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

Optics Express, Vol. 21, Issue 5, pp. 5842-5858 (2013)

http://dx.doi.org/10.1364/OE.21.005842

Acrobat PDF (4832 KB)

### 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

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

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]

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. 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]

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-LiNbO_{3}-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]

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]

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. 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]

*ultra-broadband THz DFG*with a frequency domain (FD) coupled mode theory (CMT), similar to [23

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]

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]

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]

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]

*an arbitrary number of IR modes and THz modes*. In previous works, references [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]

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]

*fs*pulse was described by

*single IR-to-single THz-mode interactions*. Furthermore, the time-domain formulations [3

**60**(6), 4918–4926 (1999). [CrossRef]

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]

*mm*point symmetry class. We apply the transfer matrix theory and mode matching methods to solve for the complex propagation constants [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]

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]

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]

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]

*fs*, 100 MHz repetition rate and with a bandwidth extending from (1504 to 1630) nm. Our results show wide bandwidths greater than 15 THz and a maximum conversion efficiency of

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]

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]

**29**(15), 1751–1753 (2004). [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]

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]

## 2. Theory

### 2.1 Frequency domain coupled mode theory

*Lorentz reciprocity theorem*together with a mirror symmetry operation it can be proved that the amplitude functions

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]

*S*is the cross-sectional area of the nonlinear section in the waveguide. Different from other coupled mode theories it is worth noting that this version

*holds even in the case of lossy materials*where both the mode constants

*1*andIf there are N modes involved in the interaction, N coupled first order partial differential equations can be derived describing the interaction. From Eqs. (2) and (5), we find that the THz mode amplitudes are given by,where the phase-mismatch is given by

^{(2)}and to the overlap integral among the modes involved in the interaction. Note that Eq. (6) accounts for all the contributions of all the possible frequency pairs

*even when multiple IR modes are involved*, which is a noted difference to our approach compared to previous descriptions in the literature.

### 2.2 TM to TM conversion for ∞ m m point symmetry class nonlinearities

### 2.3 Input beam conditions, completeness and energetic considerations

*x*coordinate, injected at

*uniformly distributed*across the

*y*direction along a strip of length

*L*Here

_{y}*x*direction at the input face of the waveguide. The value of

*E*

_{0}depends on the average power of the pump laser

*P*and the repetition rate

_{pump}*1/T*of the train of pulses produced in the laser system. To extract

_{rep}*E*we compute the energy per pulseHere

_{0}*L*is twice the beam waist in the

_{y}*y*direction and we assume that

*E*is given entirely from experimental quantities,

_{0}*1/2*that arises from the convention established through Eq. (4). We proceed to find a valid expansion for

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]

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]

**17**(16), 13502–13515 (2009). [CrossRef] [PubMed]

*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

*pseudo inner product*. It follows that:Contrary to usual conventions

*|A*

_{l}|^{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

*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.

*same-time*autocorrelation function,

## 3. Results and discussion

**29**(15), 1751–1753 (2004). [CrossRef] [PubMed]

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]

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]

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]

^{®}) and polystyrene (PS).

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]

^{®}and PS were measured with a broadband THz spectrometer in the (0.1-10) THz range [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]

^{®}showed flat indices, varying only 0.005 for TOPAS

^{®}and 0.01 for PS in the (0.1-10) THz range. Over this range TOPAS

^{®}loss is less than 3 cm

^{−1}and PS is less than 15 cm

^{−1}. We fit these data using a permittivity model with constant index and extinction coefficient. DAPC on the other hand showed several absorptions features in the THz range and was fit to a five-oscillator-Lorentz model. In Figs. 3(c) and 3(d) we show the fits used in these models for the THz. For the IR wavelengths, we measured indices for TOPAS

^{®}and DAPC using a prism coupler device Metricon model 2010/M at 1064 nm, 1311 nm and 1537 nm, and the index values over the (430-1052) nm range for PS were taken from [36

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]

^{®}and PS, see Fig. 3(a). The fit parameters

^{®}and PS and were neglected.

*modal dispersion*,

*β*, for even modes at both the THz frequencies (

_{m}(ω) = k_{0}N_{eff,m}(ω) + iα_{m}(ω)/2*Ω*) and the IR wavelengths (

*λ*) for DAPC structures with either PS or TOPAS

^{®}as the cladding material. Since we assumed symmetric input conditions, only even modes are involved in the DFG interaction. Note that for a waveguide of sub-wavelength dimensions in which the wavelength of the guided radiation is greater than the waveguide size, such as the one studied here at the

*THz frequency range*(

*Ω*), there is a

*single symmetric TM guided mode*accepted by the waveguide. Figures 3(a) and 3(c) show the modal effective index

*N*and Figs. 3(b) and 3(d) show the mode attenuation

_{eff,m}*α*. These results are obtained by finding all the solutions (

_{m}*β*’s) to the dispersion equation (Eq. (20) in Appendix A) repeatedly for 250 points at both the THz frequency range

_{m}*β*is complex. To avoid missing any solutions and facilitating the search in the complex plane we implemented an algorithm designed to account for the ill-defined dispersion equation whose entire set of solutions lies on a two-sheeted Riemannian surface. The details of this procedure are out of the scope of this work and are explained elsewhere [26

_{m}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]

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]

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]

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]

*d*. In Figs. 3(a) and 3(b) we show the modal dispersion at IR wavelengths for a DAPC structure with

*d = 1.579 μm*,

*t = 20 μm*and PS cladding. The fundamental mode (

*m =*0) is a

*core mode*since it is well confined in the core and its effective index lies between the core and cladding indices. The (

*m =*1) mode is plasmonic in nature and will not be excited in focused scenarios since its mode profile is highly confined to the cladding layers at the interface with the metal capping layers. Note in Fig. 3(b) that the attenuation for this plasmon mode is an order of magnitude larger than for the other modes. Higher order modes (

*m*= 2, 3, 4, ...) are

*cladding modes*since their effective indices are lower than the cladding index across the pump beam bandwidth. For thicker cores the (

*m*= 2) mode will become a core mode and this establishes an upper limit for the core half thickness,

*d = 1.274 μm*for structures with TOPAS

^{®}cladding and

*d = 1.579 μm*with PS cladding.

*t*is also limited to half of the minimum THz wavelength,

*t < 15 μm*. For THz frequencies the fundamental (

*m*= 0) TM THz mode is a TEM-like mode since the mode profile for the THz frequencies is uniform in the core and cladding sections and the effective index dispersion resembles the dispersion of the core. In Figs. 3(c) and 3(d), we show the calculated modal dispersion, for the fundamental (m = 0) TEM-like mode in the THz range for the most efficient structures we found for

*t*

*d,*15 μm] with both TOPAS

^{®}and PS claddings.

**2**(1), 58–76 (2007). [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]

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

*N*. The structures with TOPAS

_{eff,0}^{®}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).

*t*has a very significant effect on the effective index of the TEM-like THz mode see Fig. 3(c). Whereas for the IR range, the modal dispersion is relatively insensitive to variations in

*t*as long as the cladding thickness is comparable to the pump’s center wavelength,

*t – d ≥ λ*. This provides a useful tool to design a THz waveguide emitter achieving better phase matching by changing

_{0}*t*. For thinner claddings,

*N*for the TEM-like mode increases approaching the index of the core since most of the radiation will be guided in the higher index core and that is desired for phase-matching purposes, see Fig. 4(a). Also, thinner claddings result in better mode overlap, which increases the coupling Eqs. (6) and (7). Unfortunately, the mode attenuation increases too since the core is also more absorptive, Fig. 3(d). Surprisingly, since the conversion efficiency depends on both the effective index matching and on the mode attenuation of the THz modes, the most efficient structure is not necessarily the one that is better phase-matched at 1567 nm. THz losses are extremely important and in an optimized structure a perfect balance between both modal phase-matching and mode losses effects must be found. This is seen in Fig. 4(b) where the nonlinear efficiency,

_{eff,m}*t = 2.3 μm*has a lower index mismatch (Fig. 4(a)), structures with thicker PS claddings (

*t = 2.5 μm*and

*2.7 μm*) and with lower THz mode attenuation Fig. 3(d)) have higher conversion efficiencies (Fig. 4(b)).

*(i)*neglecting pump beam depletion of the IR modes (set

*(ii)*assuming that the THz modes are zero valued at the input face of the waveguide

*(iii)*performing the straightforward integral in

*x-*axis)

*w*for the TOPAS

_{0}= 1.4 μm^{®}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

*w*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

_{0}*l-th*IR mode using Eq. (13) to compute

*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.

*t*and

*d*, there exists an optimal emitter length,

*L*, for which the nonlinear efficiency

_{opt}*η*is a maximum. In Fig. 5(a) we plot the spectral power density for our most optimal structure with

_{THz}*d = 1.597 μm*,

*t = 2.5 μm*and PS cladding, for different emitter lengths

*L*. The maximum

*η*

_{THz}= 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.

**29**(15), 1751–1753 (2004). [CrossRef] [PubMed]

^{−6}W

^{−1}for the structure in [19

**29**(15), 1751–1753 (2004). [CrossRef] [PubMed]

^{(2)}values, P

_{TH}~(χ

^{(2)})

^{2}. Input conditions are also crucial since this effect is quadratic in

*η*; a proper choice of

_{THz}*w*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

_{0}**29**(15), 1751–1753 (2004). [CrossRef] [PubMed]

*L*, can effectively collect the power at all accessible THz frequencies. The structures in [19

_{opt}**29**(15), 1751–1753 (2004). [CrossRef] [PubMed]

^{−1}past 3 THz. The THz power generated above 3 THz by the structures in [19

**29**(15), 1751–1753 (2004). [CrossRef] [PubMed]

**29**(15), 1751–1753 (2004). [CrossRef] [PubMed]

*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.

^{(2)}values are slightly lower for longer wavelengths (see Fig. 1(b)), the phase-mismatch for these materials is greater for shorter wavelengths (Fig. 4(a)). In fact, above 1.7 μm there exists a wavelength for which each PS structure is completely phase-matched. However, since current commercial fiber laser systems have central wavelengths near 1567 nm, the structures we propose here are more suited for stand alone broadband waveguide THz emitters. An optimal set of materials is one for which the crossing between

*N*shifts to lower wavelengths (or closer to the resonance) and occurs for thicker claddings (which lowers the modal losses).

_{eff,0}## 5. Conclusions

*fs*IR pump laser. This frequency domain CMT model is suited to accurately incorporate both THz and IR losses. For the

*∞mm*point class type of second order nonlinearity, we applied the transfer matrix theory and mode-matching methods to model the symmetric five-layered metal/cladding/nonlinear-core waveguide. We applied a design strategy based in single IR mode to single THz mode conversion, to design an efficient five-layered waveguide structure made out of a DAPC core and PS cladding layers. The most efficient THz emitter we found,

*η*

_{THz}= 1.2 × 10^{−4}W^{−1}, yields an efficiency comparable to state of the art CW THz waveguide emitters but the wide bandwidth ~15 THz constitutes a record high for broadband THz waveguide emitters. We found that for an optimized structure there must be a perfect balance between both modal phase-matching and mode losses effects. This can be achieved by careful selection of the cladding, cladding thickness and emitter length. The theoretical method presented here is entirely general and can be applied to any material system provided the appropriate material refractive index and absorption dispersion information is known.

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

*z*direction with a mode constant

*β*such that

*k*for each

_{0}= ω/c*i*-th layer in the structure (

*i*=

*0,...,4).*Note that for the symmetric five-layer waveguide

*n*and

_{0}= n_{4}*n*

_{1}

*= n*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

_{3}.*E = E*for TE modes or

_{y}*E = E*for TM modes, the remaining components of the mode for both TM and TE are given in terms of the principal field component

_{x}*E*[38]. For example, TM modes are given by

*E*is given by

*i*-th layer, here

*x*

_{i},

### 2.3 The dispersion relation

*dispersion relation*for both TE and TM modes. Its solutions yield the allowed mode constants

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]

*even*modes and the lower term to

*odd*modes.

### 2.4 Mode profiles

*x = 0*at the center of the waveguide we find for the even modes,

## 7. Appendix B

**29**(15), 1751–1753 (2004). [CrossRef] [PubMed]

*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

*r*3.6 pm/V which was the value measured from corona poled films of DR1-MMA 16.5% measured at 810 nm [30

_{33}=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]

*r*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

_{33}=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]

*t =*5 μm). The waveguide was pumped with a mode-locked Ti:Sapphire laser oscillator with a pulse width of

*P*= 3 mW and

_{pump}*L*= 6 mm which correspond to the reported values. We assumed that the input pump beam was focused to a beam waist

_{y}*w*1.5 μm in the

_{0}=*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.

**29**(15), 1751–1753 (2004). [CrossRef] [PubMed]

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]

*α =2κω/2*with

*κ =*0.034 constant, which is consistent with our assumption that these materials exhibit constant indices in the entire (0.1-10) THz. For the IR range we used the function

*λ*is given in μm. This function fits the experimental values from 500 nm to 1.2 μm for DR1-MMA 10% and for NOA81 we used 1.557 for the index since it was found that this material is dispersionless at 820 nm [19

**29**(15), 1751–1753 (2004). [CrossRef] [PubMed]

*w*= 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). 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 (

_{0}*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).

*L*= 1, 2 and 3 mm are shown in Fig. 7. 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

*Ω*= 1.5 THz and

_{0}/2π*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

^{®}code developed to produce the results for this paper and to C. R. Menyuk and A. Docherty for interesting discussions on numerical mode solvers and modal analysis that helped to determine the approach taken in this work and S. Wall for measuring the IR indices of DAPC.

## 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. | K. Liu, J. Xu, and X.-C. Zhang, “GaSe crystals for broadband terahertz wave detection,” Appl. Phys. Lett. |

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 |

4. | Y. J. Ding, “Terahertz parametric converters by use of novel metallic-dielectric hybrid waveguides,” J. Opt. Soc. Am. B |

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 |

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. |

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 |

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 |

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. |

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 |

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 |

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 |

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 |

14. | F. M. Pigozzo, D. Modotto, and S. Wabnitz, “Second harmonic generation by modal phase matching involving optical and plasmonic modes,” Opt. Lett. |

15. | S. B. Bodrov, I. E. Ilyakov, B. V. Shishkin, and A. N. Stepanov, “Efficient terahertz generation by optical rectification in Si-LiNbO |

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 |

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 |

18. | V. Berger and C. Sirtori, “Nonlinear phase matching in THz semiconductor waveguides,” Semicond. Sci. Technol. |

19. | H. Cao, R. A. Linke, and A. Nahata, “Broadband generation of terahertz radiation in a waveguide,” Opt. Lett. |

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 |

21. | F. Peter, S. Winnerl, H. Schneider, and M. Helm, “Excitation wavelength dependence of phase matched terahertz emission from a GaAs slab,” Opt. Express |

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. |

23. | V. A. Kukushkin, “Generation of terahertz pulses from tightly focused single near-infrared pulses in double-plasmon waveguides,” J. Opt. Soc. Am. B |

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 |

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 |

27. | X. Ying and I. Katz, “A simple reliable solver for all the roots of a nonlinear function in a given domain,” Computing |

28. | R. E. Smith, G. W. Forbes, and S. N. Houde-Walter, “Unfolding the multivalued planar waveguide dispersion relation,” IEEE J. Quantum Electron. |

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 |

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 |

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 |

32. | L. Mandel and E. Wolf, |

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. |

34. | E. D. Palik, ed., |

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. |

36. | S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater. |

37. | M. Dellnitz, O. Schütze, and Q. Zheng, “Locating all the zeros of an analytic function in one complex variable,” J. Comput. Appl. Math. |

38. | P. N. Robson and P. C. Kendall, eds. |

**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

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### References

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