Terahertz (THz) generation by a GaP ridge waveguide with a collinear modal phasematching scheme based on cascaded difference frequency generation (DFG) processes is theoretically analyzed. The cascaded Stokes interaction processes and the cascaded antiStokes interaction processes are investigated from coupled wave equations. THz intensities and quantum conversion efficiency are calculated. Compared with noncascaded DFG processes, THz intensities from 11order cascaded DFG processes are increased to 5.48. The quantum conversion efficiency of 177.9% in cascaded processes can be realized, exceeding the ManleyRowe limit.
I. INTRODUCTION
The terahertz (THz) radiation, which is generally referred to as the frequency from 0.1 to 10 THz, has recently drawn much attention due to its tremendous potential applications, such as imaging, material detection, environmental monitoring, communication, astronomy and national defense security
[1

4]
. For such applications, a highpower, widely tunable, and compact source of THzwave is required. Due to the interest in exploiting this region there have been many schemes proposed on source technologies over the last twenty years or so
[5

10]
. Among many electronic and optical methods for the coherent THzwave generation, difference frequency generation (DFG)
[11

14]
is of importance because it offers the advantages of relative compactness, narrow linewidth, wide tuning range, highpower output and roomtemperature working environment. In DFG, two optical pump beams, with their frequencies separated by a few THz, interact through a χ
^{(2)}
process to generate a THz beam. Unfortunately, the quantum conversion efficiency of the DFG is extremely low as the THzwave is intensely absorbed by the nonlinear optical crystal. To improve the low quantum conversion efficiency and overcome the ManleyRowe limit, cascaded DFG in which more than one THz photon is generated from the depletion of a single pump photon is a promising method. Theoretical descriptions and experimental demonstrations of an enhancement output of THz wave via cascaded DFG processes have been reported recently. Liu
et al
.
[15]
proposed a scheme for monochromatic THz generation via cascading enhanced Cherenkovtype DFG in a sandwichlike waveguide. It is predicted that THz power can be boosted by nearly 8fold with a 400 MW/cm pump in a 40mmlong SiLiNbO
_{3}
Si waveguide. Lee
et al
.
[16]
experimentally observed the fourthStokes order in cascaded Stimulated Polariton Scattering utilizing Mg:LiNbO
_{3}
. Saito
et al
.
[17]
described a scheme for efficient THz generation using a cascaded optical parametric oscillator using a GaP sheet cavity. By choosing an appropriate pump wavelength and cavity design, the cascading process contributes to efficient THzwave generation, resulting in a high output peak power of 1.8 MW and a high photon conversion efficiency of 1.086 at 3 THz.
The conversion efficiency of the cascaded DFG process is primarily determined by the effective interaction length and the absorption in the nonlinear optical crystal at THz frequencies. Usually, collinear phase matching is a preferred scheme to maximize the effective interaction length. THz wave generations by GaP rib waveguide via collinear modal phasematched DFG have been observed
[18]
. While GaP has a large optical nonlinearity for THz DFG (50 pm/V at 1.55 μm
[19]
), its absorption coefficient in the THz range is relative large and increases rapidly with increasing frequency. Despite the fact that the THz wave is intensely absorbed by the optical crystal in collinear phase matching configuration, copropagating the THz wave with the pump and signal waves in the same direction can achieve long range amplification of the THz wave in a nonlinear crystal until pump depletion
[20]
.
In this paper, we present the theoretical analysis of THz generation by GaP ridge waveguide with a collinear modal phasematching scheme based on cascaded DFG processes. We investigate the cascaded Stokes interaction processes and the cascaded antiStokes interaction processes. THz intensities and quantum conversion efficiency are calculated from coupled wave equations.
II. THEORETICAL MODEL
Figure 1
shows a schematic diagram of THz wave generation by collinear modal phasematching cascaded DFG. TMlike guided THz wave in the GaP ridge waveguide was generated through typeI phase matching when the electric fields of both pump and signal waves are along [
1
10]. THz wave (
ω
_{T}
) is generated via interactions between the incident pump (
ω
_{p}
) and signal (
ω
_{s}
) waves in the firstorder DFG process, which consumes the higher frequency pump photon and amplifies the lower frequency signal photon. The amplified signal wave also acts as a higher frequency pump wave, which amplifies the THz wave and generates a new lower frequency cascaded signal (
ω
_{cs}
) wave in the secondorder DFG process. Simultaneously, antiStokes interactions will also occur that consume the THz photon and pump photon, resulting in a higher frequency antiStokes signal (
ω
_{cp}
) wave. The cascaded Stokes processes and antiStokes processes can be continued to any high order as long as the phasematching conditions are satisfied. The intensity of the THz wave is determined by a tradeoff between the Stokes processes and the antiStokes processes.
Schematic diagram of the cascaded DFG to generate THz radiation by GaP ridge waveguide. DFG with TypeI phase matching condition, the electric field vectors of pump and signal wave E_{p} and E_{s} are along [110], and that of THz wave E_{T} is along [001] .
THz wave is guided by a GaP ridge waveguide with dimensions
t
,
h
, and
w
, as shown in
Fig. 1
. Single mode operation of the THz wave produced in the ridge waveguide is realized by satisfying the following condition
[21]
:
The collinear modal phasematching condition in the ridge waveguide is expressed as
where
n_{i}
(
i
=
p
,
s
and
T.eff
) correspond to the refractive index of the pump, signal and the TMlike guided THz wave, respectively.
λ_{i}
(
i
=
p
,
s
and
T
) correspond to the wavelength of the pump, signal and the TMlike guided THz wave, respectively. The mode effective index
n_{T.eff}
of the THz wave is calculated using the effectiveindex method
[22]
.
The coupled wave equations of cascaded DFG can be derived from common nonlinear optical threewave interaction equations, shown as
where
ω_{n}
and
ω_{T}
denote the frequency of pump and THz wave, respectively.
E_{n}
and
E_{T}
denote the electric field amplitude of pump and THz wave, respectively.
α_{n}
and
α_{T}
denote the absorption coefficient of pump and THz wave in the optical crystal, respectively.Δ
k_{n}
indicates the wave vector mismatch in the cascaded DFG process,
κ_{n}
and
κ_{T}
are the coupling coefficients.
d_{eff}
is the effective nonlinear coefficient, and the
d_{eff}
is 50 pm/V at 1.55 μm
[19]
.
c
is the speed of light in vacuum,
ε
_{0}
is the vacuum dielectric constant,
I
is the power density,
n_{n}
is the refractive index. The generation and consumption of THz photons are accomplished during the interaction between the
n
order and (
n
+1)order Stokes waves, as shown in Eq. (3). The second item in the right side of the equal sign in Eq. (4) shows the Stokes processes where THz photons and
n
order Stokes photons are generated, and the third item in the right side of the equal sign in Eq. (4) shows the antiStokes processes where THz photons and (
n
+1)order Stokes photons are consumed.
The theoretical values of refractive index are calculated using a wavelengthindependent Sellmeier equation for GaP in the IR
[23]
and THz
[24]
range, respectively. The Sellmeier equation for GaP of Madarasz
et al
.
[23]
in the IR range can be written as
The Sellmeier equation for GaP in the THz range
[24]
can be written as
where
ε
(
v
) is the complex dielectric constant,
ν
is the wavenumber,
ε
∞ is the highfrequency dielectric constant,
ν_{TO}
,
ρ
and
γ
are the eigenfrequency, oscillator strength and damping coefficient of the 367 cm
^{−1}
polariton mode in GaP, and
ε
∞= 9.07,
ρ
= 1.945,
ν_{TO}
= 367 cm
^{−1}
,
γ
=9.0 cm
^{−1}
.
III. CALCULATIONS
Here, in simulating the cascaded DFG dynamics, pump wave
ω
_{p}
and signal wave
ω
_{s}
are supposed to be 193.55 and 192.55 THz, respectively. THz frequency
ω
_{T}
is taken to be 1 THz. We set the dimensions of GaP ridge waveguide
t
,
h
, and
w
to 120, 200 and 80 μm, respectively. The mode effective index
n_{T.eff}
for 1 THz is 3.1405 to realize collinear modal phasematching DFG in GaP ridge waveguide. The wave vector mismatch Δ
k
and coherence length in cascaded DFG processes is shown in
Fig. 2
. In the cascaded Stokes processes, wave vector mismatch is less than 3.14 cm
^{−1}
during 11order cascaded processes. In the case of cascaded antiStokes processes, wave vector mismatch is less than 3.14 cm
^{−1}
during 10order cascaded processes. In the following calculations, 11order cascaded Stokes and 10order cascaded antiStokes processes are taken into account as the coherence length is larger than 1 cm. As shown in
Fig. 3
, THz intensities in GaP ridge waveguide based on cascaded DFG with cascading orders 1, 3, 5, 10 and 11 versus crystal length are calculated according to Eqs. (3) and (4). The intensity of both pump and signal wave are 20 MW/mm
^{2}
. The absorption coefficient at 1 THz is 2.5 cm
^{−1}
[25]
. From
Fig. 3
we find that THz intensities without cascading processes are extremely low. THz intensities with cascading order 3, 5, 10 and 11 are enhanced. THz intensity of 0.37 MW/mm
^{2}
can be obtained with 11order cascaded Stokes processes. Compared with noncascaded DFG processes, THz intensities from 11order cascaded DFG processes are increased to 5.48. In noncascaded DFG processes, at best, a single THz photon is generated from each pump photon. The cascaded processes can enhance the THz output, simply by generating several THz photons from each pump photon.
Wave vector mismatch and coherence length of cascaded DFG. Assuming ω_{p} =193.55 THz, ω_{s} =192.55 THz, ω_{T} = 1 THz.
THz intensities by GaP ridge waveguide based on cascaded DFG with cascading orders 1, 3, 5, 10 and 11.
As the Stokes processes generate THz photons and the antiStokes processes consume THz photons, THz intensities depend on the Stokes processes and the antiStokes processes.
Figure 4
shows the maximum intensities of the optical waves during the cascaded Stokes processes and antiStokes processes. In this figure we assume that the optical waves at interval of 1 THz with frequencies from 183.55 to 193.55 THz interact in the Stokes and antiStokes processes. The initial pump and signal waves are 188.55 and 187.55 THz, respectively, with a power density of 20 MW/mm
^{2}
. From the figure we find that the power densities of optical waves in the Stokes processes is higher than that of optical waves in the antiStokes processes, which indicates that the Stokes processes are stronger than the antiStokes processes. In the cascaded Stokes processes,
n
order Stokes photons are consumed and (
n
+1)order Stokes and THz photons are amplified. The cascaded Stokes processes continue to amplify the (
n
+1)order Stokes and THz photons as long as the phasematching conditions are satisfied. Actually, the antiStokes processes take place only if the Stokes processes generate THz photons. The antiStokes processes consume highorder Stokes and THz photons, and the processes will stop if the highorder Stokes and THz photons are exhausted.
The maximum intensity of the optical waves during the cascaded Stokes processes and antiStokes processes. Assuming the initial pump and signal waves are 188.55 and 187.55 THz with a power density of 20 MW/mm^{2}, respectively.
Figure 5
shows the relationship between the maximum THz intensities and the pump wave frequencies. In this figure we assume that the optical waves at interval of 1 THz with frequencies from 183.55 to 193.55 THz interact in the cascaded Stokes and antiStokes processes. The frequency of the pump wave is 1 THz larger than that of the signal wave. Both the pump and signal intensities are 20 MW/mm
^{2}
. From the figure we find that THz intensities are higher as the pump frequencies locate in the highfrequency area. The high THz intensities originate from the interaction of the highorder Stokes processes as the pump frequencies locate in the highfrequency area, which indicates that the Stokes processes are stronger than the antiStokes processes. As the pump frequency equals 193.55 THz, THz wave with a maximum intensity of 0.3686 MW/mm
^{2}
can be obtained. In the highfrequency area where highorder Stokes processes interact, optimal crystal lengths are longer considering cascading, which is consistent with the principle of cascaded nonlinear processes.
The relationship between the maximum THz intensities and pump frequencies. Both the pump and signal intensities are 20 MW/mm^{2}.
Pump intensity is directly related to the quantum conversion efficiency in a cascaded DFG processes. The maximum THz intensity and quantum conversion efficiency are calculated when the original pump intensities are changed from 1 MW/mm
^{2}
to 20 MW/mm
^{2}
, as shown in
Fig. 6
. In the calculations, pump wave and signal wave are supposed to be 193.55 and 192.55 THz, respectively.
Fig. 6
demonstrates that the maximum THz intensity and quantum conversion efficiency significantly increase with the pump intensity. THz wave with a maximum intensity of 0.3686 MW/mm
^{2}
can be generated as pump intensity equals to 20 MW/mm
^{2}
, corresponding to the quantum conversion efficiency of 177.9%. The quantum conversion efficiency of 177.9% in cascaded processes exceeds the ManleyRowe limit.
The maximum THz intensity and quantum conversion efficiency versus pump intensity. Assuming pump and signal frequencies are 193.55 and 192.55 THz, respectively.
THz generation with the frequency of 1 THz with a collinear modal phasematching scheme based on cascaded DFG processes is theoretically analyzed above. As for THz waves with frequencies lower than 1 THz, THz generation can be effectively enhanced based on cascaded DFG processes if the collinear modal phasematching scheme is satisfied. As for THz waves with frequencies larger than several THz, the output of the THz wave is seriously affected by the absorption by the GaP crystal as the absorption coefficients of THz wave in GaP crystal rapidly increase with the increase of frequency.
IV. CONCLUSION
THz generation by GaP ridge waveguide with a collinear modal phasematching scheme based on cascaded DFG processes is theoretically analyzed. The cascaded DFG processes comprise the Stokes interaction processes and the cascaded antiStokes interaction processes. The calculation results indicate that the Stokes processes are stronger than the antiStokes processes. Compared with noncascaded DFG processes, THz intensities from 11order cascaded DFG processes are increased to 5.48. THz wave with a maximum intensity of 0.3686 MW/mm
^{2}
can be generated as pump intensity is 20 MW/mm
^{2}
, corresponding to the quantum conversion efficiency of 177.9%. The quantum conversion efficiency of 177.9% exceeds the ManleyRowe limit, providing us an efficient way to enhance the output of THz waves.
Color versions of one or more of the figures in this paper are available online.
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant Nos. 61201101 and 61205003) and Young Backbone Teachers in University of Henan Province (Grant No. 2014GGJS065).
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