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Abstract
The challenges presented by the directly reflected field in optical feedback cavity-enhanced spectroscopy systems serve as substantial obstacles, introducing additional complexity to existing systems and compromising their sensitivity, as the underlying mechanisms of its adverse effects remain not fully understood. This study aims to address this issue by introducing a comprehensive analytical model. Additionally, frequency locking can be achieved by decreasing the feedback rate, the laser’s linewidth enhancement factor, and the directly reflected field, and by increasing the refractive index of the gain medium, the length of the laser’s resonant cavity, the electric field reflectivity of the laser’s output facet, and the resonant field. These parameters can affect the feedback coupling rate pre-factor, and for a resonant cavity with a length of 0.394 m, optical feedback can only be established when the feedback coupling rate pre-factor is less than 1.05 × 109. Through experimental validation, we successfully confirm the effectiveness of the proposed solution in eliminating the detrimental effects of the directly reflected field. Importantly, this suppression is achieved without compromising other aspects of the system's performance. The research findings not only offer the potential to optimize various cavity-enhanced spectroscopy systems that rely on optical feedback but also show promising applications in advancing the development of high-purity spectrum diode lasers utilizing optical feedback from an external high-finesse cavity.
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1. Introduction
With the continuous improvement of various diode lasers in terms of reliability, power, and wavelength coverage [1,2], optical feedback locking has become widely adopted as a powerful tool for enhancing light-gas interactions within a cavity [3]. It’s noteworthy that while the technique of optical feedback locking was initially developed to stabilize and narrow down diode laser linewidth [4,5], it has evolved into a key method for cavity-enhanced spectroscopy [6] due to its robust and simple features compared to other frequency locking methods [7,8].
In a typical setup for cavity-enhanced spectroscopy employing optical feedback, continuous-wave diode lasers function as light sources and are coupled into an external resonant cavity [6], formed by two or more high-reflectivity mirrors. This cavity serves as the platform for probing gas samples. Upon achieving locking, there is a substantial improvement in the interaction paths of laser with the gas or the exciting power [9], leading to enhanced absorption [10–14] or scattering [15–18] and consequently, exceptional detection sensitivity. It’s widely acknowledged that only the resonant field (RF) leaking from the cavity has the capability to suppress laser phase fluctuations and stabilize the laser emission frequency to align with one of the cavity longitudinal modes [19]. Conversely, the directly reflected field (DRF) by the cavity input mirror tends to significantly perturb the laser spectrum [20] and undermine the beneficial effects of RF.
Two approaches have been proposed to mitigate the adverse effects of DRF caused by the input mirror. The first approach relies on a careful configuration of the optical feedback path to separate the two light fields. The most classic and widely adopted configuration is V-shaped cavity [21], where the incident light and the cavity input mirror are oriented at a specific angle. Consequently, DRF does not return to the laser cavity. Another configuration, introduced by Hippler [15], involves placing two optical isolators between the laser and the cavity to block DRF. Simultaneously, it involves redirecting RF transmitted by the cavity output mirror back to the laser for successful stabilization. The second approach exploits mode mismatch to create differences in the spot sizes between the two light fields. An iris or spatial filters placed between the laser and the cavity will automatically attenuate the majority of DRF, while ensuring that most of RF returns to the laser cavity to establish optical feedback. This idea was first proposed by King et al. [22] and then applied to diode lasers coated with anti-reflection films. Ritchie and others further applied such a concept to lock single-mode lasers without anti-reflection coatings to an external cavity [23,24].
While these two approaches effectively suppress DRF, they generally result in a reduction in other system performance aspects such as sensitivity and complexity. The introduction of additional optical components, for instance, leads to a decrease in cavity finesse and a reduction in the coupling efficiency between the laser and the cavity. Additionally, the placement and aperture sizes of devices like the iris or spatial filters can complicate the adjustment of the optical path. However, Zhao et al. [25] concluded that DRF does not impact optical feedback and frequency locking could be achieved with the appropriate feedback phase and feedback rate. As we will discuss in the following sections of this paper, through theoretical studies, we have discovered that it’s possible to achieve the desired suppression without the need for introducing additional optical components or intentional mode mismatch by carefully configuring system parameters.
Initially, this paper delves into the impact of both RF and DRF on optical feedback frequency locking, using the laser-cavity coupling model. It explains the necessary conditions for establishing frequency locking in the linear cavity. Moreover, the influences of various factors on frequency locking are thoroughly analyzed. Finally, the paper introduces a simple yet effective model to enhance RF and attenuate DRF. This model is based on the asymmetric reflectivity of the front and back mirrors, and its efficacy is validated through experimentation.
2. Theoretical model
Accurately analyzing the phase of DRF is crucial to the linear cavity optical feedback effect. The phase analysis is shown in Fig. 1. The wavelength of the Laser is λ, and the angular frequency of the laser coupled to the cavity is ω = 2π c/λ, where c is the speed of laser. The refractive index of the glass substrate is n1, and the refractive index of the gain medium in the laser resonant cavity is nd. The cavity length ld satisfies ndld = k1λ/2. The antireflective coatings on the mirror meet n2l4 = (2k4 + 1)λ/4. The reflective coatings on the mirror meet n3l6 = (2k6 + 1)λ/4. The refractive index of the external resonance cavity is nc, and its length is l7, the condition nclc = k7λ/2 is satisfied. k1, k4, k6, and k7 are all positive integers.
In order for RF to satisfy the optical feedback phase condition, it’s required that the phase of RF and laser is in-phase. Under the in-phase requirement, there is
The laser is reflected in the II-interface to form DRF, which returns to the laser along the original optical path. And the phase when it returns to the I-interface is
While the above is a phasic analysis of RF and DRF, in the following the optical feedback will be analyzed from the view of the electromagnetic field. Assuming there is enough time in RF to accumulate within the cavity, and disregarding any loss from the cavity front mirror, the reflected field E can be expressed as follows [19]:
Under steady-state conditions, the relationship between coupled frequency ωof and free-running frequency ω can be expressed as [19]
3. Discussion
To assess whether DRF impacts the optical feedback frequency locking, the optical feedback response curves of the linear cavity under two distinct scenarios are simulated based on Eq. (2). The first scenario focuses only on RF (F1 = 0, F2 = 1), while the second scenario involves both RF and DRF (F1 = 1, F2 = 1). The simulations are conducted with the same parameters as in Ref. [25], specifically R1 = R2 = 0.99986, l0 = lc = 0.394 m, β = 3 × 10−5, α = 2, ld = 10−3 m, nd = 3.5, rd = 0.6, and the results are shown in Fig. 2. In Fig. 2(a), as the laser frequency approaches the cavity resonant frequency (the resonant point), RF occurs constructive interference because of the phase coherence effect. This results in the narrowing of the coupled frequency, revealing a flat region. When the laser frequency significantly deviates from the resonant frequency (the non-resonant point), the coupled frequency aligns with the free-running frequency. The actual laser frequency varies along the trajectory OCD (Point O is the midpoint of the optical feedback curve caused by RF, point C is the maximum point of the optical feedback curve caused by RF, and point D is the vertical point of point C on the optical feedback curve). In Fig. 2(b), at the resonant point, RF also occurs constructive interference, and DRF appears destructive interference because of the π -phase difference. But at non-resonant point, the optical feedback effect of DRF is augmented due to the phase change, inducing an oscillatory variation. The actual laser frequency varies along the trajectory ABCD (Point A is the point on the optical feedback curve caused by DRF with the maximum slope (positive infinity) before the resonant frequency moves to point O, and point B is the vertical point of point A on OC or its extension line). Thus, the locking range is linked to the half-width of the oscillatory region OB influenced by DRF, and the half-width of the flat region OC influenced by RF. Based on the oscillatory and flat variation, a preliminary conclusion can be obtained that DRF does not influence the frequency locking at the resonant point. However, at non-resonant point, DRF can impact the frequency locking, which may lead to a scenario where OB becomes larger than OC, impeding frequency locking.
Under the same conditions of Fig. 2(b), the optical feedback response curves with different values of β are simulated, as shown in Fig. 3(a). For small values of β (the blue line), the coupled frequency exhibits variation along A1B1C1D1. The region OB1 is smaller than the region OC1, resulting in a locking range B1C1. As β gradually increases to 1.5 × 10−4 (the black line), OB3 surpasses OC3, causing the coupled frequency to directly jump from point A3 to point D3, eliminating the locking range. Therefore, the presence of DRF does significantly impact the frequency locking. The achievement of frequency locking hinges on the proportional sizes of OB and OC. Frequency locking is achievable when OB is smaller than OC, but it becomes unattainable when OB exceeds OC.
Fig. 3. The optical feedback response curves of the linear cavities with different parameters: (a) varying β, (b) varying α.
According to Eqs. (5) and (6), the optical feedback effect is not only contingent on β, but also relies on the laser’s intrinsic parameters, including α, nd, ld, and rd. As depicted in Fig. 3(b), an increase in α leads to concurrent increments in OB and OC, but OB exhibits a greater change. Beyond a specific threshold, OB will surpass OC, resulting in the absence of the locking range. In such instances, achieving frequency locking can be re-implemented by decreasing the feedback rate, as proposed in Ref. [25], as shown by the blue line in Fig. 4(a). When α further increases, a much smaller β is needed; for example, the critical condition where OB equals OC is β = 9 × 10−7 at α = 20. According to the findings [26], a feedback rate below 10−6 may broaden the laser linewidth, making it challenging to lock the high-finesse resonant cavity. At the same time, achieving the feedback rate of 10−6 is not easy in experiments due to limitations such as polarizing beam-splitting prisms. Alternatively, frequency locking can be accomplished by attenuating DRF by using techniques from Refs. [10], [15], and [22]. For instance, setting F1 = 0.5 will allow the feedback formed by RF to regain dominance, resulting in the emergence of the locking range, as illustrated by the blue line in Fig. 4(b). In addition to decreasing F1, increasing F2 can also achieve frequency locking, as shown by the black line in Fig. 4(b).
Fig. 4. The common approaches for optical feedback locking range: (a) decreasing β, (b) decreasing F1 (the blue line) or increasing F2 (the black line).
Figures 5(a), (c), and (e) present simulations of the optical feedback response curves in varying nd, ld, and rd. It’s evident that with an increase in these parameters, both OB and OC decrease, but the decreased amplitude of OB is more pronounced. Hence, augmenting nd, ld, and rd promotes the achievement of frequency locking. For smaller values of nd, ld, and rd, e.g., nd = 1.5, ld = 5 × 10−4 m, and rd = 0.4, the realization of frequency locking by decreasing the feedback rate (the blue line) or the feedback coefficients of DRF (the red line) and increasing the feedback coefficients of RF (the black line) is further demonstrated in Figs. 5(b), (d), and (f). In general, achieving frequency locking in the linear cavity is facilitated by diminishing β or F1 and increasing nd, ld, rd, or F2. However, for commercial lasers, parameters such as α, nd, ld, and rd are often fixed, and β cannot be indefinitely decreased in experiments. Accordingly, the more common approach involves attenuating DRF and increasing RF.
Fig. 5. The optical feedback response curves for the linear cavities with different parameters: (a) decreasing nd, (b) decreasing β (the blue line) or F1 (the red line) and increasing F2 (the black line) at nd = 1.5, (c) decreasing ld, (d) decreasing β (the blue line) or F1 (the red line) and increasing F2 (the black line) at ld = 5 × 10−4 m, (e) decreasing rd, (f) decreasing β (the blue line) or F1 (the red line) and increasing F2 (the black line) at rd = 0.4.
By diminishing β and increasing nd, ld, and rd, K is the last to be affected according to the Eq. (6). The magnitude of K plays a crucial role in the possibility to establish optical feedback. The simulations are conducted with different values of K, and the results are presented in Fig. 6. The figure illustrates that the frequency locking range is present when K is less than 1.05 × 109, but not when K is greater than 1.05 × 109. The critical value is K = 1.05 × 109, but its size depends on lc. For example, when lc = 0.294 m, the critical value is 1.41 × 109, and when lc = 0.494 m, the critical value is 0.84 × 109. The product of the critical value of K and lc is approximately equal to 4.15 × 108.
The front mirror reflectivity, R1, also impacts the reflected field intensity, as indicated in Eq. (4). Simulations are conducted for the optical feedback response curve at R1 = R2 = 0.99986 (the red line) and R1 = 0.999, R2 = 0.99986 (the black line), as illustrated in Fig. 7. Decreasing R1 can increase the power of RF and decrease the power of DRF, which enhances OC and diminishes OB simultaneously. Moreover, decreasing R1 concurrently optimizes the impedance-matching effect, which helps improve the efficiency of laser coupling into the cavity [27]. Consequently, this paper proposes a model that can achieve frequency locking more easily. This is achieved by introducing asymmetry between the reflectivity of the cavity front and back mirrors (R1 < R2), ensuring the dominance of the RF in the frequency locking.
Fig. 7. The optical feedback response curve of the linear cavities before (the red line) and after (the black line) decreasing R1.
4. Experiment and results
To validate the correctness of this model, an optical feedback test platform is meticulously constructed, as depicted in Fig. 8(a). The platform incorporates a single-mode multiple quantum well laser (Laserwave, LWRL642) emitting at 642 nm. The ant-triangular wave is generated to modulate the laser current by a function generator (Tektronix, AFG3022C). The laser traverses several optical elements, including a mirror with a piezoelectric transducer, two mirrors, a half-wave plate, a polarized beam splitter, a Faraday rotator, a quarter-wave plate, and a mode-matching lens, and is coupled into the linear F-P cavity. Half-wave plate, polarized beam splitter, Faraday rotator and quarter-wave plate are used to control the feedback rate. The transmitted field is detected by a photodetector (Thorlabs, PDA100A-EC), and data acquisition is performed using a data acquisition card (National Instruments, USB-6363). The LabVIEW program on the personal computer analyzes the cavity mode signal for extracting the feedback phase adjustment signal. This signal is then fed into a high-voltage amplifier (Coremorrow, E01.D1), which controls the expansion and contraction movement of the piezoelectric transducer (Physik Instrumente, P-016.10 H), thereby regulating the feedback phase. This adjustment ensures that the distance between the laser and the cavity front mirror is an integer multiple of the length of the cavity.
Fig. 8. The optical feedback test platform for the asymmetric linear cavities and experimental results: (a) test platform, (b)transmitted spot, (c) cavity mode signal.
Two plane mirrors, featuring reflectivity of 0.9996 and 0.99994, are selected as the cavity front mirrors. These mirrors, in conjunction with a plano-concave mirror with R2 = 0.99994, collectively form a linear F-P cavity. The oscillations of the fundamental mode are observed at R1 = 0.9996, R2 = 0.99994, and at R1= R2 = 0.99994. The transmitted spot mode is observed as transverse electromagnetic mode 00 (TEM00) by the beam quality analyzer (Ophir, SP920s), as shown in Fig. 8(b). However, by adjusting the angle of the mirrors, stable cavity mode signals can only be observed in the former condition, while chaotic cavity mode signals are observed in the latter. Despite repeated adjustments, stable cavity mode signals can not be observed. The stable cavity mode signal monitored by the photodetector is shown in Fig. 8(c), and shows a periodic distribution with no higher-order transverse modes. This observation implies that by decreasing R1, linear cavity optical feedback frequency locking can be established more easily.
5. Conclusion
In conclusion, we have meticulously evaluated the influence of DRF on frequency locking through the constructed linear-cavity coupling model. It’s found that DRF has no impact at the resonant point but influences at the non-resonant point, causing an oscillatory variation. Optical feedback frequency locking is determined to require a flat region half-width wider than the oscillatory region, achievable by decreasing the feedback rate, laser’s linewidth enhancement factor, and DRF, while increasing the refractive index of the gain medium, length of the laser’s resonant cavity, electric field reflectivity of the laser’s output facet, and RF. These parameters can affect the feedback coupling rate pre-factor, and for a resonant cavity with a length of 0.394 m, optical feedback can only be established when the feedback coupling rate pre-factor is less than 1.05 × 109. Finally, a simple and effective frequency locking model is introduced to increase RF and attenuate DRF based on the asymmetric reflectivity of the front and back mirrors, validated through experimentation. This research offers valuable insights into advancing the understanding of linear cavity optical feedback frequency locking and can be applied to various cavity-enhanced spectroscopic techniques relying on optical feedback. Furthermore, existing research on obtaining narrow linewidth diode lasers through optical feedback techniques mostly employs V-shaped cavity structures [28,29] given the complexity of conventional linear-cavity optical feedback systems. Therefore, our study holds significant potential to advance the adoption of linear cavities in this field.
Funding
Science and Technology Foundation of State Grid Corporation of China (SGCQDK00SBJS2200274).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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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