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Abstract
We propose and demonstrate a method for characterizing the individual mirror parameters of a fiber Fabry–Perot cavity (FFPC). By measuring the reflection and transmission spectra of the FFPC with an incident laser propagating from the two mirrors of the FFPC and considering several normal or unique losses, the transmittance, reflectance, and intra-cavity loss of the individual mirrors can be determined. Due to the intrinsic limitation of cavity length, traditional powerful methods, such as the cavity ring-down technique, are not applicable to FFPCs for characterizing the parameters of individual mirrors. This scheme provides a dependable method for assessing FFPC mirrors and provides a significant capability for the implementation of strong-coupling cavity quantum electrodynamics based on FFPCs.
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1. Introduction
Optical cavities, as essential and significant optical devices, have important applications in various fields, such as precision measurement [1], quantum information [2], optical sensing [3]and quantum networks [4]. Optical cavities have high frequency resolution and can enhance intracavitary field intensity. A traditional optical cavity, called the Fabry–Perot cavity (F-P cavity), consists of two parallel mirrors with a certain reflectivity and transmissivity [5]. In particular, an F-P cavity with a high finesse and small mode volume is a powerful tool for enhancing the interaction between photons and quantum emitters to realize cavity quantum electrodynamics (CQEDs).
In recent years, efforts have been devoted to enhancing the interaction strength between light and matter by reducing the mode volume of optical cavities. Miniaturization of traditional mirror-based F-P cavities is limited due to their large physical volume and complex control systems. Thus, realizing the scalability and integration of F-P cavities is difficult. Additionally, the optical input and output of mirror-based F-P cavities are not easily achievable because complex optical alignments are needed. Researchers have developed optical resonators for CQEDs, such as microdisks [6], microtoroids [7], microspheres [8] and photonic crystal waveguides [9]. These cavities have several advantages, including small mode volume, high optical quality factor, light weight, compact structure and integration [10]. However, these cavities with whispering gallery modes or evanescent field mode structures are coupled with quantum emitters through their evanescent fields. This brings technical challenges in manipulating quantum emitters within cavity modes [11]. Fortunately, the FFPC combines the advantages of the traditional F-P cavity and waveguides and is emerging as an effective technique to enhance the coupling intensity between the cavity and quantum emitters. The FFPC can be modified by ablation of a $C{O_2}$ laser pulse at the fiber end-face. The ablation technique creates microscopic concave surfaces with a certain radius of curvature. After that, the surfaces are coated with high-reflectance films [12] [13]. Two facing-placed mirrors form an FFPC. The advantages of the FFPC include spatial tunability, resonance frequency tunability, and an open cavity [14] [15] [16]. FFPCs, as excellent CQED platforms for determining the interactions between light and matter, have been widely applied in coupling atoms [17], molecules [18], quantum dots [19], NV centers [20], and ions [21]. The CQED platform based on FFPCs has realized atomic entanglement states [22], single-photon sources [23], quantum state conversions [24], and quantum relays [25].
The measurement of FFPC parameters is highly important for its applications. The parameters include the transmittance, reflectance, and intra-cavity loss. This provides crucial information for choosing the best fiber mirrors for evaluating cavity performance, optimizing design and assessing reliability. These data serve as key data for assessing and enhancing the quality and efficiency of the cavity to meet experimental requirements. There are diverse methods for measuring and characterizing the parameters of cavity mirrors. First, the cavity ring-down (CRD) technique is a powerful method and is commonly employed [26] [27]. However, this technique faces challenges in measuring cavity mirrors with very short cavity lengths. A short cavity length leads to a short ring-down time, which places greater demands on the accuracy of laser beam switching and the time needed for the CRD technique. For the FFPC, the cavity length cannot be long enough to meet the measurement requirements of the CRD technique because of clipping losses [28]. In addition, the losses obtained based on the CRD technique are the total losses in the cavity but the individual losses in each mirror. Thus, the CRD technique is not suitable for FFPCs. Second, the optical bistability induced by mirror absorption is used to characterize cavity mirrors [29]. However, this method is applicable only to cavities with two identical mirrors, high precision, small mode volume and narrow linewidth. Third, the parameters of cavity mirrors can be determined using an accompanying coated optical plate. By analyzing the transmittance and reflectance of the coated plate with an optical spectrometer, the parameters of the cavity mirrors are considered to be the same as those of the coated plate. However, this method disregards individual differences in cavity mirrors; thus, it introduces significant errors and cannot completely ensure the similarity of the reflector films on the cavity mirrors and the coated plate. Fourth, the parameters of each mirror of an F–P cavity can be determined by the relationships among the mirror reflectance, intra-cavity loss, and transmittance by detecting the light input and output powers from both ends of an empty cavity [30]. However, this method is primarily suitable for macroscopic cavities coupling with laser beams from free space. However, for FFPCs, detecting reflected and transmitted light completely is challenging. The light cannot be fully transmitted as in free space, which is accompanied by additional coupling losses when the light is coupled into the fiber from the cavity mirrors or vice versa. Therefore, this method is not applicable to FFPCs where light propagates in the optical fiber and couples in or out of the cavity from the optical fiber.
In this paper, we propose and demonstrate a scheme to characterize the parameters of individual FFPC mirrors, including transmittance, reflectance, and intra-cavity loss. The losses are divided into two types, intra-cavity losses and extra-cavity losses. The transmission and intra-cavity loss determine the quality factor or the finesses of the FFPC. The extra-cavity losses of the FFPC include clipping loss, mode matching loss, reflection loss (represented by the reflection efficiency in the equations which equals one unit minus the reflection loss), insertion loss of the optical circulator, and splicing loss of the optical fiber in the optical path. These extra-cavity losses have no influence on the quality factor or the finesses of the FFPC. These extra-cavity losses are considered and the transmittance, reflectance, and intra-cavity loss of individual FFPC mirrors can be determined by measuring the reflection and transmission spectra. The parameters can be obtained according to the relations between the FFPC transmission/reflection and these parameters. By measuring the reflection and transmission spectra with an incident laser propagating from the two mirrors of the FFPC, the following relations are obtained. Furthermore, several key extra-cavity losses are considered: clipping loss, mode matching loss, reflection efficiency of FFPC mirrors, optical fiber transmission loss, optical splicing loss and circulator insertion loss. The transmittance, reflectance, and intra-cavity loss of individual FFPC mirrors can be determined [30].
2. Principle
Figure 1 shows a schematic of the proposed method for characterizing the transmittance, reflectance and intra-cavity loss of FFPC mirrors. The key feature of FFPC mirrors is a concave surface on the end face of an optical fiber. The red shadows represent the high-reflectivity film shown in Fig. 1. Two independent mirrors with high reflectance form an FFPC, called the left mirror and right mirror. The asymmetry of the FFPC is considered in this scheme because of the actual situations for applications. Asymmetry means that each mirror possesses unique reflectance, transmittance and intra-cavity loss. The reflectances of the left and right cavity mirrors are represented by $R_{1}$ and $R_{2}$, respectively. The transmittance is represented by $T_{1}$ and $T_{2}$ and intra-cavity loss is represented by by $l_{1}$ and $l_{2}$. The relation among them is:
where subscripts $i=1,2$ represent the left and right cavity mirrors shown in Fig. 1, respectively. To precisely determine the intra-cavity lossof each individual mirror, a laser beam is coupled from both sides of the cavity separately, and the cavity reflection and transmission spectra are measured and analyzed. When the incident laser beam propagates from the left into the cavity and the cavity resonates with the laser, the ratio between the reflected power ${p_{re}}^{le}$ and the incident power ${p_{in}}^{le}$ is expressed as follows:
Fig. 1. Schematic of an FFPC containing two fiber mirrors. $\omega _f=2.6\mu m$ represents the mode field radius of light coupled into the optical fiber, determined by the core of the fiber. $\omega _0$ is the waist radius of the cavity. $\omega _m$ is the mode radius on the cavity mirror. $r_i$ and $d_i$ ($i=1$ or $2$ represent the left and right cavity mirrors, respectively) denote the effective curvature radius and effective diameter of the left and right cavity mirrors. $R_i$, $T_i$ and $l_i$ represents the reflectance, transmittance, and intra-cavity loss, respectively. $L$ represents the cavity length, and $P$ represents the optical power of the detected light (solid blue arrow). The orange solid line arrows are shown in the fiber core to explain the reflection efficiency ($\eta _{FL}^i$).
The ratio between the transmitted ${p_{tr}}^{le}$ and incident power ${p_{in}}^{le}$ of light is:
Similarly, when the incident laser beam propagates from the right into the cavity, the ratio of the reflected power ${p_{re}}^{ri}$ to the incident power ${p_{in}}^{ri}$ is:
The ratio between the transmitted power ${p_{tr}}^{le}$ and the incident power ${p_{in}}^{le}$ is the same as in Eq. (3). ${\wp _{tr}}^{le}={\wp _{tr}}^{ri}$; hence, we denote both transmitted ratios by ${\wp _{tr}}$.
The cavity finesse is:
According to Eq. (1) – (5), the reflectivities $(R_{1}, R_{2})$, and intra-cavity loss $( l_{1}, l_{2})$ of the two individual cavity mirrors and their transmission rates $(T_{1}, T_{2})$ can be determined [30].
However, for FFPCs, five additional extra-cavity losses must be considered to experimentally determine the ratios of the reflected power and the transmitted power to the incident power.
The first loss is the clipping loss. The mode radius on the surface of the cavity mirrors increases with increasing cavity length. A mismatch between the mode radius on the cavity mirror and the effective diameter of the cavity mirror results in clipping loss. For a single reflection:
where ${\omega _{mi}}$ denotes the mode radius on the cavity mirror and $d_{i}$ denotes the effective diameter of the cavity mirrors. The mode radius on the cavity mirror is determined by:Fig. 2. Clipping loss of two mirrors as a function of cavity length.The red curve represents ${r_1} = 390\mu m$ and the black curve represents ${r_2} = 545\mu m$.
The second loss is the mode matching loss between the fiber and cavity modes. For macroscopic cavity mirrors, the coupling efficiency for the $TE{M_{00}}$ mode and higher-order modes can be fully determined by the measured cavity spectrum [30]. Fortunately, for FFPCs, the light coupling from the fiber core to the cavity or vice versa is robust and stable for single-mode fibers because of the integration of the cavity mirror and the fiber. The power coupling efficiency between the fiber and cavity modes can be calculated theoretically by the overlap integral of the fiber and cavity mode intensity distributions [31]:
The third loss is the reflection efficiency of the cavity mirrors. For macroscopic cavity mirrors, the refection surface is much larger than the size of the laser beam; thus, the laser power can be reflected totally. However, for FFPCs, the light is only reflected to couple into the fiber core when it satisfies the total internal reflection condition. The solid orange lines are shown in the core of the cavity mirror in Fig. 1 to explain the losses, which are called the reflection efficiency ($\eta _{FL}^i$). Due to the nonplanar surface of the cavity mirror, the incident light propagating in the fiber core is reflected by the surface of the fiber cavity mirror and is ultimately refracted into the fiber cladding when the angle ($\alpha$) of the reflected light is smaller than the Brewster angle. Thus, this causes losses. The reflection loss can be measured experimentally.
The fourth loss is the insertion loss of the optical circulator, which is used to obtain the reflection light from the cavity. The transmission efficiency of the optical circulator is ${\eta _{cir}}$.
The last losses are splicing loss, corresponding to the transmission efficiency of the splicing points $\eta _{sl}^i$ and the transmission efficiency of the optical fiber $\eta _{tr}^i$. However, splicing losses occur in the optical fibers of both FFPCs, with one splicing point on each side. The splicing losses are decreased by optimizing the splicing process. We suppose that the splicing losses for both mirrors are the same $\eta _{sl}^1=\eta _{sl}^2={\eta _{sl}}$. The efficiencies of light transmission in an optical fiber for both mirrors are considered to be the same $\eta _{tr}^1=\eta _{tr}^2={\eta _{tr}}$ since the lengths of the two optical fibers are very short and are almost the same. Therefore, the differences in splicing losses and fiber transmission losses are ignored. Moreover, the two losses are canceled out in the following calculation process.
Considering the four losses ${\wp _{tr}}^{le}$, ${\wp _{tr}}^{ri}$ and ${\wp _{re}}^{le}$, ${\wp _{re}}^{ri}$ can be obtained from the detected powers, ${p'_{re}}^{le}$, ${p'_{in}}^{le}$, and ${p'_{re}}^{ri}$, ${p'_{in}}^{ri}$, ${p'_{tr}}^{le}$, ${p'_{tr}}^{ri}$, where is the detected incident power.
The powers $p{'_{tr}}^{le}/p{'_{in}}^{le}$, $p{'_{tr}}^{ri}/p{'_{in}}^{ri}$ and $p{'_{re}}^{le}/p{'_{in}}^{le}$, $p{'_{re}}^{ri}/p{'_{in}}^{ri}$ can be calculated from the voltage values of the optical detectors. According to these values, the parameters of two individual cavity mirrors can be determined.
3. Experiment
We experimentally implement the proposed scheme, and the experimental setup is shown schematically in Fig. 3. A laser beam from a tunable laser (TLB 6700-LN, TLB 6700-XP) is coupled into a fiber. The wavelength is $852nm$, and the phase of the laser is modulated via an electro-optic modulator (EOM, Ke yang photoelectric (Beijing) Technology, KY-PM-08-10G-PP-FA) to generate two sidebands. The laser polarization is aligned to the intrinsic polarization of an FFPC by means of a polarization controller (Thorlabs, FPC030). The reflected light is retracted by a fiber circulator (Micro Photons (Shanghai) Technology, HPCIR-85-2-L-1-C) and is detected by a Detector 1 (Thorlabs, PDA36A2-1C). The transmitted light is detected by a Detector 2 (Thorlabs, PDA36A2-1C). The reflection spectra and transmission spectra can be measured by scanning the cavity in the time domain, as shown in Fig. 4. The phase of the laser is modulated via an electro-optic modulator to generate two sidebands. The frequency difference between the carrier and one sideband is 9 GHz, as shown in Fig. 4. The frequency difference corresponds to the time difference along the horizontal axis in Fig. 4, which is considered as a ruler. The line width of the carrier (half-height full width obtained by Lorentz fitting) corresponds to a shorter time difference. The ratio between the two time differences equals the ratio between the frequency difference and the line width of the carrier, and then the line width can be determined.
Fig. 3. Schematic diagram of the experimental setup. (a) An optical image of the FFPC. The microscopic concave surfaces of the FFPC mirror can be observed using atomic force microscopy (AFM). (b) A typical AFM image of the mirror end face. (c) The depth z as a function of x represents the profile cutting through the center of the concave surface(black solid line). Close to the center of the profile, a circular curve is used to fit the profile (red solid line), yielding the effective radius of curvature ${r_1} = 390.1\mu m$ and ${d_1} = 44.0\mu m$ the effective diameter of the mirror. For another mirror, ${r_2} = 545.0\mu m$ and ${d_2} = 56.4\mu m$.
Fig. 4. (a) The reflection spectrum (black solid line) and transmission spectrum (red solid line) when the incident laser propagates from the left cavity mirror. (b) The case in which the incident laser propagates from the right cavity mirror. The dashed blue arrows point to the symmetrically distributed sidebands and the carrier in (a) and (b), respectively, and the frequency difference between the carrier and one sideband is 9 GHz.
The detectors operate in the linear region of the laser intensity and the output voltage; thus, the ratios of the detected reflection optical power to the incident optical power are $p{'_{re}}^{le}/p{'_{in}}^{le} = {V_2}/{V_1}$, $p{'_{re}}^{ri}/p{'_{in}}^{ri} = {V'_2}/{V'_1}$, and the ratios of the transmitted optical power to the incident optical power are $p{'_{tr}}^{le}/p{'_{in}}^{le} = {V_3}/{V_1}$, $p{'_{tr}}^{ri}/p{'_{in}}^{ri} = {V'_3}/{V'_1}$. ${V_2}, {V_3}, {V'_2}, {V'_3}$ are the maximum voltage of the transmitted light and the minimum voltage of the reflected light when the laser resonates with the FFPC, as shown in Fig. 4. ${V_1}, {V'_1}$ are the voltages when the laser does not resonate with the FFPC. We obtain $p{'_{re}}^{le}/p{'_{in}}^{le} = 0.876, p{'_{re}}^{ri}/p{'_{in}}^{ri} = 0.912, p{'_{tr}}^{le}/p{'_{in}}^{le} = 0.0029, p{'_{tr}}^{ri}/p{'_{in}}^{ri} = 0.0028$.
The laser is tuned to the adjacent resonance frequency of the FFPC, the free spectral range $FSR = 3345.4GHz$ is fine-tuned, and the length of the FFPC is determined as $L = 44.8\mu m$. According to the sidebands($9GHz$), the line width is $\Delta \nu = 0.928GHz$, and the finesse is ${\cal F} = \frac {{FSR}}{{\Delta \nu }} = 3605$. Using Eqs. (7, 8, and 10), the waist and laser beam radii on both mirror surfaces can be obtained as ${\omega _0} = 5.16\mu m$, ${\omega _{m1}} = 5.34\mu m$ and ${\omega _{m2}} = 5.24\mu m$. The mode matching efficiencies for two mirrors can be calculated using Eq. (9), $\eta _{mm}^1 = 0.617$ and $\eta _{mm}^2 = 0.632$. The reflection efficiencies of the two cavity mirrors are measured directly using an optical circulator when the FFPC does not resonate with the laser, $\eta _{FL}^1 = 0.571$, $\eta _{FL}^2 = 0.474$. The efficiency of the optical circulator ${\eta _{cir}} = 0.897$.
We implement multiple measurements of the transmission and reflection spectra of the FFPC and obtain $\overline { {\wp _{re}}^{le}} = 0.8016 \pm 0.0002$, $\overline { {\wp _{re}}^{ri}} = 0.8635 \pm 0.0005$, $\overline { {\wp _{tr}}^{le}} = 0.00143 \pm 0.00003$, and $\overline { {\wp _{tr}}^{ri}} = 0.00121 \pm 0.00003$. Since ${\wp _{tr}}^{le} = {\wp _{tr}}^{ri}$, we calculate ${\wp _{tr}} = \left ( { \overline {{\wp _{tr}}^{le}} + \overline {{\wp _{tr}}^{ri}} } \right )/2$. Finally, according to Eqs. (1)–(5) and (11)–(14), we determine the reflectance for each cavity mirror ${R_1} = 0.99897 \pm 0.00009$, ${R_2} = 0.99932 \pm 0.00002$ and the transmittance ${T_1} = 88.60 \pm 0.6ppm$, ${T_2} = 61.5 \pm 0.2ppm$, with intra-cavity loss ${l_1} = 947.5 \pm 0.2ppm$, ${l_2} = 645.3 \pm 0.9ppm$. All the errors in the manuscript are statistical errors. Multiple measurements are carried out, and the standard deviations are calculated as the errors of the FFPC parameters such as the line width and finesse. Then, these errors are transferred to the transmission, reflectance, and intracavity loss.
In total, 6 FFPC mirrors are measured using this scheme, and the respective transmittance, reflectance, and intra-cavity loss are obtained, as shown in Table 1. All the mirrors are fabricated and coated in the same batch.
Table 1. transmittance, reflectance, and intra-cavity loss of cavity mirrors
4. Conclusion
In summary, we determined the transmittance, reflectance and intra-cavity loss of individual FFPC mirrors by measuring the reflection and transmission spectra of an FFPC when the incident laser propagates from two mirrors of the FFPC. Compared to macroscopic cavity mirrors, several losses (efficiencies) are considered, including clipping loss, mode matching loss, reflection efficiency of FFPC mirrors, optical fiber transmission loss, optical splicing loss, and circulator insertion loss. Several losses are unique to FFPCs. Characterizing cavity mirrors plays a crucial role in various applications. The traditional method for characterizing cavity mirrors is not suitable for FFPCs. However, this scheme provides a dependable method for assessing FFPC mirrors. This scheme has several limitations. The mode matching efficiency of the FFPC is calculated theoretically and may exhibit errors compared to macroscopic cavity mirrors. Fortunately, it is easy to obtain good mode matching results for FFPCs after adjustment. Nevertheless, the error in the mode matching efficiency does not significantly affect the measurement results; therefore, we can use this method to select the best cavity mirrors for subsequent experiments. This method can be applied to various forms of microcavities.
Funding
Innovation Program for Quantum Science and Technology (2023ZD0300400); National Key Research and Development Program of China (2021YFA1402002); National Natural Science Foundation of China (11974223, 11974225, 12104277, 12104278, 92265108, U21A20433, U21A6006); Fundamental Research Program of Shanxi Province (202203021223003); Fund for Shanxi Key Subjects Construction (the Fund for Shanxi “1331 Project” Key Subjects).
Acknowledgements
The authors thank Lijun Song for valuable discussions.
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.
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