Influence of Kerr Effect on Second-Order Nonlinearity Induced Transparency

Zhiqiang ZHANG; Yanhui ZHOU

doi:10.1051/wujns/2024291067

Open Access

Issue		Wuhan Univ. J. Nat. Sci. Volume 29, Number 1, February 2024


Page(s)		67 - 73
DOI		https://doi.org/10.1051/wujns/2024291067
Published online		15 March 2024

Wuhan University Journal of Natural Sciences, 2024, Vol.29 No.1, 67-73

Physics

CLC number: O431.2

Influence of Kerr Effect on Second-Order Nonlinearity Induced Transparency

Zhiqiang ZHANG¹ and Yanhui ZHOU²

¹ College of Architectural Engineering, Zhengzhou Business University, Zhengzhou 451200, Henan, China
² Quantum Information Research Center, Shangrao Normal University, Shangrao 334001, Jiangxi, China

Received: 9 May 2023

Abstract

We theoretically study the effect of Kerr effect on the second-order nonlinearity induced transparency in a double-resonant optical cavity system. We show that in the presence of the Kerr effect, as the strength of the Kerr effect increases, the absorption curve exhibits an asymmetric-symmetric-asymmetric transition, and the zero absorption point shifts with the increase of the Kerr effect. Furthermore, by changing the strength of the Kerr effect, we can control the width of the transparent window, and the position of the zero-absorption point and meanwhile change the left and right width of the absorption peak. The asymmetry absorption curve can be employed to improve the quality factor of the cavity when the frequency detuning is tuned to be around the right peak. The simple dependence of the zero-absorption point on the strength of Kerr effect suggests that the strength of Kerr effect can be measured by measuring the position of the zero-absorption point in a possible application.

Key words: Kerr effect / second-order nonlinearity induced transparency / nonlinear optics / quantum optics

Cite this article: ZHANG Zhiqiang, ZHOU Yanhui. Influence of Kerr Effect on Second-Order Nonlinearity Induced Transparency[J]. Wuhan Univ J of Nat Sci, 2024, 29(1): 67-73.

Biography: ZHANG Zhiqiang, male, Associate professor, research direction: computational physics and nonlinear physics. E-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.

Fundation item: Supported by the Key Scientific Research Plan of Colleges and Universities in Henan Province (23B140006), and the National Natural Science Foundation of China (11965017)

© Wuhan University 2023

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

0 Introduction

Electromagnetically induced transparency (EIT)^[1] is a technique for turning an opaque medium into a transparent one as a result of quantum interference between two excited processes from independent states. Since it is proposed in the context of atomic systems, the first experimental realization of EIT is also in atomic systems^{[2, 3]}. Recently, Weis et al^[4] realized induced transparency in an optomechanical system for controlling the transport of light through the cavity by coupling a cavity with a light pulse. This technology is called optomechanically induced transparency (OMIT)^[5-8]. Therefore, searching for possible induced transparent systems has become an active research topic in quantum optics^{[9, 10]}.

With recent technological advances, second-order nonlinearities have become a hot research area in modern quantum optics and quantum information processing. Second-order nonlinearity can be achieved with high- $χ^{(2)}$ Mathematical equation nonlinear materials such as III-V semiconductors (e.g., GaAs, GaN, GaP, etc.)^{[11, 12]}. In 2017, Zhou et al^[13] theoretically predicted another form of induced transparency caused by second-order nonlinearity, which is called second-order nonlinearity induced transparency (SONIT). They discussed the impact of second-order nonlinearity on SONIT and found that, unlike EIT and OMIT, the absorption spectrum of SOINT exhibits asymmetric features due to the presence of photon asymmetric switching induced by second-order nonlinearity, and a potential application is the measurement of second-order nonlinear coefficients.

In particular, it should be pointed out that introduction of Kerr medium in the optomechanical system can not only enhance the optomechanical coupling, but also prevent the suppression of nonlinearity^[14], which makes the system more excellent in multiple dimensions^[15-17]. Jiao et al^[18] found that the frequency shift induced by the Kerr effect leads to an asymmetric OMIT signal absorption window, which may enhance the performance of OMIT devices with various nonlinear optical cavities. Xiong et al^[15] introduced the cross-Kerr effect in the optomechanical system, and found that the cross-Kerr effect leads to the asymmetry of the absorption curve, which can be used to design a cavity with a higher quality factor.

Inspired by these studies, we introduce the Kerr medium in the double-resonant optical cavity system to study the influence of the Kerr effect on SONIT. Firstly, the Hamiltonian of the SOINT system including the Kerr effect is given. Secondly, the dispersion and absorption coefficients of the SOINT system are solved by analytical methods. Finally, the influence of the Kerr effect on SOINT is discussed. We find that the absorption curve exhibits an asymmetric-symmetric-asymmetric transition, and the zero absorption point shifts as the Kerr effect increases. The width of the transparent window and the position of the zero-absorption point can be controlled by changing the strength of the Kerr effect. Possible applications of these results are to improve the quality factor of the cavity and to measure the strength of the Kerr effect.

The structure of this paper is as follows. In Section 1, the theoretical model is introduced, the absorption coefficient and dispersion coefficient of the system is obtained by theoretical methods. In Section 2, the influence of the Kerr effect on the SOINT, the position of the zero-absorption point and the width of the transparent window are discussed. In Section 3, a short summary is given.

1 Model and Theoretically Analysis

We consider two single-mode cavities A and B, coupled through a second-order nonlinear medium $χ^{(2)}$ Mathematical equation , where the second-order nonlinear medium $χ^{(2)}$ mediates the conversion of two photons in cavity mode A into one photon in cavity mode B. The frequencies of the two cavity modes are $ω_{A}$ and $ω_{B}$ , respectively, and we have $ω_{B} = 2 ω_{A}$ . The scheme of a doubly resonant cavity is shown in Fig. 1.

Fig. 1

Scheme of a doubly resonant cavity mode of $χ^{(2)}$ Mathematical equation nonlinearity material

$E_{L}$ Mathematical equation is the driving strength of strong control light, the frequency is $ω_{L}$ , and $E_{P}$ is the driving strength of weak probe light, and the frequency is $ω_{P}$

Considering the Kerr effect coefficient of the medium is $μ$ Mathematical equation , the Hamiltonian of the system can be written as

$\begin{array}{l} \hat{H} = ω_{A} {\hat{a}}^{†} \hat{a} + ω_{B} {\hat{b}}^{†} \hat{b} + g [{({\hat{a}}^{†})}^{2} \hat{b} + {\hat{b}}^{†} {\hat{a}}^{2}] \\ + μ {\hat{a}}^{†} {\hat{a}}^{†} \hat{a} \hat{a} + i [E_{L} {\hat{a}}^{†} e^{- i ω_{L} t} + E_{P} {\hat{b}}^{†} e^{- i ω_{P} t} + H . c .] \end{array}$ Mathematical equation (1)

where ${\hat{a}}^{†} (\hat{a})$ Mathematical equation denotes the creation (annihilation) operator of cavity mode A, ${\hat{b}}^{†} (\hat{b})$ denotes the creation (annihilation) operator of cavity mode B, $g$ denotes the coefficient of second-order nonlinear interactions, which can be calculated directly from $χ^{(2)}$ ^[13], $μ$ Mathematical equation denotes the Kerr effect coefficient of the medium. The last term of Eq. (1) represents the two drives of the external light to the cavity, one is the strong control light, the intensity is $E_{L}$ , the frequency is $ω_{L}$ , and the other is the weak probe light, the intensity is $E_{P}$ Mathematical equation , and the frequency is $ω_{P}$ .

To study the dynamics of the system, we want to turn to a rotating framework dominated by operators $\hat{U} (t) = e x p [i ω_{L} t ({\hat{a}}^{†} \hat{a} + 2 {\hat{b}}^{†} \hat{b})]$ Mathematical equation , then we get an effective Hamiltonian ${\hat{H}}_{e f f} = \hat{U} \hat{H} {\hat{U}}^{†} - i \hat{U} \frac{d {\hat{U}}^{†}}{d t}$ . The effective Hamiltonian can be written as follows:

$\begin{array}{l} {\hat{H}}_{e f f} = Δ {\hat{a}}^{†} \hat{a} + 2 Δ {\hat{b}}^{†} \hat{b} + g [{({\hat{a}}^{†})}^{2} \hat{b} + {\hat{a}}^{2} {\hat{b}}^{†}] + μ {({\hat{a}}^{†})}^{2} {(\hat{a})}^{2} \\ + i (E_{L} {\hat{a}}^{†} + E_{P} e^{- i Ω t} {\hat{b}}^{†} + H . c .) \end{array}$ Mathematical equation (2)

where $Δ = ω_{A} - ω_{L}$ Mathematical equation , $Ω = ω_{P} - 2 ω_{L}$ .

Using the Heisenberg equations $\frac{\partial \hat{a}}{\partial t} = \frac{1}{i ℏ} [a, H_{e f f}]$ Mathematical equation for the cavity variables, here $a = U \hat{a} U^{†}$ . For convenience, we set $ℏ = 1$ , then we get

${\begin{array}{l} \frac{\partial \hat{a}}{\partial t} = - i [Δ \hat{a} + 2 g {\hat{a}}^{†} \hat{b} + 2 μ {\hat{a}}^{†} {(\hat{a})}^{2} + i E_{L}] - \frac{γ_{a}}{2} \hat{a} + {\hat{a}}_{i n} \\ \frac{\partial \hat{b}}{\partial t} = - i (2 Δ \hat{b} + g {\hat{a}}^{2} + i E_{P} e^{- i Ω t}) - \frac{γ_{b}}{2} \hat{b} + {\hat{b}}_{i n} \end{array}$ Mathematical equation (3)

where $γ_{a}$ Mathematical equation and $γ_{b}$ are decay rates of cavity mode A and cavity mode B, respectively, ${\hat{a}}_{i n}$ and ${\hat{b}}_{i n}$ are the corresponding input fields. It is important to point out that the input field has a mean value of zero, that is $〈 {\hat{a}}_{i n} 〉 = 0$ , and $〈 {\hat{b}}_{i n} 〉 = 0$ Mathematical equation .

We take the mean values of Eq. (3), and take the notation $〈 \hat{a} 〉 = a$ Mathematical equation , $〈 {\hat{a}}^{†} 〉 = a^{*}$ , and $〈 \hat{b} 〉 = b$ , here the asterisk "^*" represents complex conjugation. Then Eq. (3) can be rewritten as

${\begin{array}{l} \frac{\partial a}{\partial t} = - i [Δ a + 2 g a^{*} b + 2 μ a^{2} a^{*} + i E_{L}] - \frac{γ_{a}}{2} a \\ \frac{\partial b}{\partial t} = - i (2 Δ b + g a^{2} + i E_{P} e^{- i Ω t}) - \frac{γ_{b}}{2} b \end{array}$ Mathematical equation (4)

Then we apply the linearization approach by assuming that each operator mean value in the system can be written as a sum of its steady-state mean value and deviations from the stationary state, i.e., $a = α + δ a$ Mathematical equation and $b = β + δ b$ . We should note that both $δ a$ and $δ b$ are small quantity. Considering the control field is much stronger than the probe field, which is $E_{L} > > E_{P}$ , $E_{P}$ is regarded as a small quantity, too.

The steady-state mean values of the system can be obtained from Eq. (4) as

${\begin{array}{l} i Δ α + i 2 g α^{*} β + i 2 μ α^{2} α^{*} - E_{L} + \frac{γ_{a}}{2} α = 0 \\ i 2 Δ β + i g α^{2} + \frac{γ_{b}}{2} β = 0 \end{array}$ Mathematical equation (5)

when the high-order small quantity is ignored, the deviations from the stationary state can be obtained as

${\begin{array}{l} \frac{\partial (δ a)}{\partial t} = - i Δ δ a - i 2 g (α^{*} δ b + β δ a^{*}) \\ - i μ 2 α (2 α^{*} δ a - α δ a^{*}) - \frac{γ_{a}}{2} δ a \\ \frac{\partial (δ b)}{\partial t} = - i 2 Δ δ b - i 2 g α δ a + E_{P} e^{- i Ω t} - \frac{γ_{b}}{2} δ b \end{array}$ Mathematical equation (6)

For Eq. (5), it is assumed that the solution can be expressed as $α = A_{1} + i A_{2}$ Mathematical equation and $β = B_{1} + i B_{2}$ , where $A_{1}$ , $A_{2}$ , $B_{1}$ , and $B_{2}$ are real numbers, and substitute them into Eq. (5), then separate the real and imaginary part, we can get

${\begin{array}{l} Δ A_{1} + 2 g A_{1} B_{1} + 2 g A_{2} B_{2} + 2 μ A_{1}^{3} + 2 μ A_{1} A_{2}^{2} + \frac{γ_{a}}{2} A_{2} = 0 \\ Δ A_{2} + 2 g A_{1} B_{2} - 2 g A_{2} B_{1} + 2 μ A_{1}^{2} A_{2} + 2 μ A_{2}^{3} + E_{L} - \frac{γ_{a}}{2} A_{1} = 0 \\ 2 Δ B_{1} + g A_{1}^{2} - g A_{2}^{2} + \frac{γ_{b}}{2} B_{2} = 0 \\ 2 Δ B_{2} + g 2 A_{1} A_{2} - \frac{γ_{b}}{2} B_{1} = 0 \end{array}$ Mathematical equation (7)

To our knowledge, the analytical solution of Eq. (7) is difficult to obtain, we can solve it numerically.

For Eq. (6), we suppose the solutions take the form of

${\begin{array}{l} δ a = A^{+} E_{P} e^{- i Ω t} + A^{-} E_{P}^{*} e^{i Ω t} \\ δ a^{*} = A^{+ *} E_{P}^{*} e^{i Ω t} + A^{- *} E_{P} e^{- i Ω t} \\ δ b = B^{+} E_{P} e^{- i Ω t} + B^{-} E_{P}^{*} e^{i Ω t} \\ δ b^{*} = B^{+ *} E_{P}^{*} e^{i Ω t} + B^{- *} E_{P} e^{- i Ω t} \end{array}$ Mathematical equation (8)

By substituting Eq. (8) into Eq. (6), we obtain

${\begin{array}{l} (\frac{γ_{a}}{2} - i Ω + i Δ + i 4 μ α α^{*}) A^{+} + i 2 g α^{*} B^{+} \\ + (i 2 g β + i 2 μ α^{2}) A^{- *} = 0 \\ (\frac{γ_{a}}{2} + i Ω + i Δ + i 4 μ α α^{*}) A^{-} + i 2 g α^{*} B^{-} \\ + (i 2 g β + i 2 μ α^{2}) A^{+ *} = 0 \\ (\frac{γ_{b}}{2} - i Ω + i 2 Δ) B^{+} + i 2 g α A^{+} - 1 = 0 \\ (\frac{γ_{b}}{2} + i Ω + i 2 Δ) B^{-} + i 2 g α A^{-} = 0 \end{array}$ Mathematical equation (9)

The transmission rate of the probe field depends only on $B^{+}$ Mathematical equation . By solving Eq. (9), we get

$B^{+} = \frac{1}{f_{3}} - \frac{f_{5} f_{7} {(f_{7} f_{5} - f_{2} f_{4})}^{*}}{f_{3} [(f_{5} f_{7} - f_{1} f_{3}) {(f_{5} f_{7} - f_{2} f_{4})}^{*} - f_{3} f_{6} {(f_{4} f_{6})}^{*}]}$ Mathematical equation (10)

where $f_{1} = \frac{γ_{a}}{2} - i Ω + i Δ + i 4 μ α α^{*}$ Mathematical equation , $f_{2} = \frac{γ_{a}}{2} + i Ω + i Δ + i 4 μ α α^{*}$ , $f_{3} = \frac{γ_{b}}{2} - i Ω + i 2 Δ$ , $f_{4} = \frac{γ_{b}}{2} + i Ω + i 2 Δ$ , $f_{5} = i 2 g α^{*}$ , $f_{6} = i 2 g β + i 2 μ α^{2}$ , and $f_{7} = i 2 g α$ .

To simplify the discussion, we can consider that the output of probe light in the cavity is equal to the input probe light minus the light dissipated in the cavity. Applying the input-output relationship of the cavity^{[4, 19]}, we can get the output field

$b_{o u t} = b_{i n} - \sqrt[]{γ_{b}} b = E_{P} e^{- i Ω t} - \sqrt[]{γ_{b}} (β + B^{+} E_{P} e^{- i Ω t} + B^{-} E_{P}^{*} e^{i Ω t})$ Mathematical equation (11)

Here, we refer to the probe transmission in the general sense of the ratio of the probe field returned from the system divided by the sent probe field. The transmission rate of the probe field is given by^{[4, 19]}

$t_{P} = \frac{E_{P} - \sqrt[]{γ_{b}} E_{P} B^{+}}{E_{P}} = 1 - \sqrt[]{γ_{b}} B^{+}$ Mathematical equation (12)

which could be measured by the homodyne technique^[20]. The corresponding quadrature field can be defined as^[21]

$ε_{T} = \sqrt[]{γ_{b}} B^{+}$ Mathematical equation (13)

Its real part $R e [ε_{T}]$ Mathematical equation is the absorption coefficient, and the imaginary part $I m [ε_{T}]$ is the dispersion coefficient.

2 Results and Discussions

We have learned that, as the Kerr coefficient for third-order nonlinear effects in nonlinear materials^[22-28], it is usually a small quantity. In order to discuss its effect on second-order nonlinearity induced transparency, we use available experimental data to estimate the magnitude of the Kerr nonlinearity coefficient. The Kerr coefficient can be expressed as $μ = \frac{3 {(ℏ ω_{0})}^{2}}{4 ε_{0} V_{e f f}} \frac{χ^{(3)}}{ε_{r}^{2}}$ Mathematical equation ^[29], where $V_{e f f}$ is the effective cavity mode volume, $ω_{0}$ is the frequency of the optical mode, and $ε_{r}$ is the relative dielectric permittivity.

The highly nonlinear materials currently used in experiments mainly include silicon (Si), germanium (Ge), silicon dioxide (SiO₂) or gallium arsenide (GaAs)^[30]. Typical orders of magnitude for $χ^{(3)}$ Mathematical equation of these materials are in the range of $4 \times 10^{- 19} - 4 \times 10^{- 18}$ m²/V². In addition, larger $χ^{(3)}$ values can be found in certain nanoparticles-doped glasses, chalcogenide glasses, or other polymeric materials^{[31, 32]}. For a typical parameter setting, such as the data used in Ref. [29], $ℏ ω_{0} = 1 e V$ Mathematical equation , $V_{e f f} ≃ 0.01 μ m^{3}$ , $ε_{r} = 2.0$ , $χ^{(3)} = 4 \times 10^{- 18}$ m²/V², then the Kerr coefficient is about $μ = 3.4 \times 10^{- 7} e V$ . As a typical value for the decay rate of cavity mode A, taking $γ_{a} = 10^{9} s^{- 1}$ , we get $\frac{μ}{ℏ γ_{a}} = 1.04$ Mathematical equation . Considering that the smaller the value of $μ$ is, the better it can be realized experimentally, we set the value of $μ$ in the range of 0- $0.25 γ_{a}$ in the follow-up discussion of this article.

Now, we can discuss the influence of Kerr effect on the second-order nonlinearity induced transparency. We plot absorption coefficient $R e [ε_{T}]$ Mathematical equation and dispersion coefficient $I m [ε_{T}]$ as a function of frequency detuning $Ω / γ_{b}$ with different values of Kerr effect coefficient $μ$ , as shown in Fig. 2.

Fig. 2

The absorption coefficient $R e [ε_{T}]$ Mathematical equation and dispersion coefficient $I m [ε_{T}]$ as a function of $Ω / γ_{b}$ with different Kerr effect coefficient $μ$

(a) $μ = 0$ Mathematical equation , (b) $μ = 0.05 γ_{a}$ , (c) $μ = 0.1 γ_{a}$ , (d) $μ = 0.15 γ_{a}$ , (e) $μ = 0.2 γ_{a}$ , and (f) $μ = 0.25 γ_{a}$ . Values of other parameters are $γ_{a} / γ_{b} = 0.002 5$ , $g / γ_{b} = 0.004$ , $Δ / γ_{b} = 0.25$ , and $E_{L} / γ_{b} = 7.5$ . For convenience, we rescale all the parameters with respect to the decay rate $γ_{b}$ Mathematical equation

First of all, we set $μ = 0$ Mathematical equation , that is to say, there is no Kerr effect, the corresponding graph of absorption and dispersion coefficients as a function of $Ω / γ_{b}$ is shown in Fig. 2(a). From Fig. 2(a), it can be concluded that the second-order nonlinearity induced transparency can be observed at point $Ω / γ_{b} = 4$ Mathematical equation in the system without Kerr effect. At that point, both absorption coefficient $R e [ε_{T}]$ and dispersion coefficient $I m [ε_{T}]$ are equal to zero, which means that the absorption of the medium is almost zero, and the refractive index of the medium is close to 1, so the medium becomes transparent under the action of a strong coherent field. We have also found that, unlike electromagnetically induced transparency and optomechanically induced transparency, the absorption curve of second-order nonlinearity induced transparency is asymmetric. This is due to the presence of asymmetric switching of photons caused by the second-order nonlinearity.

When Kerr effect is introduced, we set $μ = 0.05 γ_{a}$ Mathematical equation , the position of the zero-absorption point is shifted to $Ω / γ_{b} = 7.48$ and the absorption curve becomes almost symmetric about the zero-absorption point, as shown in Fig. 2(b). This suggests that the Kerr effect can balance the second-order nonlinearity, resulting in a left-right symmetric absorption curve. When the strength of Kerr effect increases, we set $μ = 0.1 γ_{a}$ Mathematical equation , the position of the zero-absorption point moves to $Ω / γ_{b} = 9.04$ , as shown in Fig. 2(c), and the absorption peak on the left becomes wider and the absorption peak on the right becomes narrower, the absorption curve becomes asymmetric again. When we further increase the strength of Kerr effect to $μ = 0.15 γ_{a}$ Mathematical equation , the corresponding absorption and dispersion curves are shown in Fig. 2(d). At this time, the position of the zero-absorption point moves to $Ω / γ_{b} = 10.16$ , and the absorption peak on the left side continues to widen and the one on the right side further narrows.

We continue to increase the strength of Kerr effect to $μ = 0.2 γ_{a}$ Mathematical equation , in which case the position of the zero-absorption point is moved to $Ω / γ_{b} = 11.08$ . When we set Kerr effect $μ = 0.25 γ_{a}$ , the zero-absorption point is at $Ω / γ_{b} = 11.88$ . The corresponding images are shown in Fig. 2(e) and 1(f), respectively. At the same time, we can also see from the Fig. 2(e) and 1(f) that the left absorption peak is getting wider and wider, while the right absorption peak is getting narrower and narrower. Many interesting applications of EIT arise from the spectral properties of dark resonances^[33], i.e., narrow transmission windows in opaque media. This result suggests that the Kerr effect may be employed to improve the quality factor of the cavity when the frequency detuning $Ω / γ_{b}$ Mathematical equation is tuned to be around the right peak.

Based on the above discussion, we find that the zero-absorption point moves to the right side with the increase of the strength of Kerr effect $μ$ Mathematical equation . Therefore, we plot the zero-absorption point $Ω_{p}$ as a function of $μ / γ_{a}$ in Fig. 3. Here, we define the point where the absorption coefficient $R e [ε_{T}]$ and the dispersion coefficient $I m [ε_{T}]$ are both equal to zero as zero-absorption point $Ω_{p}$ Mathematical equation . It can be seen from Fig. 3 that the zero absorption point $Ω_{p}$ first increases rapidly with the enhancement of the Kerr effect strength $μ$ , and then slowly increases with the increase of $μ$ . These results show that by adjusting the strength of Kerr effect, the position of the zero-absorption point can be controlled. The simple dependence shown in Fig. 3 indicates that the strength of Kerr effect can be measured by measuring the position of the zero-absorption point in a possible application.

Fig. 3

The zero-absorption point $Ω_{p}$ Mathematical equation as a function of $μ / γ_{a}$

The other parameters used are the same as those in Fig. 2

Observing Fig. 2, we can see that the left and right absorption peaks have a maximum value respectively, and the positions of these two maximum values which can be evaluated by

$\frac{d R e [ε_{T}]}{d Ω} |_{Ω = Ω_{L}} = 0, \frac{d R e [ε_{T}]}{d Ω} |_{Ω = Ω_{R}} = 0$ Mathematical equation (14)

where $Ω_{L}$ Mathematical equation and $Ω_{R}$ are the two points corresponding to the maximums of the left and right absorption peaks. Here, we define the width of transparent window $d = (Ω_{R} - Ω_{L}) / γ_{b}$ . In order to discuss the variation of the width of transparent window with the Kerr effect strength, we plot the width of transparent window $d$ Mathematical equation as a function of $μ / γ_{a}$ in Fig. 4. We can see from Fig. 4 that, at the beginning, the width of transparent window $d$ decreases rapidly with the increase of $μ$ . The minimum value of the width of transparent window is $d = 4.836 8$ , and the corresponding Kerr effect strength is $μ / γ_{a} = 0.125$ Mathematical equation . After that, the width of transparent window increases almost linearly with Kerr effect strength $μ$ . This result also reminds us of the possibility of detecting the Kerr effect strength $μ$ by measuring the width of transparent window $d$ in the absorption spectrum of the output field, which may have important applications in the future.

Fig. 4

The width of transparent window $d$ Mathematical equation as a function of $μ / γ_{a}$

The other parameters used are the same as those in Fig. 2

3 Conclusion

In conclusion, we theoretically investigate the impacts of the Kerr effect on the second-order nonlinearity induced transparency in a double-resonant optical cavity coupled by a second-order nonlinearity material. Concerning the second-order nonlinearity induced transparency phenomenon, we have predicted an asymmetric-symmetric-asymmetric absorption curve transition and a shift of the zero-absorption point with the increasing of Kerr effect strength. This asymmetry absorption curve can be employed to improve the quality factor of the cavity when the frequency detuning is tuned to be around the right peak. The strength of Kerr effect can be measured by measuring the position of the zero-absorption point. Therefore, by changing the strength of the Kerr effect, we can control the width of the transparent window, the position of the zero-absorption point and change the left and right width of the absorption peak, which may have potential practical applications in the future.

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Ramezanpour S, Bogdanov A. Tuning exceptional points with Kerr nonlinearity[J]. Physical Review A, 2021, 103(4): 043510. [CrossRef] [PubMed] [Google Scholar]
Sheng J T, Yang X H, Wu H B, et al. Modified self-Kerr-nonlinearity in a four-level N-type atomic system[J]. Physical Review A, 2011, 84(5): 053820. [CrossRef] [Google Scholar]
Wang H, Goorskey D, Xiao M. Enhanced Kerr nonlinearity via atomic coherence in a three-level atomic system[J]. Physical Review Letters, 2001, 87(7): 073601. [CrossRef] [PubMed] [Google Scholar]
Wang H, Goorskey D J, Xiao M. Atomic coherence induced Kerr nonlinearity enhancement in Rb vapour[J]. Journal of Modern Optics, 2002, 49(3/4): 335-347. [Google Scholar]
Khoa D X, Bang N H, Le Thuy An N, et al. An analytical model for cross-Kerr nonlinearity in a four-level N-type atomic system with Doppler broadening[J]. Chinese Physics B, 2022, 31(2): 024201. [NASA ADS] [CrossRef] [Google Scholar]
Liao Q H, Wang X Q, He G Q, et al. Tunable optomechanically induced transparency and fast-slow light in a loop-coupled optomechanical system[J]. Chinese Physics B, 2021, 30(9): 094205. [NASA ADS] [CrossRef] [Google Scholar]
Qin X T, Jiang Y, Ma W X, et al. Observation of V-type electromagnetically induced transparency and optical switch in cold Cs atoms by using nanofiber optical lattice[J]. Chinese Physics B, 2022, 31(6): 064216. [NASA ADS] [CrossRef] [Google Scholar]
Ferretti S, Gerace D. Single-photon nonlinear optics with Kerr-type nanostructured materials[J]. Physical Review B, 2012, 85(3): 033303. [CrossRef] [PubMed] [Google Scholar]
Notomi M. Manipulating light with strongly modulated photonic crystals[J]. Reports on Progress in Physics, 2010, 73(9): 096501. [NASA ADS] [CrossRef] [Google Scholar]
Galli M, Gerace D, Welna K, et al. Low-power continuous-wave generation of visible harmonics in silicon photonic crystal nanocavities[J]. Optics Express, 2010, 18(25): 26613. [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
Combrié S, De Rossi A, Tran Q V, et al. GaAs photonic crystal cavity with ultrahigh Q: Microwatt nonlinearity at 155 μm[J]. Optics Letters, 2008, 33(16): 1908. [CrossRef] [PubMed] [Google Scholar]
Fleischhauer M, Imamoglu A, Marangos J P. Electromagnetically induced transparency: Optics in coherent media[J]. Reviews of Modern Physics, 2005, 77(2): 633-673. [NASA ADS] [CrossRef] [Google Scholar]

All Figures

Fig. 1

Scheme of a doubly resonant cavity mode of $χ^{(2)}$ Mathematical equation nonlinearity material

$E_{L}$ Mathematical equation is the driving strength of strong control light, the frequency is $ω_{L}$ , and $E_{P}$ is the driving strength of weak probe light, and the frequency is $ω_{P}$

In the text

Fig. 2

The absorption coefficient $R e [ε_{T}]$ Mathematical equation and dispersion coefficient $I m [ε_{T}]$ as a function of $Ω / γ_{b}$ with different Kerr effect coefficient $μ$

In the text

Fig. 3

The zero-absorption point $Ω_{p}$ Mathematical equation as a function of $μ / γ_{a}$

The other parameters used are the same as those in Fig. 2

In the text

Fig. 4

The width of transparent window $d$ Mathematical equation as a function of $μ / γ_{a}$

The other parameters used are the same as those in Fig. 2

In the text

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