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Theoretical analysis of difference frequency generation for terahertz generation in a sheet microcavity from the CO2 laser
Optik - International Journal for Light and Electron Optics 172 (2018) 1111–1116
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Theoretical analysis of difference frequency generation for terahertz generation in a sheet microcavity from the CO2 laser Shijia Zeng, Fangsen Xie, Zhiming Rao
T
⁎
College of Physics and Communication Electronics, Jiangxi Normal University, 99 Ziyang Road, Nanchang, 330022, China
A R T IC LE I N F O
ABS TRA CT
Keywords: Terahertz generation Difference frequency generation Cavity phase matching Nonlinear optics
A scheme for efficient terahertz generation in a GaAs sheet microcavity from the CO2 laser is presented, in which the process of difference frequency generation that considered cavity phase matching is derived theoretically. This model achieves its optimum working condition when the cavity length corresponds to a coherent length and the resonant condition of pump wave is satisfied. The results show that the efficiency can realize an effective improvement when the CO2 lasers pump wavelengths are 9.55 μm and 9.79 μm, respectively, with a peak power conversion efficiency of 2.04%, which corresponds to a photon conversion efficiency of 83%. This work is expected to provide a theoretical basis for relevant experiments involving doubly and even triply resonant structures.
1. Introduction Difference frequency generation (DFG) has become an important technology for the generation of terahertz (THz) wave in recent years due to its many advantages such as high power, continuous tuning, reduced structure and no threshold condition [1–3]. An efficient THz generation may be realized using the CO2 laser. According to the Manley-Rowe relation, the maximum conversion efficiency can be improved by one order of magnitude with longer wavelength running at 10 μm [4,5]. The key to the DFG is compensating the phase mismatch in a consistent manner, which can be realized using cavity phase matching (CPM) by resonance recirculation in microcavities, such as Fabry-Perot (F–P) microcavities, because the traveling wave and reflected wave are exactly in phase for every circling. This method was first predicted in 1962 by Armstrong et al. [6] followed by some relevant theoretical studies [7–9], but the method was not demonstrated experimentally until 2011 by Xie et al. [10]. In 2013, Lin et al. [11] experimentally realized a higher conversion efficiency by reflecting the pump beam in a KTiOPO4 crystal. The principle for CPM is similar to the well-known quasi phase matching (QPM) [12] method, with the two methods differing in that the CPM does not require reversal of the orientation for the nonlinear polarization and superimposed several crystals to extend the length of the nonlinear interaction, which simplifies the crystal fabrication, makes the structure more compact and reduces costs. Currently, Saito et al. [13] realized an estimated power of 4 MW at 3 THz using optical parametric oscillation, in which the pulse duration and cavity configuration in a GaP sheet cavity was optimized in theory. Soon after, Saito et al. [14] proposed a theoretical scheme that both the cascaded effect and CPM are considered, and generated an output peak power of 1.8 WM with a photon conversion efficiency of 1.086 at 3 THz, which exceeds the Manley- Rowe limit [15]. In this paper, a scheme for DFG based on CPM is proposed using the structure of sheet microcavity, in which the crystal absorption for a THz wave is greatly reduced. By considering both of the standing wave condition for PW and the extremum condition for the
⁎
Corresponding author. E-mail address: [email protected] (Z. Rao).
https://doi.org/10.1016/j.ijleo.2018.07.130 Received 30 May 2018; Accepted 27 July 2018 0030-4026/ © 2018 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 172 (2018) 1111–1116
S. Zeng et al.
Fig. 1. Schematic structure of terahertz generation by cascaded DFG based on CPM.
THz wave generated by DFG, the optimum cavity length is determined. Meanwhile, the influence of temperature on this model is considered. Finally, the efficiencies for DFG based on CPM and normally DFG are compared. 2. Theory As illustrated in Fig.1, the THz wave is generated in the cavity via DFG (ωT = ω1−ω2) using two collinear pump wave (PW). For the continuous nonlinear interactions, a resonant cavity for PW without the THz wave is considered, in which an undepleted pump approximation can be applied and a higher efficiency can be achieved by reducing the absorption of nonlinear crystal at the THz wave in the cavity [14,16]. The cavity contains a nonlinear medium (M) with a length of L and two coatings C1 and C2 on both parallel sides. The two beams in the primary PW are PW1 with amplitude A01 and PW2 with amplitude A02 , which are normally incident from the C1 end. The reflectances of C1 for PW1 and PW2 are R1 and R2 , respectively. The reflected wave amplitudes for PW1 and PW2 are R1 A01 and R2 A02 , respectively. The amplitudes of PW1 and PW2 inside the cavity are A1 and A2 , respectively. The reflectances of C2 for PW1 and PW2 should be near 100% to avoid PW output from C2; therefore, we can consider approximately A1, l = −A1, r and A2, l = −A2, r , where r and l indicate propagation of the PW to C2 and C1, respectively. A1, t and A2, t are the amplitudes of PW1 and PW2 transmitted outside of the cavity. Therefore, A1, t = (T1 n1/ n 01 )1/2A1, l and A2, t = (T2 n2/ n 02 )1/2A2, l , where T1 and T2 are the transmittance of the C1, and n1, n2 are the refractive indexes of M for PW1 and PW2, respectively. n 01 and n 02 are, respectively, the refractive indexes of PW1 and PW2 in air, respectively. The effect of coating on PW is not considered because the thickness of the coating is ignored for an ideal model. To prevent leakage of the pump energy from the cavity, the destructive interference [16] between A1, t , A2, t and R1 A 01, R2 A 02 can be considered in the following form
R1 A01 + A1, t = 0,
(1)
R2 A02 + A2, t = 0.
(2)
The reflectance for the THz wave that generated by DFG is close to 100% and RT at C1 and C2, respectively. RT should be sufficiently small to ensure the THz wave can be transmitted outside from the C2 and reduce the loss in the cavity. ATL and ATR are the amplitudes of THz wave that propagates to C1 and C2, respectively, in the cavity. The amplitude of the THz wave that is transmitted to the outside is AT , with AT = (TT nT / n 03 )1/2ATR , where TT , nT and n 03 are the transmittance of the C2 and the refractive indices for M and air for the THz wave, respectively. The expression for the THz electric field generated by DFG must be obtained before considering the cascading effect. According to the Maxwell equation, the THz electric field intensity ET inside the cavity is given by [7,16]
∂2ET ∂2 + (kT )2ET = μ0 2 (2ε0 deff E1 E2 *), ∂x 2 ∂t
(3)
where kT = 2πnT / λT is the wave vector of THz wave and E1, E2 are the electric field intensities for the PW1 and PW2, respectively, inside the cavity. The effective nonlinear optical coefficient is
πl deff = d sinc ⎛ cav ⎞ ⎝ 2lcoh ⎠ ⎜
⎟
(4)
where d is the nonlinear coefficient of the medium M, lcav is the cavity length and lcoh is the coherence length. Here, we consider lcav ≈ L for an ideal model. The solution of Eq. (3) without crystal absorption is given by
ET = AT 1 exp (ikT x ) + UAB exp (ikB x ) + AT 2 exp (−ikT x ) + UAB exp(−ikB x ),
U=−
2μ0 ε0 ωT
2d
(5)
eff
kT 2−kB 2
(6)
where kB = k1−k2, k1, k2 are the wave vector of PW1 and PW2, respectively, and AB = A1 A2 . The amplitude for the THz wave inside the 1112
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S. Zeng et al.
cavity are given by
ATR = AT 1 exp (ikT x ) + UAB exp (ikB x ),
(7)
ATL = AT 2 exp(−ikT x ) + UAB exp(−ikB x ).
(8)
The boundary condition at C1 and C2 for the THz wave in the cavity are given by Eqs. (9) and (10), respectively. (9)
AT 1 + U AB = −(AT 2 + U AB) AT 2 exp(−ikT L) + UAB exp(−ikB L) = − RT (AT 1 exp (ikT L) + UAB exp (ikB L)).
(10)
According to these boundary conditions, expressions for AT1 and AT2 in terms of the amplitude of the PW can be obtained. Combining Eqs. (7), (9) and (10) with considering the standing wave condition for THz wave, we obtain (11)
|ATR | = |UAB Z|,
Z=
2 exp (iΔKL)−exp (i2ΔKL)−1 RT −exp (i2ΔKL)
(12)
where ΔK = k1−k2−kT . According to the formula derived above, it is enough to discuss the strength of ATR about ΔKL when both the PW and the orders of cascading effect are determined, which indicates that UAB can be viewed as a constant according to Eq. (13). For the first order, ATR as a function of ΔKL calculate using Eqs. (11) and (12) with different RT is shown in Fig. 2. ATR achieves a maximum when ΔKL = (2m + 1) π (m is an integer) because the three-wave coupling propagates the length L with a phase mismatch of (2m + 1) π , which reaches a maximum in the cavity via the second-order nonlinear conversion effects, and the PW is reflected by C1 or C2 results in a phase shift of π such that the nonlinear conversion effects can be continued. Meanwhile, a larger RT leads to stronger AT because of the resonance of the THz wave in the cavity. 3. Numerical simulations According to the above discussion, the cavity length should synchronously meets two conditions for a steady nonlinear effect, one is the standing wave condition of PW wave and the other is the phase condition which make the amplitude of the THz wave to achieve a maximum. It is similar to QPM that the conversion efficiency of CPM can achieve a maximum at first-order phase matching (ΔKL = π ). Therefore, the optimum cavity length L can be calculated using the following expression:
k1 L = pπ,
(13)
k2 L = qπ
(14)
ΔKL = π .
(15)
Where p and q are integer when the standing wave condition of PW is satisfied. Out of practical consideration, we consider the CO2 lasers lines, with 〈111〉 GaAs (= 10−10 m/V [17]) as the working medium; two lines that can approximately satisfy Eqs. (13)–(15) are 9.5524 μm [9P(20)] and 9.7938 μm [9P(46)], with a resulting THz wavelength of 387.65 μm and a cavity length of 756.6 μm . The value of p = 518.995 and q = 506.010 can be also calculated using Eqs. (13)–(15), with the result of ΔKL/ π ≈ 1. The distance between p and the integer near p is less than 0.1, and q has the similar situation. Therefore, a doubly resonant model for the PW can be applied approximately. Moreover, there is kT L = 13.98π , which indicates that a potential triply resonant model can be applied. It is sufficient to calculate the efficiency of difference frequency generation for this model using PT = |AT |2 nT A/2Z0 and ηT = PT / P1 when the PW wavelength and cavity length are determined. Where Z0 and A are the vacuum impedance and facula area, respectively, and we can obtain
Fig. 2. Amplitude of the THz wave ATR as a function of ΔKL with different RT . 1113
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S. Zeng et al.
Fig. 3. Power conversion efficiency for various λ2 with different R when λ1 = 9.5524 μm .
ηT = R
R=
8(2π ) 4deff 2L4 nT P2 ε0 cλT 4n1 n2 (2n
+
1)2A
sin4
(
1 ΔKL 2
1 ( 2 ΔKL) 4
), (16)
R1R2 . (1−R1)(1−R2)( RT −1)2
(17)
Where n = 14 because kT L ≈ 14π , and λT is the THz wavelength. The PW reflectance will be approximated as R1 = R2 = R 0 for convenience of discussion. Here, it is absence of the crystal absorption for THz wave by assume RT = 0 , which can avoid the loss of THz wave in the cavity, meanwhile, it is better to meet the design of this model. For a pump power P1 = P2 = 100 kW/cm2 and PW1 wavelength λ1= 9.5524 μm [9P(20)], the power conversion efficiency for THz generation by DFG based on CPM varied with PW2 wavelength can be calculated, as shown in Fig.3 with different R. The efficiency of DFG is increasing gradually with the increase of reflectance because of stronger resonance for PW. This picture also shows that the efficiency can reach a maximum value of 2.04% when PW2 wavelength λ2= 9.7938 μm [9P(46)] and R = 0.99. The growth of efficiency cannot be achieved by increasing the cavity length L because L is restricted by the phase conditions of resonance and nonlinear interaction, which can be illustrated by Fig.4. It is somewhat different from QPM, whose efficiency can be increased directly by superimposing the periodically poled crystal that achieve a larger length of nonlinear interaction. Efficiency of normally DFG without CPM can be expressed by the following form [18]
η=
2 2 32deff L P2 . ε0 cn1 n2 nT λT2 A
(18)
It is assumed that the conditions of normally DFG are the same as DFG that based on CPM, which means the crystal length is equal to the coherent length. The numerical simulations show that the order of magnitude of DFG efficiency is only 10−6 %, which is far less than the efficiency of DFG that based on CPM. It shows that the efficiency of DFG can be improved effectively in this model, which is studied in this paper. For practical applications, the influence of changing temperature on this model (without the influence of thermal expansion coefficient) should be considered. The refractive index of GaAs is given by [19]
Fig. 4. Power conversion efficiency varied with cavity length. 1114
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S. Zeng et al.
Fig. 5. Effect of temperature on the optimum cavity length and the degree of nonlinear interaction between the PW and THz.
n2 (λ ) = b +
g1 b1−2−λ−2
+
g2 b2−2−λ−2
+
g3 b3−2−λ−2
(19)
10−7ΔT 2 ,
ΔT−4.882 × b1 (μm) = 0.4431307 + 0.000050564ΔT , b2 (μm)= 0.8746453 + 0.0001913 where b = 5.382514, b3 (μm) = 36.9166−0.011622ΔT , g1 = 27.83972 , g2 = 0.031764 + 4.35 × 10−5ΔT + 4.664 × 10−7ΔT 2 , g3 = 0.00143436, and ΔT = T −22 °C indicates the deviation of the actual temperature to the room temperature as used in the calculations above. For a selected PW wavelength, the effect of temperature on the optimum cavity length should be considered at first, which can be illustrated in Fig. 5 using Eq. (13). In this picture, the deviation to the optimum cavity of 756.6 μm will reach 13 μm when the temperature changes from 22 °C to 300 °C. It also shows that the degree of nonlinear interaction between the PW and THz wave decreases gradually due to the increased temperature. Therefore, the efficiency of DFG may be affected to some extent by the varied temperature, as shown in Fig. 6. The changes of efficiency from 2.04% to 1.6% approximately with a varied temperature from 0 °C to 300 °C when L = 756.6 μm in this picture. It also shows that the efficiency is approximately unaffected by temperature when temperature is near 22 °C, which can be applied in a normal temperature environment for a small THz source. 4. Conclusion In conclusion, a scheme of difference frequency generation based on cavity phase matching with the structure of a sheet microcavity is proposed. In particular, the conditions for both the resonance for pump wave and nonlinear interaction between pump and THz are discussed. The results show that the optimum working condition can be approximately satisfied when the wavelengths of the two laser beams and cavity length are 9.5524 μm [9P(20)], 9.7938 μm [9 P(46)] and 756.6 μm , respectively; these conditions lead to a generated THz wavelength of 387.65 μm (0.77 THz) using GaAs as working medium. The nonlinear interaction among the pump and THz wave can be effective enhanced by contrasting with normally difference frequency generation, with a peak power conversion efficiency of 2.04%, corresponding to a photon conversion efficiency of 83%. We believe that our results are anticipated to play a positive role in such experiments involving doubly even triply resonant structures. Meanwhile, what can be expected is that the realization of small THz source will contribute to the development of THz science. Funding This work was supported by the National Natural Science Foundation of China [grant numbers 11664017]; the Natural Science Foundation of Jiangxi Province [grant number 20161BAB202052].
Fig. 6. Influence of changing temperature on the power conversion efficiency. 1115
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References [1] M.P. Fischer, J. Bühler, G. Fitzky, T. Kurihara, S. Eggert, A. Leitenstorfer, D. Brida, Coherent field transients below 15 THz from phase-matched difference frequency generation in 4H-SiC, Opt. Lett. 42 (2017) 2687. [2] A. Hamadou, J.L. Thobel, S. Lamari, Dynamic modeling of a terahertz quantum cascade laser based on difference frequency generation, Optik 156 (2017). [3] A.A. Boyko, P.G. Schunemann, S. Guha, N.Y. Kostyukova, D.B. Kolker, V.L. Panyutin, G.M. Marchev, V. Pasiskevicius, A. Zukauskas, F. Mayorov, Optical parametric oscillator pumped at ∼1 μm with intracavity mid-IR difference-frequency generation in OPGaAs, Opt. Mater. Express 8 (2018) 549. [4] Y. Lu, X. Wang, L. Miao, D. Zuo, Z. Cheng, Efficient and widely step-tunable terahertz generation with a dual-wavelength CO2 laser, Appl. Phys. B 103 (2010) 387–390. [5] Z. Rao, X. Wang, Y. Lu, Tunable terahertz generation from one CO2 laser in a GaSe crystal, Opt. Commun. 284 (2011) 5472–5474. [6] J.A. Armstrong, N. Bloembergen, J. Ducuing, P.S. Pershan, Interactions between light waves in a nonlinear dielectric, Phys. Rev. 127 (1962) 1918–1939. [7] E. Rosencher, B. Vinter, V. Berger, Second‐harmonic generation in nonbirefringent semiconductor optical microcavities, J. Appl. Phys. 78 (1995) 6042–6045. [8] V. Berger, Second-harmonic generation in monolithic cavities, J. Opt. Soc. Am. B 14 (1997) 1351. [9] R. Haidar, N. Forget, E. Rosencher, Optical parametric oscillation in microcavities based on isotropic semiconductors: a theoretical study, IEEE J. Quantum Electron. 39 (2003) 569–576. [10] Z.D. Xie, X.J. Lv, Y.H. Liu, W. Ling, Z.L. Wang, Y.X. Fan, S.N. Zhu, Cavity phase matching via an optical parametric oscillator consisting of a dielectric nonlinear crystal sheet, Phys. Rev. Lett. 106 (2011) 083901. [11] H.B. Lin, S.F. Li, Y.W. Sun, G. Zhao, X.P. Hu, X.J. Lv, S.N. Zhu, High-performance cavity-phase matching by pump reflection, Opt. Lett. 38 (2013) 1957–1959. [12] Y. Tao, S. Goh, H. Bastiaens, Pvd. Slot, S. Biedron, S. Milton, K. Boller, Temporal model for quasi-phase matching in high-order harmonic generation, Opt. Express 25 (2017) 3621. [13] K. Saito, T. Tanabe, Y. Oyama, Pump enhanced monochromatic terahertz-wave parametric oscillator toward megawatt peak power, Opt. Lett. 39 (2014) 5681–5684. [14] K. Saito, T. Tanabe, Y. Oyama, Cascaded terahertz-wave generation efficiency in excess of the Manley–Rowe limit using a cavity phase-matched optical parametric oscillator, J. Opt. Soc. Am. B 32 (2015) 617. [15] J.E. Schaar, K.L. Vodopyanov, P.S. Kuo, M.M. Fejer, X. Yu, A. Lin, J.S. Harris, D. Bliss, C. Lynch, V.G. Kozlov, W. Hurlbut, Terahertz sources based on intracavity parametric down-conversion in quasi-phase-matched gallium arsenide, IEEE J. Sel. Top. Quant. 14 (2008) 354–362. [16] S. Lei, Y. Yao, Z. Li, T. Yu, Z. Zou, Design and theoretical analysis of resonant cavity for second-harmonic generation with high efficiency, Appl. Phys. Lett. 98 (2011) 031102. [17] R.W. Boyd, Nonlinear Optics, Academic Press, New York, 2008. [18] Z.M. Rao, X.B. Wang, Y.Z. Lu, D.L. Zuo, T. Wu, Two schemes for generating efficient terahertz waves in nonlinear optical crystals with a mid-infrared CO2 laser, Chin. Phys. Lett. 28 (2011) 074215. [19] T. Skauli, P.S. Kuo, K.L. Vodopyanov, T.J. Pinguet, O. Levi, L.A. Eyres, J.S. Harris, M.M. Fejer, B. Gerard, L. Becouarn, E. Lallier, Improved dispersion relations for GaAs and applications to nonlinear optics, J. Appl. Phys. 94 (2003) 6447–6455.
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Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Theoretical analysis of difference frequency generation for terahertz generation in a sheet microcavity from the CO2 laser Shijia Zeng, Fangsen Xie, Zhiming Rao
T
⁎
College of Physics and Communication Electronics, Jiangxi Normal University, 99 Ziyang Road, Nanchang, 330022, China
A R T IC LE I N F O
ABS TRA CT
Keywords: Terahertz generation Difference frequency generation Cavity phase matching Nonlinear optics
A scheme for efficient terahertz generation in a GaAs sheet microcavity from the CO2 laser is presented, in which the process of difference frequency generation that considered cavity phase matching is derived theoretically. This model achieves its optimum working condition when the cavity length corresponds to a coherent length and the resonant condition of pump wave is satisfied. The results show that the efficiency can realize an effective improvement when the CO2 lasers pump wavelengths are 9.55 μm and 9.79 μm, respectively, with a peak power conversion efficiency of 2.04%, which corresponds to a photon conversion efficiency of 83%. This work is expected to provide a theoretical basis for relevant experiments involving doubly and even triply resonant structures.
1. Introduction Difference frequency generation (DFG) has become an important technology for the generation of terahertz (THz) wave in recent years due to its many advantages such as high power, continuous tuning, reduced structure and no threshold condition [1–3]. An efficient THz generation may be realized using the CO2 laser. According to the Manley-Rowe relation, the maximum conversion efficiency can be improved by one order of magnitude with longer wavelength running at 10 μm [4,5]. The key to the DFG is compensating the phase mismatch in a consistent manner, which can be realized using cavity phase matching (CPM) by resonance recirculation in microcavities, such as Fabry-Perot (F–P) microcavities, because the traveling wave and reflected wave are exactly in phase for every circling. This method was first predicted in 1962 by Armstrong et al. [6] followed by some relevant theoretical studies [7–9], but the method was not demonstrated experimentally until 2011 by Xie et al. [10]. In 2013, Lin et al. [11] experimentally realized a higher conversion efficiency by reflecting the pump beam in a KTiOPO4 crystal. The principle for CPM is similar to the well-known quasi phase matching (QPM) [12] method, with the two methods differing in that the CPM does not require reversal of the orientation for the nonlinear polarization and superimposed several crystals to extend the length of the nonlinear interaction, which simplifies the crystal fabrication, makes the structure more compact and reduces costs. Currently, Saito et al. [13] realized an estimated power of 4 MW at 3 THz using optical parametric oscillation, in which the pulse duration and cavity configuration in a GaP sheet cavity was optimized in theory. Soon after, Saito et al. [14] proposed a theoretical scheme that both the cascaded effect and CPM are considered, and generated an output peak power of 1.8 WM with a photon conversion efficiency of 1.086 at 3 THz, which exceeds the Manley- Rowe limit [15]. In this paper, a scheme for DFG based on CPM is proposed using the structure of sheet microcavity, in which the crystal absorption for a THz wave is greatly reduced. By considering both of the standing wave condition for PW and the extremum condition for the
⁎
Corresponding author. E-mail address: [email protected] (Z. Rao).
https://doi.org/10.1016/j.ijleo.2018.07.130 Received 30 May 2018; Accepted 27 July 2018 0030-4026/ © 2018 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 172 (2018) 1111–1116
S. Zeng et al.
Fig. 1. Schematic structure of terahertz generation by cascaded DFG based on CPM.
THz wave generated by DFG, the optimum cavity length is determined. Meanwhile, the influence of temperature on this model is considered. Finally, the efficiencies for DFG based on CPM and normally DFG are compared. 2. Theory As illustrated in Fig.1, the THz wave is generated in the cavity via DFG (ωT = ω1−ω2) using two collinear pump wave (PW). For the continuous nonlinear interactions, a resonant cavity for PW without the THz wave is considered, in which an undepleted pump approximation can be applied and a higher efficiency can be achieved by reducing the absorption of nonlinear crystal at the THz wave in the cavity [14,16]. The cavity contains a nonlinear medium (M) with a length of L and two coatings C1 and C2 on both parallel sides. The two beams in the primary PW are PW1 with amplitude A01 and PW2 with amplitude A02 , which are normally incident from the C1 end. The reflectances of C1 for PW1 and PW2 are R1 and R2 , respectively. The reflected wave amplitudes for PW1 and PW2 are R1 A01 and R2 A02 , respectively. The amplitudes of PW1 and PW2 inside the cavity are A1 and A2 , respectively. The reflectances of C2 for PW1 and PW2 should be near 100% to avoid PW output from C2; therefore, we can consider approximately A1, l = −A1, r and A2, l = −A2, r , where r and l indicate propagation of the PW to C2 and C1, respectively. A1, t and A2, t are the amplitudes of PW1 and PW2 transmitted outside of the cavity. Therefore, A1, t = (T1 n1/ n 01 )1/2A1, l and A2, t = (T2 n2/ n 02 )1/2A2, l , where T1 and T2 are the transmittance of the C1, and n1, n2 are the refractive indexes of M for PW1 and PW2, respectively. n 01 and n 02 are, respectively, the refractive indexes of PW1 and PW2 in air, respectively. The effect of coating on PW is not considered because the thickness of the coating is ignored for an ideal model. To prevent leakage of the pump energy from the cavity, the destructive interference [16] between A1, t , A2, t and R1 A 01, R2 A 02 can be considered in the following form
R1 A01 + A1, t = 0,
(1)
R2 A02 + A2, t = 0.
(2)
The reflectance for the THz wave that generated by DFG is close to 100% and RT at C1 and C2, respectively. RT should be sufficiently small to ensure the THz wave can be transmitted outside from the C2 and reduce the loss in the cavity. ATL and ATR are the amplitudes of THz wave that propagates to C1 and C2, respectively, in the cavity. The amplitude of the THz wave that is transmitted to the outside is AT , with AT = (TT nT / n 03 )1/2ATR , where TT , nT and n 03 are the transmittance of the C2 and the refractive indices for M and air for the THz wave, respectively. The expression for the THz electric field generated by DFG must be obtained before considering the cascading effect. According to the Maxwell equation, the THz electric field intensity ET inside the cavity is given by [7,16]
∂2ET ∂2 + (kT )2ET = μ0 2 (2ε0 deff E1 E2 *), ∂x 2 ∂t
(3)
where kT = 2πnT / λT is the wave vector of THz wave and E1, E2 are the electric field intensities for the PW1 and PW2, respectively, inside the cavity. The effective nonlinear optical coefficient is
πl deff = d sinc ⎛ cav ⎞ ⎝ 2lcoh ⎠ ⎜
⎟
(4)
where d is the nonlinear coefficient of the medium M, lcav is the cavity length and lcoh is the coherence length. Here, we consider lcav ≈ L for an ideal model. The solution of Eq. (3) without crystal absorption is given by
ET = AT 1 exp (ikT x ) + UAB exp (ikB x ) + AT 2 exp (−ikT x ) + UAB exp(−ikB x ),
U=−
2μ0 ε0 ωT
2d
(5)
eff
kT 2−kB 2
(6)
where kB = k1−k2, k1, k2 are the wave vector of PW1 and PW2, respectively, and AB = A1 A2 . The amplitude for the THz wave inside the 1112
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cavity are given by
ATR = AT 1 exp (ikT x ) + UAB exp (ikB x ),
(7)
ATL = AT 2 exp(−ikT x ) + UAB exp(−ikB x ).
(8)
The boundary condition at C1 and C2 for the THz wave in the cavity are given by Eqs. (9) and (10), respectively. (9)
AT 1 + U AB = −(AT 2 + U AB) AT 2 exp(−ikT L) + UAB exp(−ikB L) = − RT (AT 1 exp (ikT L) + UAB exp (ikB L)).
(10)
According to these boundary conditions, expressions for AT1 and AT2 in terms of the amplitude of the PW can be obtained. Combining Eqs. (7), (9) and (10) with considering the standing wave condition for THz wave, we obtain (11)
|ATR | = |UAB Z|,
Z=
2 exp (iΔKL)−exp (i2ΔKL)−1 RT −exp (i2ΔKL)
(12)
where ΔK = k1−k2−kT . According to the formula derived above, it is enough to discuss the strength of ATR about ΔKL when both the PW and the orders of cascading effect are determined, which indicates that UAB can be viewed as a constant according to Eq. (13). For the first order, ATR as a function of ΔKL calculate using Eqs. (11) and (12) with different RT is shown in Fig. 2. ATR achieves a maximum when ΔKL = (2m + 1) π (m is an integer) because the three-wave coupling propagates the length L with a phase mismatch of (2m + 1) π , which reaches a maximum in the cavity via the second-order nonlinear conversion effects, and the PW is reflected by C1 or C2 results in a phase shift of π such that the nonlinear conversion effects can be continued. Meanwhile, a larger RT leads to stronger AT because of the resonance of the THz wave in the cavity. 3. Numerical simulations According to the above discussion, the cavity length should synchronously meets two conditions for a steady nonlinear effect, one is the standing wave condition of PW wave and the other is the phase condition which make the amplitude of the THz wave to achieve a maximum. It is similar to QPM that the conversion efficiency of CPM can achieve a maximum at first-order phase matching (ΔKL = π ). Therefore, the optimum cavity length L can be calculated using the following expression:
k1 L = pπ,
(13)
k2 L = qπ
(14)
ΔKL = π .
(15)
Where p and q are integer when the standing wave condition of PW is satisfied. Out of practical consideration, we consider the CO2 lasers lines, with 〈111〉 GaAs (= 10−10 m/V [17]) as the working medium; two lines that can approximately satisfy Eqs. (13)–(15) are 9.5524 μm [9P(20)] and 9.7938 μm [9P(46)], with a resulting THz wavelength of 387.65 μm and a cavity length of 756.6 μm . The value of p = 518.995 and q = 506.010 can be also calculated using Eqs. (13)–(15), with the result of ΔKL/ π ≈ 1. The distance between p and the integer near p is less than 0.1, and q has the similar situation. Therefore, a doubly resonant model for the PW can be applied approximately. Moreover, there is kT L = 13.98π , which indicates that a potential triply resonant model can be applied. It is sufficient to calculate the efficiency of difference frequency generation for this model using PT = |AT |2 nT A/2Z0 and ηT = PT / P1 when the PW wavelength and cavity length are determined. Where Z0 and A are the vacuum impedance and facula area, respectively, and we can obtain
Fig. 2. Amplitude of the THz wave ATR as a function of ΔKL with different RT . 1113
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Fig. 3. Power conversion efficiency for various λ2 with different R when λ1 = 9.5524 μm .
ηT = R
R=
8(2π ) 4deff 2L4 nT P2 ε0 cλT 4n1 n2 (2n
+
1)2A
sin4
(
1 ΔKL 2
1 ( 2 ΔKL) 4
), (16)
R1R2 . (1−R1)(1−R2)( RT −1)2
(17)
Where n = 14 because kT L ≈ 14π , and λT is the THz wavelength. The PW reflectance will be approximated as R1 = R2 = R 0 for convenience of discussion. Here, it is absence of the crystal absorption for THz wave by assume RT = 0 , which can avoid the loss of THz wave in the cavity, meanwhile, it is better to meet the design of this model. For a pump power P1 = P2 = 100 kW/cm2 and PW1 wavelength λ1= 9.5524 μm [9P(20)], the power conversion efficiency for THz generation by DFG based on CPM varied with PW2 wavelength can be calculated, as shown in Fig.3 with different R. The efficiency of DFG is increasing gradually with the increase of reflectance because of stronger resonance for PW. This picture also shows that the efficiency can reach a maximum value of 2.04% when PW2 wavelength λ2= 9.7938 μm [9P(46)] and R = 0.99. The growth of efficiency cannot be achieved by increasing the cavity length L because L is restricted by the phase conditions of resonance and nonlinear interaction, which can be illustrated by Fig.4. It is somewhat different from QPM, whose efficiency can be increased directly by superimposing the periodically poled crystal that achieve a larger length of nonlinear interaction. Efficiency of normally DFG without CPM can be expressed by the following form [18]
η=
2 2 32deff L P2 . ε0 cn1 n2 nT λT2 A
(18)
It is assumed that the conditions of normally DFG are the same as DFG that based on CPM, which means the crystal length is equal to the coherent length. The numerical simulations show that the order of magnitude of DFG efficiency is only 10−6 %, which is far less than the efficiency of DFG that based on CPM. It shows that the efficiency of DFG can be improved effectively in this model, which is studied in this paper. For practical applications, the influence of changing temperature on this model (without the influence of thermal expansion coefficient) should be considered. The refractive index of GaAs is given by [19]
Fig. 4. Power conversion efficiency varied with cavity length. 1114
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Fig. 5. Effect of temperature on the optimum cavity length and the degree of nonlinear interaction between the PW and THz.
n2 (λ ) = b +
g1 b1−2−λ−2
+
g2 b2−2−λ−2
+
g3 b3−2−λ−2
(19)
10−7ΔT 2 ,
ΔT−4.882 × b1 (μm) = 0.4431307 + 0.000050564ΔT , b2 (μm)= 0.8746453 + 0.0001913 where b = 5.382514, b3 (μm) = 36.9166−0.011622ΔT , g1 = 27.83972 , g2 = 0.031764 + 4.35 × 10−5ΔT + 4.664 × 10−7ΔT 2 , g3 = 0.00143436, and ΔT = T −22 °C indicates the deviation of the actual temperature to the room temperature as used in the calculations above. For a selected PW wavelength, the effect of temperature on the optimum cavity length should be considered at first, which can be illustrated in Fig. 5 using Eq. (13). In this picture, the deviation to the optimum cavity of 756.6 μm will reach 13 μm when the temperature changes from 22 °C to 300 °C. It also shows that the degree of nonlinear interaction between the PW and THz wave decreases gradually due to the increased temperature. Therefore, the efficiency of DFG may be affected to some extent by the varied temperature, as shown in Fig. 6. The changes of efficiency from 2.04% to 1.6% approximately with a varied temperature from 0 °C to 300 °C when L = 756.6 μm in this picture. It also shows that the efficiency is approximately unaffected by temperature when temperature is near 22 °C, which can be applied in a normal temperature environment for a small THz source. 4. Conclusion In conclusion, a scheme of difference frequency generation based on cavity phase matching with the structure of a sheet microcavity is proposed. In particular, the conditions for both the resonance for pump wave and nonlinear interaction between pump and THz are discussed. The results show that the optimum working condition can be approximately satisfied when the wavelengths of the two laser beams and cavity length are 9.5524 μm [9P(20)], 9.7938 μm [9 P(46)] and 756.6 μm , respectively; these conditions lead to a generated THz wavelength of 387.65 μm (0.77 THz) using GaAs as working medium. The nonlinear interaction among the pump and THz wave can be effective enhanced by contrasting with normally difference frequency generation, with a peak power conversion efficiency of 2.04%, corresponding to a photon conversion efficiency of 83%. We believe that our results are anticipated to play a positive role in such experiments involving doubly even triply resonant structures. Meanwhile, what can be expected is that the realization of small THz source will contribute to the development of THz science. Funding This work was supported by the National Natural Science Foundation of China [grant numbers 11664017]; the Natural Science Foundation of Jiangxi Province [grant number 20161BAB202052].
Fig. 6. Influence of changing temperature on the power conversion efficiency. 1115
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