n1ocos(θ1f(s))L, and ϕ2(p)=2k0n2f(θ2f)cos(θ2f)L. Equation (4) can be rewritten using the identity n2fcγ2c2γ/(n2ac2a)=tfBa(2p)/taBf(2p). Note that in comparison with the HH theory [10

10. W. N. Herman and L. M. Hayden, “Maker fringes revisited: second-harmonic generation from birefringent or absorbing materials,” J. Opt. Soc. Am. B 12(3), 416–427 (1995). [CrossRef]

], Eq. (4) reduces to the HH equation when the Fresnel reflection coefficient of the fundamental wave rfTS(1η) is zero and the buffer and TCO layers do not exist. The detailed derivations are given in Appendix B. Finally, the transmitted SH power is given by
P2ω=c8π(tSa(2p))2|taS(ηp)|2Across,
(6)
where tSa(2p) is the p-polarized transmission coefficient of the SH wave at the substrate/air interface and Across is the cross-sectional area of the fundamental wave.

We notice that the SH transmission coefficient is proportional to taS(2p)/taBf(2p) and to (taS(1η)/tfTS(1η))2. Thus, the SH transmission can be maximized by increasing taS(2p) and taS(1η)while taBf(2p) and tfTS(1η) are decreased. As seen from Eqs. (B1)-(B4), the mathematical expression can easily be modified to apply to a multilayered structure containing additional layers on either side of the NLO film by simply substituting the appropriate transmission and reflection coefficients at virtual layers [23

23. D. Guo, R. Lin, and W. Wang, “Gaussian-optics-based optical modeling and characterization of a Fabry-Perot microcavity for sensing applications,” J. Opt. Soc. Am. A 22(8), 1577–1588 (2005). [CrossRef] [PubMed]

] and adding a phase term introduced by the additional layers. Then, we have
taS(ηp)=e2ik0n2Sc2S(L+dT+lAD)4πik0Ln2fcγ2c2γ[taS(1η)E1atfS(1η)]2taS(2p)taf(2p)×{deff(ηp)[eiΦ(ηp)+rfa(2p)(rfS(1η))2eiΦ(ηp)]sinc(Ψ(ηp))+deffr(ηp)[(rfS(1η))2eiΨ(ηp)+rfa(2p)eiΨ(ηp)]sinc(Φ(ηp))},
(7)
where rjk(iη) represents an η-polarization Fresnel reflection coefficient from j layer to k layer as described in Eqs. (B5)-(B8). lAD is the thickness of an additional layer located between the NLO film and the substrate, but the phase factor resulting from the additional layer doesn’t affect the magnitude of SH transmittance. Again, each side of the multilayer structure on a NLOP film can be treated as a single virtual interface as shown in Fig. 1(b). We note again that these equations do not apply to a sample containing multiple nonlinear film layers.

As briefly mentioned in Section 1, a metal electrode layer such as gold is usually removed after poling for the transmission type of Maker fringe experiment. It can however be used as a reflector without a metal removal process, which allows us to re-pole the sample or to use it for another reflection type of measurement. A formula for the SH reflection coefficient can be obtained from Eq. (B3) and is given by
raS(ηp)=eiϕ2aB(taS(1η)E1atfTS(1η))24πik0Ln2fcγ2c2γtaBf(2p)×{raS(2p)[deff(ηp)(eiΦ(ηp)+rfBa(2p)(rfTS(1η))2eiΦ(ηp))sinc(Ψ(ηp))+deffr(ηp)(rfBa(2p)eiΨ(ηp)+(rfTS(1η))2eiΨ(ηp))sinc(Φ(ηp))][deff(ηp)(raBf(2p)eiΦ(ηp)+(raBf(2p)rfBa(2p)taBf(2p)tfBa(2p))(rfTS(1η))2eiΦ(ηp))sinc(Ψ(ηp))+deffr(ηp)((raBf(2p)rfBa(2p)taBf(2p)tfBa(2p))eiΨ(ηp)+raBf(2p)(rfTS(1η))2eiΨ(ηp))sinc(Φ(ηp))]},
(8)
where raBf(2p)rfBa(2p)taBf(2p)tfBa(2p) becomes −1 in the case of no buffer layer, that is, a single interface.

3. Results

For the first experimental example, we measured a Maker fringe pattern (pp) for X-cut quartz (rotation is about z-axis), which is commonly used for reference. The focused incident laser spot size used in the experiment is a few tens of microns, which is much smaller than the thickness of the X-cut quartz (533 μm). Therefore, multiply reflected beams at the quartz/air interface are spatially localized, so they don’t interfere. On the other hand, Eq. (4) is derived based on the plane wave analysis. One can simply make reflections rfBa(2p) and rfTS(1η) in Eq. (4)null in order to get a formulation that does not contain a back reflection effect. As shown in Fig. 2(a)
Fig. 2 Second harmonic intensities of X-cut quartz at varying angles of incidence in three cases (Rotation is about the z axis): (a) Quartz only, (b) Quartz/Au, and (c) Au/Quartz/Au. Experimental results (red dots) are plotted with two different simulated Maker fringe patterns (blue: multiple reflections, cyan: no multiple reflections).
, a simulation (blue curve) using Eq. (4) shows large rapid SH intensity fluctuations as expected from multiple reflections in such a thick slab structure, but the measured SHG data (red dots) doesn’t show these fluctuations and is well described by the modified Eq. (4) (cyan curve), where rfBa(2p) and rfTS(1η)are set equal to zero. In order to verify our formula for a multilayer structure (number of layers ≥ 4), ~5 nm thick gold was deposited on quartz (air/quartz/Au/air) using a sputtering system and the SH intensity was measured as a function of angle. Subsequently, another thin gold layer was deposited on the other side of quartz (air/Au/quartz/Au/air) and SH data again collected. The SH transmitted intensity decreases as a gold metal layer is added because of the transmission loss and the reflection caused by the gold layer. We note that the experimental Maker fringes showed a good agreement with simulations taking into account multilayer structures as shown in Figs. 2(b) and (c). Experimental curves in Figs. 2(b) and (c) show large difference in both SHG magnitude and fringe shape with that in Fig. 1(a) where no additional gold layers are present. This indicates that it is important to include multilayer structures in analysis. The same simulation result as the cyan curve in Fig. 1(a) is obtained from the HH method that is based on three layer structure. The SH multiple reflections included in the HH method also cannot contribute to the SHG because the beam size is small compared with the quartz thickness and multiply reflected SH beams cannot interfere with each other. Note that the experimental data deviate slightly from the theoretical estimation at large angles because, for a thick sample, the laser beam translates on the back surface during rotation and this introduces sensitivity to both surface and thickness variations.

A nonlinear polymer thin film was fabricated by doping 30 wt.% of AJL49, provided by the University of Washington, with an amorphous polycarbonate (APC) host. The nonlinear polymer solution was spin-coated on a 180 nm thick indium tin oxide (ITO)/glass substrate manufactured by Delta Technologies (Product #: CB-40IN-0111), to give a 1.2 μm thick film. Then 100 nm thick gold was deposited on the film using a thermal evaporator. After contact poling at the glass transition temperature of 135 °C using a poling voltage of ~100 V/μm, the gold poling electrode was etched off using a gold etchant KI + I2 + H2O (4:1:40) for a transmission type Maker fringe experiment. The optical properties of ITO and the NLOP film were measured using a spectroscopic ellipsometer (VASE®, J. A. Woollam Co., Inc.) as shown in Figs. 3(a) and (b)
Fig. 3 (a) Complex index of refraction of ITO, (b) Complex anisotropic indices of refraction of NLOP measured using ellipsometer (inset figure: absorption profile of the multilayered sample by UV-VIS spectrophotometer), (c) pp and (d) sp Maker fringe profiles. Experimental results (red circles) were plotted with simulation results by various types of configurations. For line colors in (c) and (d), Cyan: multilayer structure with multiple reflections of both fundamental and SH. Black: multilayer structure without any multiple reflections. Green dash: multilayer structure with multiple reflections of SH only. Blue: three layer model (air/NLOP/substrate) without any multiple reflections.
. Note that the large linear absorption of ITO in the infrared region can lead to considerable back reflections of fundamental waves.

Multiply reflected beams inside a few micron thick NLOP film interfere because of the larger ratio of beam size to NLO film thickness. Therefore, back reflections of both fundamental and SH waves should be included in the analysis when back reflections at the interface of NLOP/TCO/substrate are not negligible. As shown in Figs. 3(c) and (d), our derived formulas in Eqs. (4) and (6) (cyan curve) agree very well with experimental results (red circles) in both pp and sp cases. Noteworthy is that one can see a slight dip around 50° angle of incidence caused by back reflections in both the experimental and simulation data. However, this dip is not accounted for properly when back reflections of fundamental waves are ignored (green dash and black curve), which demonstrates that back reflections of fundamental waves should be taken into account in the analysis of SH transmission. In fact, the fundamental wave is more reflective than the SH at the ITO layer because of free carrier absorption of ITO in the infrared region as shown in Fig. 3(a). Depending on the thickness and optical properties of the NLOP film and the ITO layers, the shape and the dip position may change drastically. The linear absorption of the NLO film and the ITO is taken into account in the analysis based on the complex index of refraction in Figs. 3(a) and (b), but the nonparametric nonlinear absorption of the NLO film (imaginary part of d-tensor component) is ignored for this specific nonlinear polymer because it is expected to be small compared with the real part of the d-tensor component. The effect of a nonlinear absorption up to 20%

of the real part of the d-tensor component in the analysis is negligible in the fitting results, whereas the linear absorption of ITO and NLO film does affect the fitting. We also see that the analysis by the three layer model (blue curves in Figs. 3(c) and (d)), ignoring back reflection terms, is not suitable for the analysis of Maker fringe data from the multilayered structures involving reflective layers. Therefore, inclusion of the back reflection of fundamental waves and multilayer structures in the analysis is crucial in order to reliably estimate SH d-tensor components for the poled polymer thin films.

For the AJL49 NLOP film, we measured a SH d33 of 360 ± 5 pm/V with the ratio of d31/d330.31. Using the two-level model [24

24. K. D. Singer, M. D. Kuzyk, and J. E. Sohn, “Second-order nonlinear-optical processes in orientationally ordered materials: relationship between molecular and macroscopic properties,” J. Opt. Soc. Am. B 17, 566–571 (2000).

], the EO coefficient r33 is estimated to be 140 ± 30 pm/V and 100 ± 20 pm/V at 1300 nm and 1550 nm respectively, where the large error bound results from imprecise knowledge of local field effects. We were unable to directly measure EO coefficients of this sample, because the 1.2 μm thick film can support only one guided mode, which doesn’t allow use of the Attenuated Total Reflection (ATR) method for an accurate characterization. Direct measurement of EO coefficients of a thicker AJL49/APC sample with the higher weight percentage of 45 wt.% and lower poling field of 70 V/μm gave an EO coefficient r33 of 110 pm/V and 80 pm/V at 1300 nm and 1550 nm, respectively, which are comparable to our estimation from the measured SH d33.

4. Conclusions

We have presented new closed-form expressions for obtaining SH d-coefficients from Maker fringe SHG experiments on poled polymer thin films embedded in a multilayer structure with reflective layers, which can be dielectric or non-dielectric. The multilayer analysis including back reflections is necessary for second-order NLOP thin films prepared by electrode contact poling. We have shown that a TCO layer as a poling electrode has to be included in the analysis because it can enhance multiple reflections. We expect the proposed formulation to be useful to characterize Maker fringes of a multilayered poled polymer thin film or thick organic material [25

25. R. C. Hoffman, A. G. Mott, M. J. Ferry, T. M. Pritchett, W. Shensky, J. A. Orlicki, G. R. Martin, J. Dougherty, J. L. Leadore, A. M. Rawlett, and D. H. Park, “Poling of visible chromophores in millimeter-thick PMMA host,” Opt. Mater. Express 1(1), 67–77 (2011). [CrossRef]

] and to investigate the relation between SHG and EO properties on the same sample for contact poled films. Our formulations can also be applied to NLO crystals [26

26. J. Zhang, G. Wang, Z. Liu, L. Wang, G. Zhang, X. Zhang, Y. Wu, P. Fu, and Y. Wu, “Growth and optical properties of a new nonlinear Na3La9O3(BO3)8 crystal,” Opt. Express 18(1), 237–243 (2010). [CrossRef] [PubMed]

,27

27. C. Chen, Z. Shao, J. Jiang, J. Wei, J. Lin, J. Wang, N. Ye, J. Lv, B. Wu, M. Jiang, M. Yoshimura, Y. Mori, and T. Sasaki, “Determination of the nonlinear optical coefficients of YCa4O(BO3)3 crystal,” J. Opt. Soc. Am. B 17(4), 566–571 (2000). [CrossRef]

] by modifying, in a straightforward manner, the reflectance, transmittance, and effective d coefficients depending on the spatial symmetry and crystal cut.

Appendix A: Bound waves

We assume no pump depletion of the fundamental wave because, in the absence of phase matching, it is assumed to be negligible. From Maxwell’s equations, the nonlinear wave equation in cgs unit convention is
××E2(2k0)2(n2o2000n2o2000n2e2)E2=(2k0)24πPNL,
(A1)
where the nonlinear polarization is given by

PNL=d:E1E1.
(A2)

Because the fundamental field E1 in Eq. (A2) contains both forward and backward electric fields resulting from multiple reflections, the SH field E2 inside the NLO film has three particular solutions for the bound waves, ebe2ik1fr, ebre2ik1frr, and emixei(k1f+k1fr)r along with two homogeneous solutions e^2feik2fr and e^2freik2frr for the free waves. However, we neglect the last term emixei(k1f+k1fr)r because of the large phase mismatch. Two homogenous and two particular solutions are included in the field expressions in Eq. (2). Substituting E2 for the bound wave eb or ebr and multiplying e^2f or e^2fr in Eq. (A1) gives, after some algebraic manipulations (see Ref. [10

10. W. N. Herman and L. M. Hayden, “Maker fringes revisited: second-harmonic generation from birefringent or absorbing materials,” J. Opt. Soc. Am. B 12(3), 416–427 (1995). [CrossRef]

]), the following relations:
n2f(θ2f)cosγ2febx±cos(θ2fγ2f)hby=4πk0Lϕ1(η)12ϕ2(p){e^2fPNLe^2frPNL
(A3)
and
n2f(θ2f)cosγ2febxr±cos(θ2fγ2f)hbyr=4πk0Lϕ1(η)12ϕ2(p){e^2fPNLre^2frPNLr,
(A4)
where ebx and hby are x- and y-components of the forward bound electric and magnetic field, respectively, and the superscript r denotes the reflected bound field. The phase terms are ϕ1(p)=k0n1f(θ1f)cos(θ1f)L, ϕ1(s)=k0n1ocos(θ1f(s))L, and ϕ2(p)=2k0n2f(θ2f)cos(θ2f)L. The inner products of the unit vector nonlinear polarization in the right hand side of Eqs. (A3) and (A4) are given as
e^2fPNL=E12e^2fd:e^1fe^1f=E12deff(pp)e^2frPNL=E12e^2frd:e^1fe^1f=E12deffr(pp)e^2fPNLr=E1r2e^2fd:e^1fre^1fr=E1r2deffr(pp)e^2frPNLr=E1r2e^2frd:e^1fre^1fr=E1r2deff(pp),
(A5)
where the effective SH d-coefficients for forward and backward waves are
deff(pp)=2d15c2γs1γc1γ+s2γ(d31c1γ2+d33s1γ2)deffr(pp)=2d15c2γs1γc1γ+s2γ(d31c1γ2+d33s1γ2)
(A6)
with

ciγ=cos(θifγif),siγ=sin(θifγif).
(A7)

For the sp case, we have

deff(sp)=d31s2γ=deffr(sp).
(A8)

Note that if Kleinman symmetry is not valid (d31d15), then an additional input and output polarization scheme, e.g. αs (α-polarized fundamental input to s-polarized SH), should be considered in order to determine d15 separately. In this case, the fundamental electric field inside nonlinear medium can be expressed as

E1f=E1sy^eik1fsr+E1sry^eik1fs,rr+E1pe^1peik1fpr+E1pre^1preik1fp,rr.
(A9)

Then, we obtain
y^d:E1fE1f=2d15s1γ(E1sE1pei(k1fs+k1fp)r+E1sE1prei(k1fs+k1fp,r)r+E1srE1pei(k1fs,r+k1fp)r+E1srE1prei(k1fs,r+k1fp,r)r),
(A10)
where we ignore terms such as ei(k1fs+k1fp,r)r and ei(k1fs,r+k1fp)rbecause they don’t contribute to SHG. The s-polarized SH transmission coefficient for α-polarized fundamental input is given in Appendix C.

Appendix B: Boundary conditions

As the number of layers becomes larger, the analysis becomes more complicated. A matrix method facilitates solving complicated boundary conditions [22

22. P. Yeh, Optical Waves in Layered Media (Wiley, 1988)

]. We use the 2×2 matrix method to obtain the equations resulting from the boundary conditions and calculate the SH transmission and reflection coefficients. Continuity of tangential E and H components at each interface provides boundary conditions, which can be expressed in the matrix form
I/II:eiϕ2aBDa(0raS(pp))=DBB(AB),II/III:DB(AB)=Df(CD)+(ebxhby)+(ebxrhbyr),III/IV:Df21(CD)+e2iϕ1(ebxhby)+e2iϕ1(ebxrhbyr)=DT2TL1(EF),IV/V:DTT12TL1(EF)=ei(ϕ2SL+ϕ2ST)DS(taS(pp)0),
(B1)
where the phase terms are ϕ2aB=2k0n2ac2adB, ϕ2SL=2k0n2Sc2SL, and ϕ2ST=2k0n2Sc2SdT. The dynamic and propagation matrices D and are given as
Dl=(cos(θ2l)cos(θ2l)n2ln2l),Df=(cos(θ2fγ2f)cos(θ2fγ2f)n2cos(γ2)n2cos(γ2)),m=(eiϕ2m00eiϕ2m),f=(eiϕ2f00eiϕ2f),and2TL=(eiϕ2TL00eiϕ2TL),
(B2)
where l=a,B,T,S and m=B,T. The walk-off angle should be applied to the dynamic matrix when the layer is anisotropic as included in the matrix Df for the nonlinear film. The phase terms are ϕ2m=2k0n2mc2mdm (m=B,T) and ϕ2TL=2k0n2Tc2TL. From Eq. (B1), we can construct a matrix equation
ei(ϕ2SL+ϕ2ST)DaS(taS(pp)0)=eiϕ2aB(0raS(pp))12n2fcγ2c2γDaBf×{(1ei(2ϕ1(p)ϕ2(p))001ei(2ϕ1(p)+ϕ2(p)))(n2fcγ2ebx+c2γhbyn2fcγ2ebx+c2γhby)+(1ei(2ϕ1(p)+ϕ2(p))001ei(2ϕ1(p)ϕ2(p)))(n2fcγ2ebxr+c2γhbyrn2fcγ2ebxr+c2γhbyr)},
(B3)
where cγ2=cosγ2f. The dynamic matrix for the multilayered stack is given as

DaBf=1taBf(2p)(1rfBa(2p)raBf(2p)taBf(2p)tfBa(2p)raBf(2p)rfBa(2p)),DaS=1taS(2p)(1rSa(2p)raS(2p)taS(2p)tSa(2p)raS(2p)rSa(2p)).
(B4)

Assuming that the incident light is a plane wave, iteration of the Airy formula [21

21. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).

,22

22. P. Yeh, Optical Waves in Layered Media (Wiley, 1988)

] gives the reflection and the transmission coefficients in the multilayered structure. The resulting expressions for reflection and transmission coefficient have the form
raS(iη)=raB(iη)+rBS(iη)e2iβB(iη)dB1+raB(iη)rBS(iη)e2iβB(iη)dBrBS(iη)=rBf(iη)+rfTS(iη)e2iβf(iη)L1+rBf(iη)rfTS(iη)e2iβf(iη)LriTS(iη)=rfT(iη)+rTS(iη)e2iβT(iη)dT1+rfT(iη)rTS(iη)e2iβT(iη)dT
(B5)
and
taS(iη)=taB(iη)tBS(iη)eiβB(iη)dB1+raB(iη)rBS(iη)e2iβB(iη)dBtBS(iη)=tBf(iη)tfTS(iη)eiβf(iη)L1+rBf(iη)rfTS(iη)e2iβf(iη)LtiTS(iη)=tfT(iη)tTS(iη)eiβT(iη)dT1+rfT(iη)rTS(iη)e2iβT(iη)dT,
(B6)
where the s- and p- propagation constants are given by βl(is)=ikonscos(θs) and βl(ip)=ikonp(θp)cos(θp) and these can be applied at each layer with appropriate anisotropic indices and internal angles. np is given in Eq. (3), ns=no, and θs and θp are the complex propagation angles inside the medium. The corresponding reflection and transmission coefficients from layer j to k are given by
rjk(is)=Zk(is)Zj(is)Zk(is)+Zj(is),rjk(ip)=Zj(ip)Zk(ip)Zj(ip)+Zk(ip),tjk(iη)=2Zj(iη)Zj(iη)+Zk(iη)nj(θj)cosγijnk(θk)cosγik,
(B7)
with the s- and p- wave impedances of each anisotropic and/or absorbing layer given by [28

28. S. Ramo, J. R. Whinnery, and T. V. Duzer, Fields and Waves in Communication Electronics (Wiley, New York, 1965), Chap. 6.

]
Z(is)=1nio2sin2θ,Z(ip)=1nio1(sinθnie)2,
(B8)
where θ is the external angle of incidence from air, and i = 1 or 2 represents fundamental and SH wavelengths, respectively. These can be applied to other layers with corresponding refractive indices and angle of incidence from air.

Appendix C: α-polarized fundamental to s-polarized SHG

We assume the input fundamental wave is polarized at an angle α (e.g. 0° and 90° mean s- and p-polarizations, respectively). Similar to the pp calculations in Appendix B, we can solve boundary conditions such that Ey and Hx are continuous at each interface and obtain the s-polarized SH transmission coefficient in the form
taS(αs)=e2ik0n2Sc2S(L+dT+lAD)12n2oc2ftaS(2s)taf(2s)×[(1ei(ϕ1(s)+ϕ1(p)ϕ2(s)))(n2oc2febyhbx)rfa(2s)(1ei(ϕ1(s)+ϕ1(p)+ϕ2(s)))(n2oc2feby+hbx)+(1ei(ϕ1(s)+ϕ1(p)+ϕ2(s)))(n2oc2febyrhbxr)rfa(2s)(1ei(ϕ1(s)+ϕ1(p)ϕ2(s)))(n2oc2febyr+hbxr)],
(C1)
where c2f=cosθ2f(s) and θ2f(s) is the refractive angle of s-polarized light inside the film. Substituting E2 for the bound wave ebyei(k1fs+k1fp)r or ebyrei(k1fs,r+k1fp,r)r and taking the dot product with e^2fs=e^2fs,r=y^ in Eq. (A1) gives
n2oc2feby±hbx=8πk0Lϕ1(s)+ϕ1(p)±ϕ2(s)y^PNL,
(C2)
and
n2oc2fbyr±hbxr=±8πk0Lϕ1(s)+ϕ1(p)ϕ2(s)y^PNLr,
(C3)
where

y^PNL=2E1sE1pd15s1γ,y^PNLr=2E1srE1prd15s1γ.
(C4)

The fundamental s and p electric fields, E1η and E1ηr, inside the nonlinear medium can be calculated from the input fundamental wave and Airy’s formula. Assuming the α-polarized input fundamental wave is E1a=E1a(y^cosα+e^1asinα), the electric fields for forward and backward fundamental waves inside the NLO film can be expressed as

E1f=E1a(y^cosαtaS(1s)tfS(1s)eiϕ1(s)+e^1fsinαtaS(1p)tfS(1p)eiϕ1(p))=y^E1s+e^1fE1pE1fr=E1a(y^cosαtaS(1s)tfS(1s)rfTS(1s)eiϕ1(s)+e^1frsinαtaS(1p)tfS(1p)rfTS(1p)eiϕ1(p))=y^E1sr+e^1frE1pr.
(C5)

Substitution into Eq.(C4) gives

y^PNL=(E1a)2sin(2α)taS(1s)tfS(1s)taS(1p)tfS(1p)eiϕ1(s)eiϕ1(p)d15s1γy^PNLr=(E1a)2sin(2α)taS(1s)tfS(1s)taS(1p)tfS(1p)rfS(1s)rfS(1p)eiϕ1(s)eiϕ1(p)d15s1γ.
(C6)

From Eqs. (C2)-(C6), we thus obtain the s-polarized SH transmission coefficient in a similar form to Eq. (7):
taS(αs)=e2ik0n2Sc2S(L+dT+lAD)4πik0Ln2oc2f[taS(1s)tfS(1s)taS(1p)tfS(1p)E1a2]taS(2s)taf(2s)sin(2α)×d15s1γ{[eiΦ(αs)+rfa(2s)(rfS(1s)rfS(1p))eiΦ(αs)]sinc(Ψ(αs))+[eiΨ(αs)(rfS(1s)rfS(1p))+rfa(2s)eiΨ(αs)]sinc(Φ(αs))},
(C7)
where Ψ(αs) and Φ(αs) are defined as

Ψ(αs)=ϕ1(s)+ϕ1(p)ϕ2(s)2,Φ(αs)=ϕ1(s)+ϕ1(p)+ϕ2(s)2.
(C8)

For the cases that α = 0° (ss) and α = 90° (ps) for the input polarization, Eq. (C7) gives the well-known result of zero SH intensity because of the sin(2α) term in Eq. (C7).

Acknowledgments

We would like to thank Professor Alex K.-Y. Jen and Dr. Jingdong Luo at the University of Washington for providing the nonlinear polymers.

References and links

1.

G. A. Lindsay and K. D. Singer, eds., Polymers for Second-Order Nonlinear Optics, Vol 601 of ACS Symposium Series (ACS, 1995).

2.

W. N. Herman, S. R. Flom, and S. H. Foulger, eds., Organic Thin Films for Photonic Applications, Vol. 1039 of ACS Symposium Series (ACS, 2010).

3.

D. M. Burland, R. D. Miller, and C. A. Walsh, “Second-order nonlinearity in poled-polymer systems,” Chem. Rev. 94(1), 31–75 (1994). [CrossRef]

4.

C. C. Teng and H. T. Man, “Simple reflection technique for measuring the electro-optic coefficient of poled polymers,” Appl. Phys. Lett. 56(18), 1734–1736 (1990). [CrossRef]

5.

J. S. Schildkraut, “Determination of the electro optic coefficient of a poled polymer film,” Appl. Opt. 29(19), 2839–2841 (1990). [CrossRef] [PubMed]

6.

D. H. Park, C. H. Lee, and W. N. Herman, “Analysis of multiple reflection effects in reflective measurements of electro-optic coefficients of poled polymers in multilayer structures,” Opt. Express 14(19), 8866–8884 (2006). [CrossRef] [PubMed]

7.

P. D. Maker, R. W. Terhune, M. Nisenhoff, and C. M. Savage, “Effects of Dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8(1), 21–22 (1962). [CrossRef]

8.

J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41(4), 1667–1681 (1970). [CrossRef]

9.

N. Okamoto, Y. Hirano, and O. Sugihara, “Precise estimation of nonlinear-optical coefficients for anisotropic nonlinear films with C∞v symmetry,” J. Opt. Soc. Am. B 9(11), 2083–2087 (1970). [CrossRef]

10.

W. N. Herman and L. M. Hayden, “Maker fringes revisited: second-harmonic generation from birefringent or absorbing materials,” J. Opt. Soc. Am. B 12(3), 416–427 (1995). [CrossRef]

11.

T. K. Lim, M.-Y. Jeong, C. Song, and D. C. Kim, “Absorption effect in the calculation of a second-order nonlinear coefficient from the data of a maker fringe experiment,” Appl. Opt. 37(13), 2723–2728 (1998). [CrossRef] [PubMed]

12.

H. Hellwig and L. Bohaty, “Multiple reflections and Fabry-Perot interference corrections in Maker fringe experiments,” Opt. Commun. 161(1-3), 51–56 (1999). [CrossRef]

13.

N. A. Sanford and J. A. Aust, “Nonlinear optical characterization of LiNbO3. I. Theoretical analysis of Maker fringe patterns for x-cut wafers,” J. Opt. Soc. Am. B 15(12), 2885–2908 (1998). [CrossRef]

14.

M. Abe, I. Shoji, J. Suda, and T. Kondo, “Comprehensive analysis of multiple-reflection effects on rotational Maker-fringe experiments,” J. Opt. Soc. Am. B 25(10), 1616–1624 (2008). [CrossRef]

15.

M. Braun, F. Bauer, Th. Vogtmann, and M. Schwoerer, “Precise second-harmonic generation Maker fringe measurements in single crystals of the diacetylene NP/4-MPU and evaluation by a second-harmonic generation theory in 4×4 matrix formulation and ray tracing,” J. Opt. Soc. Am. B 14(7), 1699–1706 (1997). [CrossRef]

16.

M. Braun, F. Bauer, Th. Vogtmann, and M. Schwoerer, “Detailed analysis of second-harmonic-generation Maker fringes in biaxially birefringent materials by a 4×4 matrix formulation,” J. Opt. Soc. Am. B 15(12), 2877–2884 (1998). [CrossRef]

17.

V. Rodriguez and C. Sourisseau, “General Maker-fringe ellipsometric analyses in multilayer nonlinear and linear anisotropic optical media,” J. Opt. Soc. Am. B 19(11), 2650–2664 (2002). [CrossRef]

18.

S. Lee, B. Park, S.-D. Lee, G. Park, and Y. D. Kim, “Second-harmonic generation in poled films of nonlinear optical polymer composites,” Opt. Quantum Electron. 27(5), 411–420 (1995). [CrossRef]

19.

I. P. Kaminow, An Introduction to Electrooptic Devices (Academic, 1974).

20.

J. P. Drummond, S. J. Clarson, J. S. Zetts, F. K. Hopkins, and S. J. Caracci, “Enhanced electro-optic poling in guest–host systems using conductive polymer-based cladding layers,” Appl. Phys. Lett. 74(3), 368–370 (1999). [CrossRef]

21.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).

22.

P. Yeh, Optical Waves in Layered Media (Wiley, 1988)

23.

D. Guo, R. Lin, and W. Wang, “Gaussian-optics-based optical modeling and characterization of a Fabry-Perot microcavity for sensing applications,” J. Opt. Soc. Am. A 22(8), 1577–1588 (2005). [CrossRef] [PubMed]

24.

K. D. Singer, M. D. Kuzyk, and J. E. Sohn, “Second-order nonlinear-optical processes in orientationally ordered materials: relationship between molecular and macroscopic properties,” J. Opt. Soc. Am. B 17, 566–571 (2000).

25.

R. C. Hoffman, A. G. Mott, M. J. Ferry, T. M. Pritchett, W. Shensky, J. A. Orlicki, G. R. Martin, J. Dougherty, J. L. Leadore, A. M. Rawlett, and D. H. Park, “Poling of visible chromophores in millimeter-thick PMMA host,” Opt. Mater. Express 1(1), 67–77 (2011). [CrossRef]

26.

J. Zhang, G. Wang, Z. Liu, L. Wang, G. Zhang, X. Zhang, Y. Wu, P. Fu, and Y. Wu, “Growth and optical properties of a new nonlinear Na3La9O3(BO3)8 crystal,” Opt. Express 18(1), 237–243 (2010). [CrossRef] [PubMed]

27.

C. Chen, Z. Shao, J. Jiang, J. Wei, J. Lin, J. Wang, N. Ye, J. Lv, B. Wu, M. Jiang, M. Yoshimura, Y. Mori, and T. Sasaki, “Determination of the nonlinear optical coefficients of YCa4O(BO3)3 crystal,” J. Opt. Soc. Am. B 17(4), 566–571 (2000). [CrossRef]

28.

S. Ramo, J. R. Whinnery, and T. V. Duzer, Fields and Waves in Communication Electronics (Wiley, New York, 1965), Chap. 6.

OCIS Codes
(160.5470) Materials : Polymers
(190.4710) Nonlinear optics : Optical nonlinearities in organic materials
(310.6860) Thin films : Thin films, optical properties

ToC Category:
Thin Films

History
Original Manuscript: November 7, 2011
Revised Manuscript: December 9, 2011
Manuscript Accepted: December 12, 2011
Published: December 19, 2011

Citation
Dong Hun Park and Warren N. Herman, "Closed-form Maker fringe formulas for poled polymer thin films in multilayer structures," Opt. Express 20, 173-185 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-1-173


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References

  1. G. A. Lindsay and K. D. Singer, eds., Polymers for Second-Order Nonlinear Optics, Vol 601 of ACS Symposium Series (ACS, 1995).
  2. W. N. Herman, S. R. Flom, and S. H. Foulger, eds., Organic Thin Films for Photonic Applications, Vol. 1039 of ACS Symposium Series (ACS, 2010).
  3. D. M. Burland, R. D. Miller, and C. A. Walsh, “Second-order nonlinearity in poled-polymer systems,” Chem. Rev.94(1), 31–75 (1994). [CrossRef]
  4. C. C. Teng and H. T. Man, “Simple reflection technique for measuring the electro-optic coefficient of poled polymers,” Appl. Phys. Lett.56(18), 1734–1736 (1990). [CrossRef]
  5. J. S. Schildkraut, “Determination of the electro optic coefficient of a poled polymer film,” Appl. Opt.29(19), 2839–2841 (1990). [CrossRef] [PubMed]
  6. D. H. Park, C. H. Lee, and W. N. Herman, “Analysis of multiple reflection effects in reflective measurements of electro-optic coefficients of poled polymers in multilayer structures,” Opt. Express14(19), 8866–8884 (2006). [CrossRef] [PubMed]
  7. P. D. Maker, R. W. Terhune, M. Nisenhoff, and C. M. Savage, “Effects of Dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett.8(1), 21–22 (1962). [CrossRef]
  8. J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys.41(4), 1667–1681 (1970). [CrossRef]
  9. N. Okamoto, Y. Hirano, and O. Sugihara, “Precise estimation of nonlinear-optical coefficients for anisotropic nonlinear films with C∞v symmetry,” J. Opt. Soc. Am. B9(11), 2083–2087 (1970). [CrossRef]
  10. W. N. Herman and L. M. Hayden, “Maker fringes revisited: second-harmonic generation from birefringent or absorbing materials,” J. Opt. Soc. Am. B12(3), 416–427 (1995). [CrossRef]
  11. T. K. Lim, M.-Y. Jeong, C. Song, and D. C. Kim, “Absorption effect in the calculation of a second-order nonlinear coefficient from the data of a maker fringe experiment,” Appl. Opt.37(13), 2723–2728 (1998). [CrossRef] [PubMed]
  12. H. Hellwig and L. Bohaty, “Multiple reflections and Fabry-Perot interference corrections in Maker fringe experiments,” Opt. Commun.161(1-3), 51–56 (1999). [CrossRef]
  13. N. A. Sanford and J. A. Aust, “Nonlinear optical characterization of LiNbO3. I. Theoretical analysis of Maker fringe patterns for x-cut wafers,” J. Opt. Soc. Am. B15(12), 2885–2908 (1998). [CrossRef]
  14. M. Abe, I. Shoji, J. Suda, and T. Kondo, “Comprehensive analysis of multiple-reflection effects on rotational Maker-fringe experiments,” J. Opt. Soc. Am. B25(10), 1616–1624 (2008). [CrossRef]
  15. M. Braun, F. Bauer, Th. Vogtmann, and M. Schwoerer, “Precise second-harmonic generation Maker fringe measurements in single crystals of the diacetylene NP/4-MPU and evaluation by a second-harmonic generation theory in 4×4 matrix formulation and ray tracing,” J. Opt. Soc. Am. B14(7), 1699–1706 (1997). [CrossRef]
  16. M. Braun, F. Bauer, Th. Vogtmann, and M. Schwoerer, “Detailed analysis of second-harmonic-generation Maker fringes in biaxially birefringent materials by a 4×4 matrix formulation,” J. Opt. Soc. Am. B15(12), 2877–2884 (1998). [CrossRef]
  17. V. Rodriguez and C. Sourisseau, “General Maker-fringe ellipsometric analyses in multilayer nonlinear and linear anisotropic optical media,” J. Opt. Soc. Am. B19(11), 2650–2664 (2002). [CrossRef]
  18. S. Lee, B. Park, S.-D. Lee, G. Park, and Y. D. Kim, “Second-harmonic generation in poled films of nonlinear optical polymer composites,” Opt. Quantum Electron.27(5), 411–420 (1995). [CrossRef]
  19. I. P. Kaminow, An Introduction to Electrooptic Devices (Academic, 1974).
  20. J. P. Drummond, S. J. Clarson, J. S. Zetts, F. K. Hopkins, and S. J. Caracci, “Enhanced electro-optic poling in guest–host systems using conductive polymer-based cladding layers,” Appl. Phys. Lett.74(3), 368–370 (1999). [CrossRef]
  21. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).
  22. P. Yeh, Optical Waves in Layered Media (Wiley, 1988)
  23. D. Guo, R. Lin, and W. Wang, “Gaussian-optics-based optical modeling and characterization of a Fabry-Perot microcavity for sensing applications,” J. Opt. Soc. Am. A22(8), 1577–1588 (2005). [CrossRef] [PubMed]
  24. K. D. Singer, M. D. Kuzyk, and J. E. Sohn, “Second-order nonlinear-optical processes in orientationally ordered materials: relationship between molecular and macroscopic properties,” J. Opt. Soc. Am. B17, 566–571 (2000).
  25. R. C. Hoffman, A. G. Mott, M. J. Ferry, T. M. Pritchett, W. Shensky, J. A. Orlicki, G. R. Martin, J. Dougherty, J. L. Leadore, A. M. Rawlett, and D. H. Park, “Poling of visible chromophores in millimeter-thick PMMA host,” Opt. Mater. Express1(1), 67–77 (2011). [CrossRef]
  26. J. Zhang, G. Wang, Z. Liu, L. Wang, G. Zhang, X. Zhang, Y. Wu, P. Fu, and Y. Wu, “Growth and optical properties of a new nonlinear Na3La9O3(BO3)8 crystal,” Opt. Express18(1), 237–243 (2010). [CrossRef] [PubMed]
  27. C. Chen, Z. Shao, J. Jiang, J. Wei, J. Lin, J. Wang, N. Ye, J. Lv, B. Wu, M. Jiang, M. Yoshimura, Y. Mori, and T. Sasaki, “Determination of the nonlinear optical coefficients of YCa4O(BO3)3 crystal,” J. Opt. Soc. Am. B17(4), 566–571 (2000). [CrossRef]
  28. S. Ramo, J. R. Whinnery, and T. V. Duzer, Fields and Waves in Communication Electronics (Wiley, New York, 1965), Chap. 6.

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