Simultaneous Generation of Two Orthogonally Polarized Terahertz Waves by Stimulated Polariton Scattering with a Periodically Poled LiNbO

We present a theoretical investigation of the simultaneous generation of two orthogonally polarized terahertz (THz) waves by stimulated polariton scattering (SPS) with a periodically poled LiNbO3 (PPLN) crystal. The two orthogonally polarized THz waves are generated from SPS with A1 and E symmetric transverse optical (TO) modes in a LiNbO3 crystal, respectively. The parallel polarized THz wave is generated from A1 symmetric TO modes with type-0 phase-matching of e = e + e, and the perpendicular polarized THz wave is generated from E symmetric TO modes with type-I phase-matching of e = o + o. The two types of phase-matching of e = e + e and e = o + o can be almost satisfied simultaneously by accurately selecting the poling period of the PPLN crystal. We calculate the photon flux density of the two orthogonally polarized THz waves by solving the coupled wave equations. The calculation results indicate that the two orthogonally polarized THz waves can be efficiently generated, and the relative intensities between the two orthogonally polarized THz waves can be modulated.


Introduction
Stimulated polariton scattering (SPS) has proven to be an efficient scheme to generate terahertz (THz) waves [1][2][3][4][5][6][7][8].A polariton is a coupled quantum between the pump laser and the infrared-and Raman-active transverse optical (TO) modes in a crystal, and it behaves like phonons near the resonant frequency associated with the TO mode and exhibits photon-like behavior for lower non-resonant frequencies [1].SPS consists of second-order and third-order nonlinear frequency conversion processes where a pump photon stimulates a Stokes photon at the difference frequency between the pump photon and the polariton.At the same time, a THz wave is generated by the parametric process due to the nonlinearity arising from both electronic and vibrational contributions of the crystal.The TO phonon resonances can contribute substantially to the magnitude of the second-and third-order nonlinearities, which are beneficial to the THz generation via SPS.
MgO:LiNbO 3 has been the most widely used crystal for THz wave generation via SPS [1][2][3][4][5].MgO:LiNbO 3 has strong second-order nonlinear response, as well as TO phonon resonances for efficient SPS [9].MgO:LiNbO 3 has five A 1 symmetric infrared-and Raman-active TO modes polarized parallel to the c-axis with frequencies of 248 cm −1 , 274 cm −1 , 307 cm −1 , 628 cm −1 , and 692 cm −1 [10].MgO:LiNbO 3 has eight E symmetric infrared-and Raman-active TO modes polarized perpendicular to the c-axis with frequencies of 152 cm −1 , 236 cm −1 , 265 cm −1 , 322 cm −1 , 363 cm −1 , 431 cm −1 , 586 cm −1 , and 670 cm −1 [10].A 1 symmetric TO modes have been the most widely used for THz wave generation via SPS [1][2][3][4][5].However, E symmetric TO modes can be also employed to generate THz waves via SPS.In 1969, Yarborough reported the observation of tunable SPS from A 1 and E symmetric TO modes with a pump wave in a LiNbO 3 crystal [11].If the SPS from A 1 and E symmetric TO modes can be simultaneously excited, then two orthogonally polarized THz waves can be simultaneously generated.Orthogonally polarized THz waves are useful for imaging [12].Yu et al. [12] showed that the addition or subtraction of two images, which were taken with a perpendicularly polarized THz wave and parallel polarized THz wave, was effective to enhance the contrast of terahertz images.
In this work, we theoretically study the simultaneous generation of two orthogonally polarized THz waves by SPS with a periodically poled LiNbO 3 (PPLN) crystal.The two orthogonally polarized THz waves are generated from SPS with A 1 and E symmetric TO modes in a MgO:LiNbO 3 crystal, respectively.We calculate the photon flux density of the two orthogonally polarized THz waves by solving the coupled wave equations.

Phase-Matching Characteristics
Due to the different eigenfrequency, oscillator strength, and bandwidth of the TO modes, the two orthogonally polarized THz waves, THz e and THz o , have different dispersion and absorption characteristics.The value of the refractive index in the THz range is much larger than that in the optical range, so the collinear phase-matching is impossible to realize.The absorption coefficients α Te and α To are very large, especially in the high THz frequency range.In the optical SPS processes, the THz waves are generated, and the seed waves are amplified.The amplified seed waves are Stokes waves.In order to achieve efficient conversion of the SPS processes from the pump wave to the THz waves, a precise phase-matching condition must be satisfied.For the forward SPS processes, the pump, Seed e , and THz e waves satisfy the type-0 phase-matching of e = e + e, and the phase mismatch ∆ k e is as follows: The pump, Seed o , and THz o waves satisfy the type-I phase-matching of e = o + o, and the phase mismatch ∆ k o is as follows: where k p is the wave vector of the pump wave, k se and k so are the wave vectors of the two Seed e and Seed o waves, respectively, and k Te k To are the wave vectors of the two THz e and THz o waves, respectively.k Λ = 2π/Λ is the grating vector, and Λ is the poling period of the PPLN crystal.
For the backward SPS processes, the pump, Seed e and THz e waves satisfy the type-0 phase-matching of e = e + e, and the phase mismatch ∆ k e is as follows: The pump, seed o , and THz o waves satisfy the type-I phase-matching of e = o + o, and the phase mismatch ∆ k o is as follows: The energy conservation condition has to be fulfilled according to the following: where λ p is the wavelength of pump wave, λ se and λ so are the wavelengths of the two Seed e and Seed o waves, respectively, and λ Te λ To are the wavelengths of the two THz e and THz o waves, respectively.
If both the phase mismatches ∆ k e and ∆ k o are small enough, two perpendicular THz waves THz e and THz o can be generated simultaneously with a single pump wave.
For the SPS processes, we calculate the phase mismatches ∆ k e and ∆ k o according to Equations ( 1) and (2), respectively, at a fixed pump wavelength.The wavelengths of the two seed waves and the two THz waves are dependent on Equations ( 5) and (6).The sum phase mismatch If the sum phase mismatch ∆k s is small enough, the two phase mismatches ∆ k e and ∆ k o are small enough to realize the two phase-matching conditions of e = e + e and e = o + o.
Figure 3 shows the phase-matching characteristics for the forward SPS processes with a pump wavelength of 1550 nm.ν Te and ν To are the frequencies of the THz e and THz o waves, respectively.The theoretical values of the refractive indices are calculated using a Sellmeier equation for LiNbO 3 in the infrared range [13].From Figure 3a, we find that as Λ varies from 9 to 18 µm, there are many points of ∆k s with values below π cm −1 , which indicates that the two SPS processes generating the THz e and THz o waves can be efficiently realized.Most frequencies from 4.6 to 6 THz of THz e and most frequencies from 0.4 to 2.8 THz of THz o can be efficiently generated.The minimum value of ∆k s is 0.064 cm −1 with Λ of 17.1 µm, corresponding to ν Te of 4.66 THz and ν To of 0.56 THz. Figure 3b shows the detailed phase-matching characteristics with Λ from 17.096 to 17.100 µm.As Λ varies from 17.0981 to 17.0983 µm, ∆k s with a value of 0.0369 cm −1 is small enough to stimulate the two SPS processes.In particular, as Λ is 17.0982 µm, ∆k e equals ∆k o , which indicates that the two SPS processes can be realized to equal degrees.Figure 4 shows the phase-matching characteristics for the backward SPS processes with a pump wavelength of 1550 nm.From Figure 4a, we find that as Λ varies from 20 to 100 µm, there are also many points of ∆k s with values below π cm −1 , particularly below 1 cm −1 .Most frequencies from 0.45 to 2.03 THz of THz e and most frequencies from 2.01 to 3 THz of THz o can be efficiently generated.The minimum value of ∆k s is 0.344 cm −1 with Λ of 80.98 µm, corresponding to ν Te of 0.52 THz and ν To of 2.06 THz. Figure 4b shows the detailed phase-matching characteristics with Λ around 80.98 µm.As Λ varies from 80.960 to 80.995 µm, ∆k s with a value of 0.344 cm −1 is small enough to stimulate the two SPS processes.In particular, as Λ is 80.978 µm, ∆k e equals ∆k o , which indicates that the two SPS processes can be realized to equal degrees.

THz Photon Flux Density
The coupled wave equations for the SPS processes can be found in [9,14].The coupled wave equations describe the field envelope variation of the pump, Stokes, and THz waves.The analytical expression of THz parametric gain coefficient g T under the QPM condition in the international system of units can be written as follows: where ω 0 j , S j , and Γ j denote the eigenfrequency, the oscillator strength of the polariton modes, and the bandwidth of the jth TO mode in the LiNbO 3 crystal, respectively.I p is the power density of the pump wave, and g 0 is the low-loss parametric gain.n p , n s , and n T are the refractive indices of the pump, Stokes, and THz waves, respectively.ϕ is the angle between the wavevectors of the pump wave and THz wave.α T is material absorption coefficient in THz region.d E and d Q are nonlinear coefficients related to pure parametric (second-order) and Raman (third-order) scattering processes, respectively.When THz frequencies are far below the lowest A 1 symmetry TO mode of 248 cm −1 and the lowest E symmetry TO mode of 152 cm −1 , Equation ( 8) can be rewritten as follows [9]: For SPS with type-0 phase-matching of e = e + e, the relationship between d E and d Q is given by [9,15,16] the following: For SPS with type-I phase-matching of e = o + o, the relationship between d E and d Q is as follows: where r 33 and r 13 are the linear electro-optic coefficients of LiNbO 3 .
With strong THz wave absorption and phase mismatch and without pump depletion, the coupled wave equations can be solved to give the THz photon flux density φ T with a general solution [17], given by the following: where ∆k is the phase mismatching and L is the crystal length.The initial THz photon flux density φ T is assumed to be zero, and φ s (0) is the initial seed wave photon flux density.The initial photon flux densities of Seed e and seed o are φ se (0) and φ so (0), respectively.The THz photon flux densities of THz e and THz o are φ Te and φ To , respectively.The ratio R of φ so (0) to φ se (0) is as follows: Figure 5 shows the THz wave photon flux densities φ Te and φ To for the forward SPS processes.From Figure 5a-d , we find that when Λ varies from 15 to 19 µm, φ Te and φ To increase first and then decrease.When Λ is equal to 17.0982 µm, φ Te and φ To reach their maximum values as the phase mismatches ∆k e and ∆k o reach their minimum values.The maximum value of φ Te is 5.74 × 10 −6 s −1 cm −2 .The value of φ Te is so small, because the THz absorption coefficient of 4.66 THz is very large.φ To increases with the increase of R. The relative photon flux densities between φ Te and φ To can be tuned by varying R. When R is 0.0037, the maximum values of φ Te and φ To are approximately equal.From Figure 5e, we find that when crystal length L varies from 0 to 30 mm, φ Te and φ To increase rapidly and smoothly.When R is 0.00323, the values of φ Te and φ To are approximately equal, as L is larger than 20 mm.
Figure 6 shows the THz wave photon flux densities φ Te and φ To for the backward SPS processes.From Figure 6a, we find that when Λ varies from 74 to 88 µm, φ Te and φ To increase first and then decrease.When Λ is equal to 80.978 µm, φ Te and φ To reach their maximum values as the phase mismatches ∆k e and ∆k o reach their minimum values.The maximum value of φ Te is 43.62 s −1 cm −2 .The maximum value of φ Te in the backward SPS processes is larger than that in the forward SPS processes, because the THz absorption coefficient of 0.52 THz in the backward SPS processes is smaller than that of 4.66 THz in the forward SPS processes.The relative photon flux densities between φ Te and φ To can be tuned by varying R. When R is 2.5 × 10 7 , the maximum values of φ Te and φ To are approximately equal.From Figure 6b, we find that when crystal length L varies from 0 to 50 mm, φ Te and φ To increase rapidly and smoothly.When R is 5.21 × 10 7 and L is larger than 40 mm, the values of φ Te and φ To are approximately equal.
The intensities of generated THz waves are very low, because the THz waves are heavily absorbed by the PPLN crystal.However, the intensities of the THz waves can be enhanced by injection intense seed waves, as shown in Equation ( 13).Moreover, one can use organic crystals with QPM, because organic crystals have larger nonlinear optical coefficients and lower absorption coefficients in the THz region [18].Furthermore, the enhancement of the THz intensities can be realized by cryogenic cooling.At liquid nitrogen temperature, the gain coefficients of the THz waves in the SPS processes are enhanced.At the same time, the absorption coefficients of the THz waves decrease.The scheme in this work of generating two orthogonally polarized THz waves by SPS processes has certain advantages.First of all, the two orthogonally polarized THz waves are simultaneously generated by a pump wave, which means that the two THz waves are phase-conjugate.Second, the two orthogonally polarized THz waves are generated only by a PPLN crystal.Third, the intensities of the two orthogonally polarized THz waves can be tuned by varying the intensities of the input seed waves.

Conclusions
We present the simultaneous generation of two orthogonally polarized THz waves by forward and backward SPS processes with a PPLN crystal.The minimum values of ∆k s of 0.064 cm −1 in the forward SPS processes and 0.344 cm −1 in the backward SPS processes indicate that the type-0 phase-matching generating parallel polarized THz wave and the type-I phase-matching generating perpendicular polarized THz wave can almost be satisfied simultaneously.In particular, the two SPS processes can be excited to equal degrees by accurately selecting the poling period of the PPLN crystal.We calculate the photon flux densities of the two orthogonally polarized THz waves by solving the coupled wave equations.The theoretical calculations show that the photon flux densities of the two orthogonally polarized THz waves are very small.The relative intensities between the two orthogonally polarized THz waves can be modulated by varying the intensities of the input seed waves.

Figure 1
Figure 1 shows a schematic diagram of THz wave generation by the SPS processes by a PPLN crystal with a quasi-phase-matching (QPM) condition.A pump wave and two seed waves (Seed e and Seed o ) propagate along the x-axis of the PPLN crystal.The electric field of the pump wave and Seed e is along the z-axis of the PPLN crystal, whereas the electric field of Seed o is perpendicular to the z-axis of the PPLN crystal.The z-axis is the optical axis of the LiNbO 3 crystal.The poling period of the PPLN crystal is Λ.Two orthogonally polarized THz waves (THz e and THz o ) are generated by the SPS processes.The electric field of THz e is along the z-axis of the PPLN crystal, whereas the electric field of THz o is perpendicular to the z-axis of the PPLN crystal.The pump, Seed e , and THz e waves satisfy the type-0 phase-matching of e = e + e, whereas the pump, Seed o , and THz o waves satisfy the type-I phase-matching of e = o + o.The above two types of phase-matching can also be applied to the forward SPS processes and backward SPS processes by accurately selecting the poling period Λ of the PPLN crystal.The generated THz waves are deflected by parabolic mirrors, which transmit the pump and two seed waves.

Figure 1 .
Figure 1.Schematic diagram of terahertz (THz) wave generation by stimulated polariton scattering (SPS) processes in a periodically poled LiNbO 3 (PPLN) crystal with a quasi-phase-matching (QPM) condition.Λ is the poling period of the PPLN crystal.P 1 and P 2 are parabolic mirrors which transmit the pump and two seed waves, and couple out the two THz waves.

Figure 2
shows the dispersion and absorption characteristics of the THz e and THz o waves.n Te and n To are the refractive indices of THz e and THz o , respectively, and α Te and α To are the absorption coefficients of THz e and THz o , respectively.The curves of n Te and α Te are below the lowest A 1 symmetric TO mode, 248 cm −1 , and the curves of n To and α To are below the lowest E symmetric TO mode, 152 cm −1 .The theoretical parameters of the refractive index and absorption coefficient for LiNbO 3 in the THz range are cited in [9].From the figure, we find that n Te and n To are larger than 5.

Figure 2 .
Figure 2. The dispersion and absorption characteristics of the two orthogonally polarized THz waves, THz e and THz o .n Te and n To are the refractive indices of THz e and THz o , respectively, and α Te and α To are the absorption coefficient of THz e and THz o , respectively.

Figure 3 .
Figure 3.The phase-matching characteristics for the forward SPS processes.ν Te and ν To are the frequencies of the THz e and THz o waves, respectively.The sum phase mismatch ∆k s = ∆ k e + ∆ k o , and λ p = 1550 nm.(a) The phase-matching characteristics with Λ from 9 to 18 µm.(b) The detailed phase-matching characteristics with Λ from 17.096 to 17.100 µm.

Figure 4 .
Figure 4.The phase-matching characteristics for the backward SPS processes, λ p = 1550 nm.(a) The phase-matching characteristics with Λ from 20 to 100 µm.(b) The detailed phase-matching characteristics with Λ from 80.93 to 81.02 µm.