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Abstract
White light cavities with broadband resonance are usually filled with negative dispersion medium, which inevitably leads to gain. In this article, pure passive white light cavities are designed, in which negative dispersion medium is no longer necessary. Theoretically, if the reflection phase of the cavity wall can exhibit a negative dispersion slope, then it can also satisfy white light cavities conditions without medium. In practice, the negative dispersion property of the cavity wall can be realized by two metal coatings with different reflection coefficients. Therefore, our white light cavities are composite cavities, in which the main cavity provides resonance while the auxiliary cavity forms the cavity wall, providing negative dispersion reflection phase. Further, atomic gas can be employed to improve the performance of the white light cavities. Atomic gas exploits effects such as Electromagnetic Induced Transparency (EIT), enabling the white light cavities to be controlled by coherent driving field. With the passive characters, our design can be realized and implemented much more easily.
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1. Introduction
Cavities with broadband resonance are called white light cavities. To detect gravitational wave, white light cavities (WLC) [1] were first demanded in 1997. Generally, WLC are achieved by filling the cavity with negative dispersion medium. So, the negative dispersion behavior is the key of WLC [2]. According to Kramers-Kronig relationship [3–5], however, positive dispersion is accompanied by absorption, while negative dispersion is accompanied by gain. As well, medium with negative dispersion slope cannot be found in nature. In order to realize negative dispersion behavior, coherent Raman scattering [6–8] can be employed. Unfortunately, this method suffers from obvious gain and weak coherent property of the transmission. Accordingly, other methods instead investigated WLC based on EIT with population inversion [9–11]or added Kerr nonlinear dispersion [12] to obtain negative dispersion without gain. However, it is difficult to prepare a practical EIT atomic system in the state of population inversion and also Kerr nonlinear dispersion requires strong light intensity. So, previously cited methods, such as broadband optical switches [10] and low-power ultra-fast optical switching [11], are still remained to be tested in experiment. Besides, Jacob Scheuer [13]constructed negative dispersion by means of medium with different impedance. Yet, when extending to the visible light region, extremely high relative permittivity is required, which is impractical. Although there exists significant challenges in designing WLC in visible light region, its promising application, such as ring laser gyroscopes [14], optical switches [15], hypersensitive sensor [16,17] and distortion free pulse delay system [18], still drives researches on white light cavities.
In this thesis, we report a new design of WLC, which is purely passive and workable in visible light region. We identify the cavity resonant condition and notice that in order to satisfy WLC conditions, negative dispersion of reflection phase (NDRP) generated by means of cavity walls can be an alternative to negative dispersion medium. Such cavity walls can be realized by asymmetry cavity, which is more practical by using several metal coatings.
Nevertheless, focusing on the metal coating, high accuracy of the reflection coefficient is required to satisfy WLC conditions, which may bring obstacles when putting into practice. To address this problem, we optimize our system to make it controllable. By filling the main cavity with EIT gas, we can satisfy WLC conditions by adjusting the intensity of driving field.
The article is organized as follows: In Sec. 2, new WLC conditions are presented and the cavity wall with NDRP is addressed. Sec. 3 is devoted to optimize the WLC by filling it with EIT gas. Sec. 4 finally gives the conclusion of the paper.
2. Pure passive WLC
Taking simple Frabry-Perot cavities [Fig. 1(a)] as an example, in which the cavity length is $d$, the refractive index of the medium is $n$, the reflection (transmission) coefficients of the left and right cavity walls are $r_a$ ($t_a$)and $r_b$ ($t_b$) respectively. Thus, the total transmissivity of the cavities is
Fig. 1. Asymmetric cavity with metal coatings (MC) for $d=\lambda _0/2$, $\omega _0=2\pi \times 7.5\times 10^9Hz$ and $r_b=-0.9$ (a) schematic diagram of a vacuum cavity(b) equivalent reflection coefficient and (c) dispersion slope for different $r_a$.
For commonly used cavities with metal coatings as cavity walls, the resonant condition is a well-known formula, i.e. $n\omega d/c=k\pi$, $k\in Z$. To meet WLC requirements, the medium needs to exhibit a negative dispersion slope and meet $\frac {dn}{d\omega }\mid _{\omega =\omega _0}^{}=-\frac {n\left ( \omega _0 \right )}{\omega _0}$ [6–11]. However, if the reflection coefficient of the cavity wall is a complex number, which varies with frequency, i.e, $r_a=\left | r_a \right |\exp \left ( i\varphi _a \right )$ and $r_b=\left | r_b \right |\exp \left ( i\varphi _b \right )$ , then the resonance conditions change as follows
Equation (3) is the starting point of our work. If we can find a special cavity wall whose reflection phase ($\varphi _a$ or $\varphi _b$) decreases monotonically with frequency $\omega$, thereby compensating the increasing of round-trip phase shift ($2n\omega d/c$ ), WLC is thus available without negative dispersion medium, or even without any medium.
Does there truly exist a cavity wall that can generate negative dispersion of reflection phase(NDRP)? Actually, we shall see in the following discuss that empty asymmetric F-P cavities can bring NDRP. Take empty cavity in Fig. 1(a) as an example, two cavity walls are filmed with metal coatings of different reflection coefficients, i.e., $r_a\ne r_b$, and no medium is added, i.e. $n=1$. When incident field is transmitted from the left side, the reflection coefficient of the whole system is
The reflection coefficient of metal coating is negative and the sign of dispersion is governed by $(r_b-r_a)$, which means that $\varphi _{eff}$ exhibits positive dispersion if $\left | r_b \right |>\left | r_a \right |$, while $\varphi _{eff}$ exhibits negative dispersion if $\left | r_b \right |<\left | r_a \right |$. As well, the numerical value of $r_a$ and $r_b$ determines the strength of $\varphi _{eff}$ dispersion. For symmetric cavity, phase transition occurs near the resonant frequency, i.e. $\frac {d\varphi _{eff}}{d\omega }\mid _{\omega =\omega _0}^{}\rightarrow \infty$. For example, for $r_b=-0.9$, $d=\lambda _0/2$, $\left | r_{eff} \right |$ and $\varphi _{eff}$ can be respectively plotted as a function of $\omega$ with various $r_a$ in Fig. 1(b) and (c).
For $r_a=-0.85$, $\left | r_b \right |>\left | r_a \right |$ , $\varphi _{eff}$ has a positive dispersion slope and its gradient is larger than the case when $r_a=-0.8$ (comparing the red curve with black curve). Compared the case when $r_a=-0.95$ with $r_a=-0.99$, $\left | r_b \right |<\left | r_a \right |$, both $\varphi _{eff}$ has negative dispersion slope, while the gradient when $r_a=-0.95$ is larger than the gradient when $r_a=-0.99$ (comparing the blue curve with green curve). On the other hand, $\left | r_{eff} \right |$ at $\omega =\omega _0$ decreases when $r_a\rightarrow r_b$. For a vacuum asymmetric cavity with fixed cavity length, we consider it as a whole to be an effective cavity wall with its phase of effective reflection coefficient governed by frequency, i.e. $\varphi _{eff}$ $\sim$ $\omega$. On the other hand, dispersion in our work can also be determined solely by the reflection coefficient of the component mirrors, i.e. $r_a$ and $r_b$, which means we are able to tune the dispersion with fixed frequency.
Appling the theory above to WLC, we construct composite cavities by using two identical asymmetric cavities as cavity walls (see in Fig. 2). Among four metal coatings, two of them have a reflection coefficient of $r_a$ and the others have a reflection coefficient of $r_b$. The length between adjacent metal coatings is $d$. The yellow area in the middle serves as the main cavity, providing resonant conditions. On the other hand, two asymmetric cavities on both sides serve as effective cavity walls, providing effective reflection coefficients $r_{eff}=\left | r_{eff} \right |\exp \left ( i\varphi _{eff} \right )$. Thus, the resonant conditions of the entire composite cavities are
Meanwhile, the WLC conditions are
By combining Eq. (5) and Eq. (7), we can calculate the required $r_a$ with $r_b$ and $d$ already known.
Fig. 2. The schematic diagram of three-cavitiy model. The length of cavity $d$ is an integral multiple of half wavelength. Atomic gas with three-level atoms is filled into the middle cavity to construct EIT.
For example, for $\omega _0=2.890\times 10^{13}Hz$ and $d=\lambda _0/2$, $r_b =-0.99$, we obtain $r_a= -0.9999752$, which meets the WLC condition. Next, we calculated the transmission spectrum $T\left ( \omega \right )$ and the electric field intensity distribution in the composite cavities, as shown in Figs. 3(a) and (b). It is obviously illustrated in Fig. 3(a) that the transmission spectrum exhibits a very flat transmission peak within range $\left [ 2.888\times 10^{13},2.892\times 10^{13} \right ]$ Hz, with the width about $40GHz$, which is the evidence of WLC. As a comparison to Ref. [12], the transmission spectrum flat has been broadened from about 50MHz to 40GHz, exhibiting a better performance. Within the corresponding frequency range, the electric field exhibits resonance enhancement in the main cavity, as shown in the area marked within yellow dashed lines in Fig. 3(b). Notice also that enhancement occurs in the equivalent cavity wall outside the operating frequency at about $2.886\times 10^{13}Hz$ and $2.895\times 10^{13}Hz$. Focus back to the main cavity, its maximum enhancements lay outside the operating frequency of WLC, contributing barely to transmittance.
Fig. 3. (a) The transmissivity as a function of the angular frequency, (b) the distribution of light strength (without incident light) in three cavities and (c) the equivalent reflection coefficient as a function of the angular frequency and the dispersion of effective wall. The above figure with parameters $r_b=-0.99$, $r_a=-0.9999752$ and $d=\lambda _0/2$. (d) the quality factor at resonant angular frequency with different value $r_a$, where $r_b$ satisfies the WLC condition.
The operating frequency range of WLC is subject to the negative dispersion region of the reflection phase. In Fig. 3(c), we calculated $r_{eff}$ and $\varphi _{eff}$ as a function of frequency. It can be found by comparing Fig. 3(a) with Fig. 3(c) that the corresponding frequency range when $\varphi _{eff}$ has a negative dispersion slope fits the operating frequency range of WLC. Meanwhile, $\left | r_{eff} \right |$ has appreciable values, about $0.995$, at these frequencies.
In order to evaluate the performance of the cavity, quality factor $Q$ is thus introduced. As shown in Fig. 3(d), quality factor increases as $r_b\rightarrow -1$. However, the value of quality factor is subject to the lower limit of $r_a$ and $r_b$ ($r_b>r_a>-1$). Although a quite large quality factor around $300$ can be achieved with, for instance, $r_a=-0.9999752$ and $r_b=-0.99$, it is not a practical parameter in experiments.
3. WLC with electromagnetic induced transparency
As seen in the preceding section, negative dispersion is achieved by means of asymmetric cavity to construct WLC. We point out that quality factor can also increase to a considerable value as $r_a\rightarrow -1$. However, the reflection coefficient required to satisfy WLC conditions is strict and should be designed carefully. In order to make our cavities easier to be realized, we introduce electromagnetic induced transparency (EIT) to our model, which will be elaborated in this section. Ref. [9] pointed out that EIT can generate dispersion with positive slope in the absent of pump field, compensating the negative dispersion generated by asymmetric cavity, so that a steeper slope of negative dispersion becomes acceptable. Promisingly, the adjustment of dispersion by EIT generate no absorption and thus can be applied on WLC. Based on our three-cavity model, we fill the main cavity with three-level atomic gas, with excited level $|{a}\rangle$ and ground level $|{b}\rangle$ and $|{c}\rangle$, as shown in Fig. 2. $\gamma _{ab}$ is the decay rate from the excited level to the ground level. The dipole $|{a}\rangle \leftrightarrow |{b}\rangle$, defined by $\wp$, is resonant with the cavity, i.e., $\omega _0=\omega _{ab}$. Dipole $|{a}\rangle \leftrightarrow |{c}\rangle$ is coupled by a driving field with Rabi frequency $\varOmega _{\mu }$ at resonance. So far, the atomic gas is able to contribute a controllable susceptibility $\chi \left ( \omega,\varOmega _{\mu } \right )$. Refractive index is then defined by $n=\sqrt {1+\chi }$. Resonant condition is to be redefined as
By taking derivation with respect to Eq. (8), the equation becomes
Since susceptibility $\chi$ is small enough to be neglected at the resonant frequency, refractive index can be expanded as $n\left ( \omega \right ) \approx 1+\frac {\left ( \chi '+i\chi '' \right )}{2}$ and $n\left ( \omega _0 \right ) \approx 1$. The dispersion of refractive index [9,19] takes the form
To give an example, reflection coefficients take the values $r_a=-0.995$, $r_b=-0.95$ and cavity length $d=25\lambda _0$. Obviously, such empty composite cavities, whose transmissivity is shown in Fig. 4(a), are not WLC. We then fill atomic gas into the main cavity, with $A=\frac {\gamma _{ab}}{1.1}$, $\gamma _{ab}=3.18\times 10^{7}Hz$. To meet WLC conditions, Eq. (11) should be satisfied with required $\varOmega _{\mu }=248\gamma _{ab}$. If a driving field with $\varOmega _{\mu }=248\gamma _{ab}$, as indicated before, is applied in [Fig. 4(c)], a flat peak can be found in the transmission spectrum. However, if we instead apply a driving field with $\varOmega _{\mu }=27\gamma _{ab}$, as shown in [Fig. 4(b)], the peak is no longer flat. Likely, the flat cannot be observed when increasing the driving field to $382\gamma _{ab}$[Fig. 4(d)].
Fig. 4. Transmissivity as a function of the angular frequency for $r_a=-0.995$, $r_b=-0.95$ and $d=25\lambda _0$ (a) without atomic gas, (b) $\varOmega _{\mu }=27\gamma _{ab}$, (c) $\varOmega _{\mu }=248\gamma _{ab}$ and (d) $\varOmega _{\mu }=382\gamma _{ab}$.
Furthermore, the maximum width of the flat is closely related to Rabi frequency $\varOmega _{\mu }$ since the positive dispersion range of EIT [Eq. (8)] is determined by $\varOmega _{\mu }$ while the flat width must be narrower than EIT operating range. However, the value of $\varOmega _{\mu }$ is subject to $r_a$ and $r_b$. To be precise, as shown in Fig. 1(c), a considerable NDRP appears if $r_a$ is close enough to $r_b$, leading a larger positive dispersion needed to compensate the negative dispersion by means of smaller $\varOmega _{\mu }$. For a more detailed discussion, the width can be tuned by changing the length of cavities. For example, if the cavity length is tuned to be $5\lambda _0$ ($50\lambda _0$), flat width is thus around $300\gamma _{ab}$ ($70\gamma _{ab}$), as indicated in Fig. 5(b),(c). It should be mentioned that the logic between reflection coefficients and flat width is valid as well for pure WLC case. In practice, to construct a WLC, $r_b$ can be designed first with fixed $r_a$ to make the flat width wider than needed, and subsequently tuning the cavity length $d$ to adjust the flat width.
Fig. 5. (a) Real part (black solid line) and imaginary part (red dashed line) of susceptibility $\chi$ as a function of angular frequency with $\varOmega _{\mu }=248\gamma _{ab}$, (b)[(c)]Transmissivity as a function of angular frequency with $d=5\lambda _0$ ($d=50\lambda _0$), $r_a=-0.995$, $r_b=-0.95$ and $\varOmega _{\mu }=248\gamma _{ab}$.
4. Conclusion
In this article, we have provided numerical evidence that NDRP is available via asymmetric cavity. Based on this discovery, composite cavities are constructed to broaden the range of reflectivity/transmissivity frequency, which are called white light cavities. In practice, asymmetric cavity can be easily realized by means of metal coatings, which can be used to tune the reflection coefficients to satisfy WLC conditions. However, pure WLC suffer a strict request for reflection coefficients. EIT is thus introduced to lower the difficulty of constructing WLC in practice. EIT is able to generate a positive dispersion slope in the absence of pump field, compensating the negative dispersion generated by NDRP, so that a less precision coating becomes acceptable. Analytically, we obtained the Rabi frequency needed to satisfy WLC conditions with EIT. We also observed that the width of the frequency flat can be tuned by adjusting Rabi frequency and cavity length. To conclude, we have combined a classical way and a quantum way to design WLC.
Funding
National Natural Science Foundation of China (12174288, 12204352, 12274326); National Key Research and Development Program of China (2021YFA1400602).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. A. Wicht, K. Danzmann, M. Fleischhauer, M. Scully, G. Müller, and R. H. Rinkleff, “White-light cavities, atomic phasecoherence, and gravitationalwave detectors,” Opt. Commun. 134(1-6), 431–439 (1997). [CrossRef]
2. §. Wise, G. Mueller, D. Reitze, D. B. Tanner, and B. F. Whiting, “Linewidth-broadened Fabry–Perot cavities within future gravitational wave detectors,” Class. Quantum Grav. 21(5), S1031–S1036 (2004). [CrossRef]
3. R. D. L. Kronig, “On the Theory of Dispersion of X-Rays,” J. Opt. Soc. Am. 12(6), 547–557 (1926). [CrossRef]
4. J. S. Toll, “Causality and the Dispersion Relation: Logical Foundations,” Phys. Rev. 104(6), 1760–1770 (1956). [CrossRef]
5. H. C. Bolton and G. J. Troup, “The modification of the Kronig-Kramers relations under saturation conditions,” Philos. Mag. 19(159), 477–485 (1969). [CrossRef]
6. Q. Q. Sun, M. S. Shahriar, and M. S. Zubairy, “Effects of noise and parameter deviations in a bichromatic Raman white light cavity,” Phys. Rev. A 81(3), 033826 (2010). [CrossRef]
7. J. P. Zhang, X. G. Wei, G. Hernandez, and Y. F. Zhu, “White-light cavity based on coherent Raman scattering via normal modes of a coupled cavity-and-atom system,” Phys. Rev. A 81(3), 033804 (2010). [CrossRef]
8. G. S. Pati, M. Salit, K. Salit, and M. S. Shahriar, “Demonstration of a Tunable-Bandwidth White Light Interferometer using Anomalous Dispersion in Atomic Vapor,” Phys. Rev. Lett. 99(13), 133601 (2007). [CrossRef]
9. J. P. Xu, M. Al-Amri, Y. P. Yang, S. Y. Zhu, and M. S. Zubairy, “Wide-band optical switch via white light cavity,” Phys. Rev. A 86(3), 033828 (2012). [CrossRef]
10. N. Li, J. P. Xu, G. Song, C. J. Zhu, S. Y. Xie, Y. P. Yang, M. S. Zubairy, and S. Y. Zhu, “Nonlinear all-optical switch based on a white-light cavity,” Phys. Rev. A 93(4), 043819 (2016). [CrossRef]
11. A. Othman, D. Yevick, and M. Al-Amri, “Generation of three wide frequency bands within a single white-light cavity,” Phys. Rev. A 97(4), 043816 (2018). [CrossRef]
12. K. Yang, Y. P. Niu, D. J. Chen, H. W. Cai, R. H. Qu, and S. Q. Gong, “White light cavity via modification of linear and nonlinear dispersion in an N-typeatomicsystem,” Opt. Commun. 342, 189–192 (2015). [CrossRef]
13. J. Scheuer, D. Filonov, T. Vosheva, and P. Ginzburg, “Extraordinary broadband impedance matching in highly dispersive media - the white light cavity approach,” Opt. Express 30(4), 5192–5199 (2022). [CrossRef]
14. M. S. Shahriar, G. S. Pati, R. Tripathi, V. Gopal, M. Messall, and K. Salit, “Ultrahigh enhancement in absolute and relative rotation sensing using fast and slow light,” Phys. Rev. A 75(5), 053807 (2007). [CrossRef]
15. M. Skorczakowski, J. Swiderski, W. Pichola, P. Nyga, A. Zajac, M. Maciejewska, L. Galecki, J. Kasprzak, S. Gross, A. Heinrich, and T. Bragagna, “Mid-infrared Q-switched Er:YAG laser for medical applications,” Laser Phys. Lett. 7(7), 498–504 (2010). [CrossRef]
16. H. N.Yum, M. Salit, J. Yablon, K. Salit, Y. Wang, and M. S. Shahriar, “Superluminal ring laser for hypersensitive sensing,” Opt. Express 18(17), 17658–17665 (2010). [CrossRef]
17. O. Kotlicki, J. Scheuer, and M. S. Shahriar, “Theoretical study on Brillouin fiber laser sensor based on white light cavity,” Opt. Express 20(27), 28234–28248 (2012). [CrossRef]
18. H. N. Yum, M. E. Kim, Y. J. Jang, and M. S. Shahriar, “Distortion free pulse delay system using a pair of tunable white light cavities,” Opt. Express 19(7), 6705–6713 (2011). [CrossRef]
19. M. Fleischhauer, C. H. Keitel, M. O. Scully, C. Su, B. T. Ulrich, and S. Y. Zhu, “Resonantly enhanced refractive index without absorption via atomic coherence,” Phys. Rev. A 46(3), 1468–1487 (1992). [CrossRef]
