OSA's Digital Library

Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 5 — Mar. 11, 2013
  • pp: 5842–5858
« Show journal navigation

Design of ultra-broadband terahertz polymer waveguide emitters for telecom wavelengths using coupled mode theory

Felipe A. Vallejo and L. Michael Hayden  »View Author Affiliations


Optics Express, Vol. 21, Issue 5, pp. 5842-5858 (2013)
http://dx.doi.org/10.1364/OE.21.005842


View Full Text Article

Acrobat PDF (4832 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

We use coupled mode theory, adequately incorporating optical losses, to model ultra-broadband terahertz (THz) waveguide emitters (0.1-20 THz) based on difference frequency generation of femtosecond infrared (IR) optical pulses. We apply the model to a generic, symmetric, five-layer, metal/cladding/core waveguide structure using transfer matrix theory. We provide a design strategy for an efficient ultra-broadband THz emitter and apply it to polymer waveguides with a nonlinear core composed of a poled guest-host electro-optic polymer composite and pumped by a pulsed fiber laser system operating at 1567 nm. The predicted bandwidths are greater than 15 THz and we find a high conversion efficiency of 1.2 × 10−4 W−1 by balancing both the modal phase-matching and effective mode attenuation.

© 2013 OSA

1. Introduction

A widely used method to produce ultra-broadband terahertz (THz) pulses is to mix the frequency components of infrared (IR) femtosecond (fs) pulses through difference frequency generation (DFG) in organic and inorganic crystals [1

1. X. Zheng, C. V. McLaughlin, P. D. Cunningham, and L. M. Hayden, “Organic broadband terahertz sources and sensors,” J. Nanoelectron. Optoelectron. 2(1), 58–76 (2007). [CrossRef]

,2

2. K. Liu, J. Xu, and X.-C. Zhang, “GaSe crystals for broadband terahertz wave detection,” Appl. Phys. Lett. 85(6), 863–865 (2004). [CrossRef]

]. Pulsed fiber laser systems operating at telecom wavelengths are compact and have the potential to provide portable ultra-broadband THz sources outside the laboratory. However, these systems are power limited and only a few materials are well phase-matched at these wavelengths for generation in a broadband THz range (DAST, electro-optic (EO) polymers [1

1. X. Zheng, C. V. McLaughlin, P. D. Cunningham, and L. M. Hayden, “Organic broadband terahertz sources and sensors,” J. Nanoelectron. Optoelectron. 2(1), 58–76 (2007). [CrossRef]

]). Poled EO guest-host polymer films are attractive since their material properties are tunable and provide an inexpensive alternative to crystal based ultra-broadband THz emitters. Unfortunately, these EO composites are somewhat absorptive in the THz range. In addition, to achieve maximum conversion when using them as bulk emitters the IR radiation must be polarized parallel to the poling direction, normal to the film surface. This slanted geometry results in a non-optimal overlap of the input IR polarization with the emitter nonlinearity.

Guided wave geometries can overcome this restriction. Many structures have been proposed and some have been demonstrated for narrow-band nonlinear applications: THz generation through DFG [3

3. U. Peschel, K. Bubke, D. C. Hutchings, J. S. Aitchison, and J. M. Arnold, “Optical rectification in a traveling-wave geometry,” Phys. Rev. A 60(6), 4918–4926 (1999). [CrossRef]

10

10. Y. Li, X. Hu, F. Liu, J. Li, Q. Xing, M. Hu, C. Lu, and C. Wang, “Terahertz waveguide emitters in photonic crystal fiber form,” J. Opt. Soc. Am. B 29(11), 3114–3118 (2012). [CrossRef]

] and four wave mixing [11

11. Z. Wang, H. Liu, N. Huang, Q. Sun, and J. Wen, “Efficient terahertz-wave generation via four-wave mixing in silicon membrane waveguides,” Opt. Express 20(8), 8920–8928 (2012). [CrossRef] [PubMed]

], and plasmon enhanced and slow light second harmonic generation SHG [12

12. F. F. Lu, T. Li, J. Xu, Z. D. Xie, L. Li, S. N. Zhu, and Y. Y. Zhu, “Surface plasmon polariton enhanced by optical parametric amplification in nonlinear hybrid waveguide,” Opt. Express 19(4), 2858–2865 (2011). [CrossRef] [PubMed]

14

14. F. M. Pigozzo, D. Modotto, and S. Wabnitz, “Second harmonic generation by modal phase matching involving optical and plasmonic modes,” Opt. Lett. 37(12), 2244–2246 (2012). [CrossRef] [PubMed]

]. Also, phase matching strategies have been successful for THz generation in collinear and non-collinear setups with approaches relying on velocity matching [15

15. S. B. Bodrov, I. E. Ilyakov, B. V. Shishkin, and A. N. Stepanov, “Efficient terahertz generation by optical rectification in Si-LiNbO3-air-metal sandwich structure with variable air gap,” Appl. Phys. Lett. 100(20), 201114 (2012). [CrossRef]

,16

16. M. I. Bakunov, S. B. Bodrov, A. V. Maslov, and M. Hangyo, “Theory of terahertz generation in a slab of electro-optic material using an ultrashort laser pulse focused to a line,” Phys. Rev. B 76(8), 085346 (2007). [CrossRef]

], quasi-phased matching [17

17. Y.-C. Huang, T.-D. Wang, Y.-H. Lin, C.-H. Lee, M.-Y. Chuang, Y.-Y. Lin, and F.-Y. Lin, “Forward and backward THz-wave difference frequency generations from a rectangular nonlinear waveguide,” Opt. Express 19(24), 24577–24582 (2011). [CrossRef] [PubMed]

], and mode matching (modal phase matching [18

18. V. Berger and C. Sirtori, “Nonlinear phase matching in THz semiconductor waveguides,” Semicond. Sci. Technol. 19(8), 964–970 (2004). [CrossRef]

]) [3

3. U. Peschel, K. Bubke, D. C. Hutchings, J. S. Aitchison, and J. M. Arnold, “Optical rectification in a traveling-wave geometry,” Phys. Rev. A 60(6), 4918–4926 (1999). [CrossRef]

,4

4. Y. J. Ding, “Terahertz parametric converters by use of novel metallic-dielectric hybrid waveguides,” J. Opt. Soc. Am. B 23(7), 1354–1359 (2006). [CrossRef]

,19

19. H. Cao, R. A. Linke, and A. Nahata, “Broadband generation of terahertz radiation in a waveguide,” Opt. Lett. 29(15), 1751–1753 (2004). [CrossRef] [PubMed]

24

24. K. Saito, T. Tanabe, and Y. Oyama, “Elliptically polarized THz-wave generation from GaP-THz planar waveguide via collinear phase-matched difference frequency mixing,” Opt. Express 20(23), 26082–26088 (2012). [CrossRef] [PubMed]

] strategies. Unfortunately, the bandwidths demonstrated in all these works are less than 3 THz.

A symmetric, five-layer, slab waveguide, Fig. 1(a)
Fig. 1 (a) Symmetric five-layer slab waveguide. The arrows represent the dipole moment of the chromophores. (b) Second order susceptibility, χxxx(2), vs. wavelength for DAPC in the telecom range. Inset shows the chemical structure of the chromophore used in DAPC.
, similar to the one reported in [19

19. H. Cao, R. A. Linke, and A. Nahata, “Broadband generation of terahertz radiation in a waveguide,” Opt. Lett. 29(15), 1751–1753 (2004). [CrossRef] [PubMed]

], has the ideal geometry for exploiting the EO polymer nonlinearity, achieving good phase-matching conditions and managing THz attenuation. The two metal capping layers help confine the modes involved in the interaction. The IR pump is coupled to the fundamental TM mode that is bounded inside the EO core and entirely polarized parallel to the poling direction. The THz radiation is collected by the fundamental TEM-like mode that extends spatially in the entire core/cladding section of the waveguide. The low loss cladding layers help mitigate both the high THz losses introduced by the lossy core and metal capping layers.

2. Theory

2.1 Frequency domain coupled mode theory

2.2 TM to TM conversion for mm point symmetry class nonlinearities

2.3 Input beam conditions, completeness and energetic considerations

We assume a Gaussian input beam, both in time and in the x coordinate, injected at z=0 and uniformly distributed across the y direction along a strip of length Ly
Ein(r,t)=Re[x^E0exp(x2w02t22τ02iω0t)].
(9)
Here τ0 is a measure of the pulse duration, ω0 is the central frequency and w0 is the beam waist of the IR pump laser in the x direction at the input face of the waveguide. The value of E0 depends on the average power of the pump laser Ppump and the repetition rate 1/Trep of the train of pulses produced in the laser system. To extract E0 we compute the energy per pulse
Upulse=PpumpTrep=Trep/2Trep/2SEin(t)×Hin(t)dAdtπ|E0|2Lyw0τ02Z02.
(10)
Here Ly is twice the beam waist in the y direction and we assume that Trep>>τ0 and ωoτ0>>1. Under these conditions E0 is given entirely from experimental quantities, E0=(23/2Z0PpumpTrep/πw0Lyτ0)1/2and Ein(ω) is given by the Fourier transform of Eq. (9)
Ein(ω,x,z=0)=x^E0τ022πexp(x2w02((ωω0)τ0)22).
(11)
Note the extra factor of 1/2 that arises from the convention established through Eq. (4). We proceed to find a valid expansion for Ein(ω,x) in terms of the waveguide modes. Care must be taken here since when considering materials with losses the differential operator that gives the mode decomposition is non-Hermitian (non-self-adjoint) and its spectrum is complex, see Appendix A Eqs. (16) and (18). The general classification of the spectrum for non-self-adjoint operators is still an open problem and completeness is difficult to prove [31

31. Ş. E. Kocabaş, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal waveguides,” Phys. Rev. B 79(3), 035120 (2009). [CrossRef]

]. Also, the standard inner product <i,j>=ρ(x)Ei(x)Ej(x)*dx=δij, with ρ(x)=n(x)2 for TM and ρ(x)=1 for TE modes, is not necessarily valid for lossy materials. In fact, for TM modes when ρ(x) the solution space for the differential operator will lack a well-defined metric <i,i> and the solution space is not a Hilbert space. Nevertheless, for metal-insulator-metal waveguides it has been shown that its solution space forms a complete basis even when the modes are complex and result from a non-self-adjoint operator [31

31. Ş. E. Kocabaş, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal waveguides,” Phys. Rev. B 79(3), 035120 (2009). [CrossRef]

]. Fortunately, using the Lorentz reciprocity theorem combined with symmetry operations yields [8

8. Z. Ruan, G. Veronis, K. L. Vodopyanov, M. M. Fejer, and S. Fan, “Enhancement of optics-to-THz conversion efficiency by metallic slot waveguides,” Opt. Express 17(16), 13502–13515 (2009). [CrossRef] [PubMed]

,25

25. P. Bienstman, “Rigorous and efficient modeling of wavelength scale photonic components,” (Universiteit Gent 2000–2001).

]
12SdAe^i×h^j=δij=Ly2neffiZ0ρ(x)Ei(x)Ej(x)dx.
(12)
Here S is the waveguide cross section that lies on the xy plane. Equation (12) holds independently if the modes are lossy or not. This orthogonality relation provides means to normalize by using <Ei,f>=Ly/(2neffiZ0)ρ(x)Ei(x)f(x)dx as a pseudo inner product. It follows that:
AlIR(ω)=Ly2nefflZ0ρ(x)El(x)Ein(ω,x)dx.
(13)
Contrary to usual conventions |Al|2 is not the power in Watts transported by the l-th mode. The total power needs to be calculated using Eq. (14).

2.6 Energetic considerations

The ensemble formed by the train of IR pump pulses and the corresponding THz pulses generated through DFG form a stationary random process. The average THz power is
P¯THz=SE(t)×H(t)dA=1TrepTrep2Trep2dtSdAdΩdΩ(E˜(Ω)×H˜(Ω)ei(Ω+Ω)t+E˜(Ω)×H˜(Ω)*ei(ΩΩ)t+cc.)04πTrepRe[SE˜(Ω)×H˜(Ω)*dA]dΩ0dPTHz(z,Ω)dΩ.
(14)
Here, the sum runs only on the THz modes. To compute the integral in Ω' we used sin(x/ϵ)/(πx)δ(x) as ϵ0 where δ(x) is the Dirac delta function, note 2/TrepMHz and (Ω±Ω)THz. The first term in Eq. (14) cancels while computing the integral since E(Ω)=0 for Ω<0. Note that the quantity dPTHz(Ω) has units of Watt/Hz and it represents the power spectral density of the generated THz radiation. The fact that we can indeed define this “power spectrum” is not surprising due to the stochastic nature of the THz generation process. This is explicitly stated in the Wiener-Khintchine theorem: the autocorrelation function of a stationary random process and the spectral density form a Fourier transform pair, i.e. Γ(τ)=S(ω)eiωτ [32

32. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

]. In this realm, the average THz power takes the role of the same-time autocorrelation function, Γ(τ=0)=P¯THz, and S(ω)=dPTHz. For this TM to TM conversion the output THz power density generated by the structure in Fig. 1(a) is

dPTHz(z,Ω)=4πTrepLyZ0l,lRe[Al(z,Ω)Al(z,Ω)*ei(βlβl*)z(neffl)*(n(x))2El(Ω,x)El*(Ω,x)dx].
(15)

3. Results and discussion

To benchmark the model we applied the theory to the waveguide device fabricated by Cao et. al., in 2004 that served as a proof of concept for broadband THz generation in waveguides [19

19. H. Cao, R. A. Linke, and A. Nahata, “Broadband generation of terahertz radiation in a waveguide,” Opt. Lett. 29(15), 1751–1753 (2004). [CrossRef] [PubMed]

]. Our predictions for this structure, reported in detail in Appendix B, are in excellent agreement with the experimental results of the peak-to-peak ratio of the waveforms produced by emitters with different propagation lengths, the only experimental result reported in that work. We consider this to be a satisfactory benchmark since we had no knowledge of the values of extinction coefficients for the cladding and core in that experiment nor did we have access to the exact detector response function of their photoconductive antenna.

In this section, we proceed to study structures composed of DAPC, an inexpensive polymeric EO guest-host composite with a high EO coefficient at IR frequencies. The DAPC films we produce through contact poling typically have EO coefficients r3340pm/V at 830 nm [1

1. X. Zheng, C. V. McLaughlin, P. D. Cunningham, and L. M. Hayden, “Organic broadband terahertz sources and sensors,” J. Nanoelectron. Optoelectron. 2(1), 58–76 (2007). [CrossRef]

]. The corresponding susceptibility value χxxx(2)53pm/V at 1567 nm is obtained from the measured values at 830 nm assuming a two level model [29

29. K. D. Singer, M. G. 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 4(6), 968–976 (1987). [CrossRef]

], since DAPC is known to have a single resonance at 610 nm (see Fig. 1(b)). The choice of cladding materials can lead to better performance based on their optical properties at both THz and IR frequency ranges. In particular, we report here results for two polymeric cladding materials that were shown [33

33. P. D. Cunningham, N. N. Valdes, F. A. Vallejo, L. M. Hayden, B. Polishak, X.-H. Zhou, J. Luo, A. K.-Y. Jen, J. C. Williams, and R. J. Twieg, “Broadband terahertz characterization of the refractive index and absorption of some important polymeric and organic electro-optic materials,” J. Appl. Phys. 109(4), 043505 (2011). [CrossRef]

] to have low THz losses in a broadband range, (0.1-10 THz), polyethylene cyclic olefin copolymer (TOPAS®) and polystyrene (PS).

The phase-matching concept for THz generation in bulk geometries can be extended to waveguide emitters [1

1. X. Zheng, C. V. McLaughlin, P. D. Cunningham, and L. M. Hayden, “Organic broadband terahertz sources and sensors,” J. Nanoelectron. Optoelectron. 2(1), 58–76 (2007). [CrossRef]

,5

5. A. Marandi, T. E. Darcie, and P. P. M. So, “Design of a continuous-wave tunable terahertz source using waveguide-phase-matched GaAs,” Opt. Express 16(14), 10427–10433 (2008). [CrossRef] [PubMed]

,18

18. V. Berger and C. Sirtori, “Nonlinear phase matching in THz semiconductor waveguides,” Semicond. Sci. Technol. 19(8), 964–970 (2004). [CrossRef]

]. Better phase-matching (i.e., more efficient mode conversion from IR to THz) is obtained when the effective group index of the IR mode, ng,IR=Neff,0λ(dNeff,0/dλ), matches the effective index of the THz mode Neff,0. The structures with TOPAS® cladding are less efficient than the PS structures due to a higher effective index mismatch caused by the lower THz index in TOPAS®, see Fig. 4 (a)
Fig. 4 (a) Group index ng,IR for the m = 0 IR mode and the value of Neff,0 for the TEM-like mode at 5 THz for structures with different half thickness values t. For the TOPAS structure d = 1.274 μm and d = 1.579 μm for the PS structure. (b) Nonlinear efficiency ηTHz vs emitter length L for the different structures.
.

In our computations, we obtain a closed solution to Eq. (6) by decoupling the amplitudes for the THz modes by: (i) neglecting pump beam depletion of the IR modes (set AlIR(z)=AlIR(0)), (ii) assuming that the THz modes are zero valued at the input face of the waveguide AmTHz(0)=0 and (iii) performing the straightforward integral in z. We assumed a Gaussian pulse with 47-fs pulse width centered at 1567nm, 10 mW of input pump power, and repetition rate of 100 MHz. In addition, we assumed symmetric input conditions and, an input Gaussian beam focused into the core with the aid of an elliptical lens to a beam waist (in the x-axis) w0 = 1.4 μm for the TOPAS® structure and 1.8 μm for those with PS. To couple most of the input pump power to the fundamental IR mode, the beam waist of the input profile is chosen such that the expansion in Eq. (1) is dominated by the l = 0 mode. For these values of w0 about 81% of the pump power is coupled into the fundamental IR mode. To determine the power coupled to a certain IR mode we use Eq. (15) limited to the l-th IR mode using Eq. (13) to computeAlIR; integrating across the pump spectral range yields the total power coupled into the l-th mode. In our calculations we did not include coupling losses resulting from reflections at the input face of the waveguide or at the cylindrical coupling lens. Therefore, we assume 10 mW of power crossing the input interface of the waveguide emitter just after the z ≥ 0, and only around 8 mW couple into the fundamental IR mode.

For particular values of t and d, there exists an optimal emitter length, Lopt, for which the nonlinear efficiency ηTHz is a maximum. In Fig. 5(a)
Fig. 5 For a structure with d = 1.579 μm and t = 2.5 μm and PS cladding: (a) Spectral power densities dPTHz vs. THz frequencies for different emitter lengths, L. (b) Output electric field at the center of the waveguide produced by the most efficient structure we found.
we plot the spectral power density for our most optimal structure with d = 1.597 μm, t = 2.5 μm and PS cladding, for different emitter lengths L. The maximum ηTHz = 1.2 × 10−4 W−1 for this structure occurs when L = 550 μm. In Fig. 5(b) we show the pulse produced by such a structure with corresponding bandwidth >15 THz. For longer structures the higher attenuation observed for higher frequencies degrades the bandwidth of the device. This attenuation effect is less significant for the structure with t = 2.7 μm which has a lower mode attenuation but due to the greater phase-mismatch results in a slightly lower maximum efficiency than the t = 2.5 μm structure.

Unfortunately efficiency values were not reported in [19

19. H. Cao, R. A. Linke, and A. Nahata, “Broadband generation of terahertz radiation in a waveguide,” Opt. Lett. 29(15), 1751–1753 (2004). [CrossRef] [PubMed]

]. In our benchmark (see Appendix B) we predicted nonlinear conversion efficiencies ~10−6 W−1 for the structure in [19

19. H. Cao, R. A. Linke, and A. Nahata, “Broadband generation of terahertz radiation in a waveguide,” Opt. Lett. 29(15), 1751–1753 (2004). [CrossRef] [PubMed]

]. These values are 100 times smaller than the efficiency of our optimized structure. An enhancement factor 25 results just from using an EO polymer with five times higher χ(2) values, PTH ~(χ(2))2. Input conditions are also crucial since this effect is quadratic in ηTHz; a proper choice of w0 allows coupling a higher fraction of the pump power into the fundamental IR mode. Another source of enhancement is the access to power at higher THz frequencies due mainly to our shorter pump pulse duration that provides broader IR bandwidth than the one used in [19

19. H. Cao, R. A. Linke, and A. Nahata, “Broadband generation of terahertz radiation in a waveguide,” Opt. Lett. 29(15), 1751–1753 (2004). [CrossRef] [PubMed]

]. Structures with optimal emitter lengths, Lopt, can effectively collect the power at all accessible THz frequencies. The structures in [19

19. H. Cao, R. A. Linke, and A. Nahata, “Broadband generation of terahertz radiation in a waveguide,” Opt. Lett. 29(15), 1751–1753 (2004). [CrossRef] [PubMed]

] had longer propagation lengths (a few millimeters) and the mode attenuation we calculate for these structures are bigger than 50 cm−1past 3 THz. The THz power generated above 3 THz by the structures in [19

19. H. Cao, R. A. Linke, and A. Nahata, “Broadband generation of terahertz radiation in a waveguide,” Opt. Lett. 29(15), 1751–1753 (2004). [CrossRef] [PubMed]

] is considerably reduced for longer emitter lengths, similarly as in Fig. 5(a). Finally, our optimized structure also has a slightly higher overlap coefficient Eq. (7) than those in [19

19. H. Cao, R. A. Linke, and A. Nahata, “Broadband generation of terahertz radiation in a waveguide,” Opt. Lett. 29(15), 1751–1753 (2004). [CrossRef] [PubMed]

] due to higher mode confinement.

We proceed to study the effect of group velocity dispersion (GVD) in our results. A pulse coupled to the fundamental IR mode of one of our structures will be confined within the DAPC core. Due to the low curvature observed in the dispersion curve of the DAPC index (quasi-linear) in Fig. 3(a) the GVD effects should be small; in addition the emitter lengths are short (550 μm). We confirmed this by using an approximate solution to Eq. (6) where we compute analytically the integral in frequency but we ignore GVD and higher order dispersion effects. This simplified 3-wave approach produces almost identical results as those observed in Fig. 5(a) and Fig. 4(b) for such short interaction lengths.

5. Conclusions

6. Appendix A: Waveguide modes obtained through transfer matrix theory

We assume that the fields inside the structure shown in Fig. 1(a) are propagating in the z direction with a mode constant β such that E(r,t)=E(x,y)ei(βzωt) and H(r,t)=H(x,y)ei(βzωt). Since metal layers are present in the structure, the TM mode profiles will present an abrupt discontinuity at the interface between metal and cladding layers due to the high index contrast, see below Eq. (18). Mode solvers based on finite difference (FD) methods have to be carefully modified and implemented to account for this effect. Luckily for layered structures /y=0 and the mode profiles depend only on the x component, E(x,y)=E(x). In the absence of any nonlinearity an analytical expression for the modes is:

(2x2+ni2k02β2)E(i)(x)=0xi1<x<xi.
(16)

with k0 = ω/c for each i-th layer in the structure (i = 0,...,4). Note that for the symmetric five-layer waveguide n0 = n4 and n1 = n3. It must be pointed out that Eq. (16) yields the modes only when combined with appropriate boundary conditions since all of Maxwell’s equations must be simultaneously fulfilled. In the linear regime and for this type of waveguide, the evolution of any input radiation propagating longitudinally can be expanded as a superposition of two independent sets of solutions to Eq. (16), the TE {Hx,Ey,Hz} and TM {Ex,Hy,Ez} modes. Using an electric field formulation in which we solve Eq. (16) only for the principal electric component, E = Ey for TE modes or E = Ex for TM modes, the remaining components of the mode for both TM and TE are given in terms of the principal field component E [38

38. P. N. Robson and P. C. Kendall, eds. Rib Waveguide Theory by the Spectral Index Method (Electronic and Electrical Engineering Research Studies: Optoelectronics Series) (Wiley, 1990).

]. For example, TM modes are given by Hy=ni2E/(neffZ0) and Ez=iβ1(E/x) where β=k0neff and Z0=(μ0/ϵ0)1/2, and E is given by

E(x)=E(i)(x)=Aieiki(xxi)+Bieiki(xxi)            xi1<x<xi.
(17)

at the i-th layer, here ki2=ni2k02β2. To solve for the unknown coefficients {Ai,Bi} and the unknown modes constants β's we impose boundary conditions, i.e. the tangential components of both E and H must be continuous at each one of the boundaries at xi, {Ey,Hz} for TE modes and {Hy, Ez} for the TM modes. This is equivalent to

E(i)(xi)=ηi+1E(i+1)(xi)anddE(i)dx|x=xi=dE(i+1)dx|x=xi.
(18)

(0B0)=(m1,1m1,2m2,1m2,2)(A40).
(19)

2.3 The dispersion relation

Bound modes must satisfy m1,1=0, this requirement is known as the dispersion relation for both TE and TM modes. Its solutions yield the allowed mode constants {βn}’s. Bound modes can be further classified m2,1=1 for even and m2,1=1 for odd modes. The dispersion relation for the TM modes is explicitly [26

26. Y.-F. Li and J. W. Y. Lit, “General formulas for the guiding properties of a multilayer slab waveguide,” J. Opt. Soc. Am. A 4(4), 671–677 (1987). [CrossRef]

]:

0=f(β2)=n12n22α2{tanh(α2d)coth(α2d)}(1+α˜0α1tanh(α1(td)))+(α˜0+α1tanh(α1(td))).
(20)

Here αi2=β2k02ni2 and α˜i=αini+12/ni2. The upper term in the brackets applies to even modes and the lower term to odd modes.

2.4 Mode profiles

The {Ai,Bi}coefficients can be solved in terms of B0=m2,1A4. Setting x = 0 at the center of the waveguide we find for the even modes,

E=B0{sech(α1(td))eα0(|x|t)                                                                            t<|x|sech(α1(td))η1(cosh(α1(|x|t))+α˜0α1sinh(α1(|x|t)))                      d<|x|<t1η1η2[(cosh(α2(x+d))+α˜1α2tanh(α1(td))sinh(α2(x+d)))+α˜0α1(tanh(α1(td))cosh(α2(x+d))+α˜1α2sinh(α2(x+d)))]         d<x<d.
(21)

7. Appendix B

To benchmark the model, we applied the theory to the waveguide device fabricated by Cao, Linke, and Nahata in 2004 that served as a proof of concept for broadband THz generation in waveguides [19

19. H. Cao, R. A. Linke, and A. Nahata, “Broadband generation of terahertz radiation in a waveguide,” Opt. Lett. 29(15), 1751–1753 (2004). [CrossRef] [PubMed]

]. The THz output generated from this device was well documented in their paper and most of the experimental parameters necessary to apply our theory were available. The THz emitter used in this demonstration consisted of a symmetric five-layer waveguide as the one in Fig. 1(a) on top of a silicon substrate. The outer metal layers consisted of 200 nm layers of Al but we assumed that the metal capping layers were infinitely thick since the absorption depth in the metal is considerably thinner than the thickness of the metal layers. The nonlinear core was made out of a 3 μm layer (d = 1.5 μm) of the copolymer Disperse Red 1 covalently functionalized at a 10%-mol ratio to a methyl-methacrylate backbone (DR1-MMA 10%). The structure was corona poled to achieve a second order nonlinearity. We used r33 = 3.6 pm/V which was the value measured from corona poled films of DR1-MMA 16.5% measured at 810 nm [30

30. A. Nahata, J. Shan, J. T. Yardley, and C. Wu, “Electro-optic determination of the nonlinear-optical properties of a covalently functionalized Disperse Red 1 copolymer,” J. Opt. Soc. Am. B 10(9), 1553–1564 (1993). [CrossRef]

], this corresponds to r33 = 2.21 pm/V at 820 nm for DR1-MMA 10%. We obtain this value scaling by 10/16 and using a two level system model on the susceptibility with a resonance at 470 nm [30

30. A. Nahata, J. Shan, J. T. Yardley, and C. Wu, “Electro-optic determination of the nonlinear-optical properties of a covalently functionalized Disperse Red 1 copolymer,” J. Opt. Soc. Am. B 10(9), 1553–1564 (1993). [CrossRef]

]. With the 3.5 μm thick cladding layers of UV-curable acrylate (Norland optical adhesive NOA 81) the cladding/core/cladding section was 10 μm thick, (t = 5 μm). The waveguide was pumped with a mode-locked Ti:Sapphire laser oscillator with a pulse width of τFWHM= 100 fsoperating at 820 nm, ω0= 365.6 THz. The other pump beam parameters used in the model were Ppump = 3 mW and Ly = 6 mm which correspond to the reported values. We assumed that the input pump beam was focused to a beam waist w0 = 1.5 μm in the x axis at the input interface with a cylindrical lens used to couple the pump beam into the waveguide. We also assumed that the repetition rate of the laser system was 100 MHz.

For the Gaussian input beam in Eq. (9) with w0 = 1.5 μm that we used in our computations, we need about 5 modes in the mode expansion to completely describe the input mode profile. However, in our implementation only two IR modes were necessary to solve for Eq. (7) since the expansion with two terms provides a reasonable approximation, see Fig. 6(a)
Fig. 6 (a) Gaussian input profile (w0 = 1.5 μm) for the IR pump beam at 820 nm and corresponding mode decompositions using 1 and 2 modes and (b) mode profile for the fundamental TEM-like mode at 10 THz for a structure with (t = 5 μm, d = 1.5 μm).
. About 70% of the input IR pump couples to the fundamental IR mode and 7% couples to the first order mode. The waveguide structure has a single guided THz mode, this fundamental (m = 0) TM THz mode is a TEM like mode since the mode profile for the THz frequencies is uniform in the core, see Fig. 6(b).

The THz output produced by our model for Cao’s structure for the 3 waveguide devices they fabricated, with propagation length L = 1, 2 and 3 mm are shown in Fig. 7
Fig. 7 Output THz electric field for a device with (d = 1.5 μm, t = 5 μm) for different propagation lengths L. (a) THz waveforms after being convolved with the detector response function R(Ω). (b) The corresponding THz spectral fields before and after being convolved with R(Ω), shown for comparison.
. The time-domain waveforms are obtained by computing the Fourier transform in Eq. (3) after being convolved with a detector response function for a mode expansion with one THz mode and two IR modes. The fields are evaluated at the center of the waveguide (x = 0). To reproduce the experimental conditions we had to account for the response of the detector used in their experimental setup. In Cao’s setup a photo-conductive dipole antenna limited the detected signal to approximately 2 THz, yet our model predicts a bandwidth up to 6 THz. Since we did not have access to the response function of this antenna we assumed a response function R(Ω)=1/2Erfc(r(ΩΩ0))where, Erfc(x) is the complementary error function, Ω0/2π = 1.5 THz and r = 1/(1 THz) is a constant that provides a soft and continuous decay from 1 to zero. With this response function the bandwidth of the convolved THz spectra is limited to approximately 2 THz. In Fig. 7(b) we show the response function R(Ω), along with the complete generated and corresponding convolved bandwidths for the 3 devices. The waveforms shown in Fig. 7(a) correspond to the convolved bandwidths and since the response function is not precisely the one used for the detector the exact structure of the pulse does not correspond to the experimental results. However, the ratio of peak-to-peak field amplitudes obtained by our model for the 1 mm: 2 mm: 3 mm waveguides in the Cao devices, are 1:1.57:1.75 which are in very good agreement with the experimental results 1:1.56:1.77. The nonlinear conversion efficiency we predict for these devices is on the order of ~ 10−6 W−1.

Acknowledgments

References and links

1.

X. Zheng, C. V. McLaughlin, P. D. Cunningham, and L. M. Hayden, “Organic broadband terahertz sources and sensors,” J. Nanoelectron. Optoelectron. 2(1), 58–76 (2007). [CrossRef]

2.

K. Liu, J. Xu, and X.-C. Zhang, “GaSe crystals for broadband terahertz wave detection,” Appl. Phys. Lett. 85(6), 863–865 (2004). [CrossRef]

3.

U. Peschel, K. Bubke, D. C. Hutchings, J. S. Aitchison, and J. M. Arnold, “Optical rectification in a traveling-wave geometry,” Phys. Rev. A 60(6), 4918–4926 (1999). [CrossRef]

4.

Y. J. Ding, “Terahertz parametric converters by use of novel metallic-dielectric hybrid waveguides,” J. Opt. Soc. Am. B 23(7), 1354–1359 (2006). [CrossRef]

5.

A. Marandi, T. E. Darcie, and P. P. M. So, “Design of a continuous-wave tunable terahertz source using waveguide-phase-matched GaAs,” Opt. Express 16(14), 10427–10433 (2008). [CrossRef] [PubMed]

6.

M. Cherchi, A. Taormina, A. C. Busacca, R. L. Oliveri, S. Bivona, A. C. Cino, S. Stivala, S. R. Sanseverino, and C. Leone, “Exploiting the optical quadratic nonlinearity of zinc-blende semiconductors for guided-wave terahertz generation: a material comparison,” IEEE J. Quantum Electron. 46(3), 368–376 (2010). [CrossRef]

7.

C. Staus, T. Kuech, and L. McCaughan, “Continuously phase-matched terahertz difference frequency generation in an embedded-waveguide structure supporting only fundamental modes,” Opt. Express 16(17), 13296–13303 (2008). [CrossRef] [PubMed]

8.

Z. Ruan, G. Veronis, K. L. Vodopyanov, M. M. Fejer, and S. Fan, “Enhancement of optics-to-THz conversion efficiency by metallic slot waveguides,” Opt. Express 17(16), 13502–13515 (2009). [CrossRef] [PubMed]

9.

T. Chen, J. Sun, L. Li, and J. Tang, “Proposal for efficient terahertz-wave difference frequency generation in an AlGaAs photonic crystal waveguide,” J. Lightwave Technol. 30(13), 2156–2162 (2012). [CrossRef]

10.

Y. Li, X. Hu, F. Liu, J. Li, Q. Xing, M. Hu, C. Lu, and C. Wang, “Terahertz waveguide emitters in photonic crystal fiber form,” J. Opt. Soc. Am. B 29(11), 3114–3118 (2012). [CrossRef]

11.

Z. Wang, H. Liu, N. Huang, Q. Sun, and J. Wen, “Efficient terahertz-wave generation via four-wave mixing in silicon membrane waveguides,” Opt. Express 20(8), 8920–8928 (2012). [CrossRef] [PubMed]

12.

F. F. Lu, T. Li, J. Xu, Z. D. Xie, L. Li, S. N. Zhu, and Y. Y. Zhu, “Surface plasmon polariton enhanced by optical parametric amplification in nonlinear hybrid waveguide,” Opt. Express 19(4), 2858–2865 (2011). [CrossRef] [PubMed]

13.

S. B. Hasan, C. Rockstuhl, T. Pertsch, and F. Lederer, “Second-order nonlinear frequency conversion processes in plasmonic slot waveguides,” J. Opt. Soc. Am. B 29(7), 1606–1611 (2012). [CrossRef]

14.

F. M. Pigozzo, D. Modotto, and S. Wabnitz, “Second harmonic generation by modal phase matching involving optical and plasmonic modes,” Opt. Lett. 37(12), 2244–2246 (2012). [CrossRef] [PubMed]

15.

S. B. Bodrov, I. E. Ilyakov, B. V. Shishkin, and A. N. Stepanov, “Efficient terahertz generation by optical rectification in Si-LiNbO3-air-metal sandwich structure with variable air gap,” Appl. Phys. Lett. 100(20), 201114 (2012). [CrossRef]

16.

M. I. Bakunov, S. B. Bodrov, A. V. Maslov, and M. Hangyo, “Theory of terahertz generation in a slab of electro-optic material using an ultrashort laser pulse focused to a line,” Phys. Rev. B 76(8), 085346 (2007). [CrossRef]

17.

Y.-C. Huang, T.-D. Wang, Y.-H. Lin, C.-H. Lee, M.-Y. Chuang, Y.-Y. Lin, and F.-Y. Lin, “Forward and backward THz-wave difference frequency generations from a rectangular nonlinear waveguide,” Opt. Express 19(24), 24577–24582 (2011). [CrossRef] [PubMed]

18.

V. Berger and C. Sirtori, “Nonlinear phase matching in THz semiconductor waveguides,” Semicond. Sci. Technol. 19(8), 964–970 (2004). [CrossRef]

19.

H. Cao, R. A. Linke, and A. Nahata, “Broadband generation of terahertz radiation in a waveguide,” Opt. Lett. 29(15), 1751–1753 (2004). [CrossRef] [PubMed]

20.

G. Chang, C. J. Divin, J. Yang, M. A. Musheinish, S. L. Williamson, A. Galvanauskas, and T. B. Norris, “GaP waveguide emitters for high power broadband THz generation pumped by Yb-doped fiber lasers,” Opt. Express 15(25), 16308–16315 (2007). [CrossRef] [PubMed]

21.

F. Peter, S. Winnerl, H. Schneider, and M. Helm, “Excitation wavelength dependence of phase matched terahertz emission from a GaAs slab,” Opt. Express 18(19), 19574–19580 (2010). [CrossRef] [PubMed]

22.

K. Saito, T. Tanabe, Y. Oyama, K. Suto, and J.-i. Nishizawa, “Terahertz-wave generation by GaP rib waveguides via collinear phase-matched difference-frequency mixing of near-infrared lasers,” J. Appl. Phys. 105(6), 063102 (2009). [CrossRef]

23.

V. A. Kukushkin, “Generation of terahertz pulses from tightly focused single near-infrared pulses in double-plasmon waveguides,” J. Opt. Soc. Am. B 25(5), 818–824 (2008). [CrossRef]

24.

K. Saito, T. Tanabe, and Y. Oyama, “Elliptically polarized THz-wave generation from GaP-THz planar waveguide via collinear phase-matched difference frequency mixing,” Opt. Express 20(23), 26082–26088 (2012). [CrossRef] [PubMed]

25.

P. Bienstman, “Rigorous and efficient modeling of wavelength scale photonic components,” (Universiteit Gent 2000–2001).

26.

Y.-F. Li and J. W. Y. Lit, “General formulas for the guiding properties of a multilayer slab waveguide,” J. Opt. Soc. Am. A 4(4), 671–677 (1987). [CrossRef]

27.

X. Ying and I. Katz, “A simple reliable solver for all the roots of a nonlinear function in a given domain,” Computing 41(4), 317–333 (1989). [CrossRef]

28.

R. E. Smith, G. W. Forbes, and S. N. Houde-Walter, “Unfolding the multivalued planar waveguide dispersion relation,” IEEE J. Quantum Electron. 29(4), 1031–1034 (1993). [CrossRef]

29.

K. D. Singer, M. G. 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 4(6), 968–976 (1987). [CrossRef]

30.

A. Nahata, J. Shan, J. T. Yardley, and C. Wu, “Electro-optic determination of the nonlinear-optical properties of a covalently functionalized Disperse Red 1 copolymer,” J. Opt. Soc. Am. B 10(9), 1553–1564 (1993). [CrossRef]

31.

Ş. E. Kocabaş, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal waveguides,” Phys. Rev. B 79(3), 035120 (2009). [CrossRef]

32.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

33.

P. D. Cunningham, N. N. Valdes, F. A. Vallejo, L. M. Hayden, B. Polishak, X.-H. Zhou, J. Luo, A. K.-Y. Jen, J. C. Williams, and R. J. Twieg, “Broadband terahertz characterization of the refractive index and absorption of some important polymeric and organic electro-optic materials,” J. Appl. Phys. 109(4), 043505 (2011). [CrossRef]

34.

E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, 1998), Vol. I - III.

35.

M. A. Ordal, R. J. Bell, R. W. Alexander Jr, L. A. Newquist, and M. R. Querry, “Optical properties of Al, Fe, Ti, Ta, W, and Mo at submillimeter wavelengths,” Appl. Opt. 27(6), 1203–1209 (1988). [CrossRef] [PubMed]

36.

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater. 29(11), 1481–1490 (2007). [CrossRef]

37.

M. Dellnitz, O. Schütze, and Q. Zheng, “Locating all the zeros of an analytic function in one complex variable,” J. Comput. Appl. Math. 138(2), 325–333 (2002). [CrossRef]

38.

P. N. Robson and P. C. Kendall, eds. Rib Waveguide Theory by the Spectral Index Method (Electronic and Electrical Engineering Research Studies: Optoelectronics Series) (Wiley, 1990).

OCIS Codes
(190.4390) Nonlinear optics : Nonlinear optics, integrated optics
(320.7110) Ultrafast optics : Ultrafast nonlinear optics
(300.6495) Spectroscopy : Spectroscopy, teraherz

ToC Category:
Ultrafast Optics

History
Original Manuscript: December 12, 2012
Revised Manuscript: February 20, 2013
Manuscript Accepted: February 22, 2013
Published: March 1, 2013

Citation
Felipe A. Vallejo and L. Michael Hayden, "Design of ultra-broadband terahertz polymer waveguide emitters for telecom wavelengths using coupled mode theory," Opt. Express 21, 5842-5858 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-5-5842


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. X. Zheng, C. V. McLaughlin, P. D. Cunningham, and L. M. Hayden, “Organic broadband terahertz sources and sensors,” J. Nanoelectron. Optoelectron.2(1), 58–76 (2007). [CrossRef]
  2. K. Liu, J. Xu, and X.-C. Zhang, “GaSe crystals for broadband terahertz wave detection,” Appl. Phys. Lett.85(6), 863–865 (2004). [CrossRef]
  3. U. Peschel, K. Bubke, D. C. Hutchings, J. S. Aitchison, and J. M. Arnold, “Optical rectification in a traveling-wave geometry,” Phys. Rev. A60(6), 4918–4926 (1999). [CrossRef]
  4. Y. J. Ding, “Terahertz parametric converters by use of novel metallic-dielectric hybrid waveguides,” J. Opt. Soc. Am. B23(7), 1354–1359 (2006). [CrossRef]
  5. A. Marandi, T. E. Darcie, and P. P. M. So, “Design of a continuous-wave tunable terahertz source using waveguide-phase-matched GaAs,” Opt. Express16(14), 10427–10433 (2008). [CrossRef] [PubMed]
  6. M. Cherchi, A. Taormina, A. C. Busacca, R. L. Oliveri, S. Bivona, A. C. Cino, S. Stivala, S. R. Sanseverino, and C. Leone, “Exploiting the optical quadratic nonlinearity of zinc-blende semiconductors for guided-wave terahertz generation: a material comparison,” IEEE J. Quantum Electron.46(3), 368–376 (2010). [CrossRef]
  7. C. Staus, T. Kuech, and L. McCaughan, “Continuously phase-matched terahertz difference frequency generation in an embedded-waveguide structure supporting only fundamental modes,” Opt. Express16(17), 13296–13303 (2008). [CrossRef] [PubMed]
  8. Z. Ruan, G. Veronis, K. L. Vodopyanov, M. M. Fejer, and S. Fan, “Enhancement of optics-to-THz conversion efficiency by metallic slot waveguides,” Opt. Express17(16), 13502–13515 (2009). [CrossRef] [PubMed]
  9. T. Chen, J. Sun, L. Li, and J. Tang, “Proposal for efficient terahertz-wave difference frequency generation in an AlGaAs photonic crystal waveguide,” J. Lightwave Technol.30(13), 2156–2162 (2012). [CrossRef]
  10. Y. Li, X. Hu, F. Liu, J. Li, Q. Xing, M. Hu, C. Lu, and C. Wang, “Terahertz waveguide emitters in photonic crystal fiber form,” J. Opt. Soc. Am. B29(11), 3114–3118 (2012). [CrossRef]
  11. Z. Wang, H. Liu, N. Huang, Q. Sun, and J. Wen, “Efficient terahertz-wave generation via four-wave mixing in silicon membrane waveguides,” Opt. Express20(8), 8920–8928 (2012). [CrossRef] [PubMed]
  12. F. F. Lu, T. Li, J. Xu, Z. D. Xie, L. Li, S. N. Zhu, and Y. Y. Zhu, “Surface plasmon polariton enhanced by optical parametric amplification in nonlinear hybrid waveguide,” Opt. Express19(4), 2858–2865 (2011). [CrossRef] [PubMed]
  13. S. B. Hasan, C. Rockstuhl, T. Pertsch, and F. Lederer, “Second-order nonlinear frequency conversion processes in plasmonic slot waveguides,” J. Opt. Soc. Am. B29(7), 1606–1611 (2012). [CrossRef]
  14. F. M. Pigozzo, D. Modotto, and S. Wabnitz, “Second harmonic generation by modal phase matching involving optical and plasmonic modes,” Opt. Lett.37(12), 2244–2246 (2012). [CrossRef] [PubMed]
  15. S. B. Bodrov, I. E. Ilyakov, B. V. Shishkin, and A. N. Stepanov, “Efficient terahertz generation by optical rectification in Si-LiNbO3-air-metal sandwich structure with variable air gap,” Appl. Phys. Lett.100(20), 201114 (2012). [CrossRef]
  16. M. I. Bakunov, S. B. Bodrov, A. V. Maslov, and M. Hangyo, “Theory of terahertz generation in a slab of electro-optic material using an ultrashort laser pulse focused to a line,” Phys. Rev. B76(8), 085346 (2007). [CrossRef]
  17. Y.-C. Huang, T.-D. Wang, Y.-H. Lin, C.-H. Lee, M.-Y. Chuang, Y.-Y. Lin, and F.-Y. Lin, “Forward and backward THz-wave difference frequency generations from a rectangular nonlinear waveguide,” Opt. Express19(24), 24577–24582 (2011). [CrossRef] [PubMed]
  18. V. Berger and C. Sirtori, “Nonlinear phase matching in THz semiconductor waveguides,” Semicond. Sci. Technol.19(8), 964–970 (2004). [CrossRef]
  19. H. Cao, R. A. Linke, and A. Nahata, “Broadband generation of terahertz radiation in a waveguide,” Opt. Lett.29(15), 1751–1753 (2004). [CrossRef] [PubMed]
  20. G. Chang, C. J. Divin, J. Yang, M. A. Musheinish, S. L. Williamson, A. Galvanauskas, and T. B. Norris, “GaP waveguide emitters for high power broadband THz generation pumped by Yb-doped fiber lasers,” Opt. Express15(25), 16308–16315 (2007). [CrossRef] [PubMed]
  21. F. Peter, S. Winnerl, H. Schneider, and M. Helm, “Excitation wavelength dependence of phase matched terahertz emission from a GaAs slab,” Opt. Express18(19), 19574–19580 (2010). [CrossRef] [PubMed]
  22. K. Saito, T. Tanabe, Y. Oyama, K. Suto, and J.-i. Nishizawa, “Terahertz-wave generation by GaP rib waveguides via collinear phase-matched difference-frequency mixing of near-infrared lasers,” J. Appl. Phys.105(6), 063102 (2009). [CrossRef]
  23. V. A. Kukushkin, “Generation of terahertz pulses from tightly focused single near-infrared pulses in double-plasmon waveguides,” J. Opt. Soc. Am. B25(5), 818–824 (2008). [CrossRef]
  24. K. Saito, T. Tanabe, and Y. Oyama, “Elliptically polarized THz-wave generation from GaP-THz planar waveguide via collinear phase-matched difference frequency mixing,” Opt. Express20(23), 26082–26088 (2012). [CrossRef] [PubMed]
  25. P. Bienstman, “Rigorous and efficient modeling of wavelength scale photonic components,” (Universiteit Gent 2000–2001).
  26. Y.-F. Li and J. W. Y. Lit, “General formulas for the guiding properties of a multilayer slab waveguide,” J. Opt. Soc. Am. A4(4), 671–677 (1987). [CrossRef]
  27. X. Ying and I. Katz, “A simple reliable solver for all the roots of a nonlinear function in a given domain,” Computing41(4), 317–333 (1989). [CrossRef]
  28. R. E. Smith, G. W. Forbes, and S. N. Houde-Walter, “Unfolding the multivalued planar waveguide dispersion relation,” IEEE J. Quantum Electron.29(4), 1031–1034 (1993). [CrossRef]
  29. K. D. Singer, M. G. Kuzyk, and J. E. Sohn, “Second-order nonlinear-optical processes in orientationally ordered materials: relationship between molecular and macroscopic properties,” J. Opt. Soc. Am. B4(6), 968–976 (1987). [CrossRef]
  30. A. Nahata, J. Shan, J. T. Yardley, and C. Wu, “Electro-optic determination of the nonlinear-optical properties of a covalently functionalized Disperse Red 1 copolymer,” J. Opt. Soc. Am. B10(9), 1553–1564 (1993). [CrossRef]
  31. Ş. E. Kocabaş, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal waveguides,” Phys. Rev. B79(3), 035120 (2009). [CrossRef]
  32. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  33. P. D. Cunningham, N. N. Valdes, F. A. Vallejo, L. M. Hayden, B. Polishak, X.-H. Zhou, J. Luo, A. K.-Y. Jen, J. C. Williams, and R. J. Twieg, “Broadband terahertz characterization of the refractive index and absorption of some important polymeric and organic electro-optic materials,” J. Appl. Phys.109(4), 043505 (2011). [CrossRef]
  34. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, 1998), Vol. I - III.
  35. M. A. Ordal, R. J. Bell, R. W. Alexander, L. A. Newquist, and M. R. Querry, “Optical properties of Al, Fe, Ti, Ta, W, and Mo at submillimeter wavelengths,” Appl. Opt.27(6), 1203–1209 (1988). [CrossRef] [PubMed]
  36. S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater.29(11), 1481–1490 (2007). [CrossRef]
  37. M. Dellnitz, O. Schütze, and Q. Zheng, “Locating all the zeros of an analytic function in one complex variable,” J. Comput. Appl. Math.138(2), 325–333 (2002). [CrossRef]
  38. P. N. Robson and P. C. Kendall, eds. Rib Waveguide Theory by the Spectral Index Method (Electronic and Electrical Engineering Research Studies: Optoelectronics Series) (Wiley, 1990).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited