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Although conventional ultra-stable lasers have high application value, they operate in the good-cavity regime, where the linewidth of the laser cavity Γcavity is much narrower than the optical gain bandwidth Γgain, and cavity-length fluctuations strongly influence the laser frequency ν0 through the so-called cavity pulling effect9. Therefore, it is essential for the cavity to be carefully engineered. An ultrahigh finesse \({{\mathcal{F}}}\) is of great importance since it determines the resonance linewidth of the reference cavity. The typical \({{\mathcal{F}}}\) of an optical cavity has gradually increased to the order of 105 10,11,12,13,14,15,16,17,18,19,20 (see Fig. 1a), which leads to a resonance linewidth at, for example, the 10 kHz level for a cavity length of 10 cm. In addition, great efforts such as choosing crystalline silicon as cavity material, placing the cavity in a vacuum chamber, and cryogenic cooling of the cavity have to be made to maximally isolate the reference cavity from environmental perturbations (e.g., vibrations and Brownian-motion thermal noise)15,20. The linewidth of a laser stabilized to such a cryogenic reference cavity can be narrowed down to 5 mHz, corresponding to a coherence time of tens of seconds19. Nevertheless, even for the state-of-the-art stable laser using ultrahigh-finesse cavity, the inevitable cavity-length thermal noise introduces a time-integrated phase drift21,22,23,24.

Abbott, B. P. et al. Observation of gravitational waves from a binary black hole merger. Phys. Rev. Lett. 116, 061102 (2016).

In the presence of an extremely bad cavity, weak optical feedback is introduced and the 1470-nm laser power grows strongly as shown in Fig. 3a. This is understandable because of the stimulated emission by the gain medium, the feedback is enhanced, thereby raising the output power and suppressing the lasing threshold (see the inset in Fig. 3a). Remarkably, the width of the laser spectrum is dramatically suppressed down to, for example, Δν = 1.2 kHz with R = 0.485% (\({{\mathcal{F}}}=2.01\)), leading to a spectral narrowing factor of about 250 compared to the mirrorless lasing (see Fig. 3b). It may be ascribed to the fact that the coherence of the lasing signal is enhanced by coherent photons (i.e., reflected laser portion). The resultant laser frequency stability47 potentially reaches \({\sigma }_{y}(\tau )=6.8\times 1{0}^{-14}/\sqrt{\tau }\). Actually, after eliminating the common-mode noise such as cavity vibrations, pump power fluctuations, and vapor cell temperature variation, the laser linewidth may be further suppressed down to Δν = 341 Hz (see Fig. 3c and “Methods”). That is, the laser linewidth is extremely sensitive to the cavity reflectance within the range of 0 < R < 0.485%. However, due to the limitations of the experiment, it is challenging to determine the specific sensitivity relation. As illustrated in Fig. 3c, the laser linewidth Δν can still be maintained around 1.2 kHz, even though R approaches to 0. This is a new way of obtaining narrow linewidths in addition to good-cavity lasers using ultrahigh-finesse cavities.

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Here, R = r1r2 denotes the cavity reflectance with the amplitude reflection coefficients r1,2 of two cavity mirrors. Reducing R decreases \({{\mathcal{F}}}\), thereby broadening Γcavity. The laser system enters the bad-cavity regime when Γcavity surpasses Γgain. Active optical clocks are typically operated in this regime because of the strong suppression of the cavity pulling24. Nevertheless, lasing in the extremely bad-cavity regime, where the cavity reflectance R closes to zero and the cavity only provides weak optical feedback, has never been accessed. In this extreme situation, Eq. (1) is simplified as \({{\mathcal{F}}}\approx 2+\frac{8R}{\pi (1+{R}^{2})-4R}\) and \({\Gamma }_{{{\rm{cavity}}}}\approx \frac{{{\rm{FSR}}}}{2}\). It is seen that as R tends to zero, \({{\mathcal{F}}}\) approaches 2, at which the optical feedback almost completely vanishes, i.e., mirrorless, rather than zero. In what follows, we investigate the 1470-nm lasing dynamics in five extremely bad optical cavities with R ranging from 0.485% (\({{\mathcal{F}}}=2.01\)) to 22.4% (\({{\mathcal{F}}}=2.78\)) (see Table 1 in “Methods”) and \({{\mathcal{F}}}\,\approx\, 2\) (mirrorless) as well.

The temperature T of the Cs vapor cell is stabilized at 116.2 °C. Although temperature fluctuations are controlled within 0.1 °C, the vapor cell still experiences a long-term temperature drift. To evaluate the influence of temperature fluctuations on the 1470-nm laser, we measure the dependence of the 1470-nm laser frequency on T over a range of 3.5 °C. It is found that the laser central frequency shifts approximately linearly with T and the curve fitting gives a slope of −530 ± 30 kHz/°C (see Fig. 4c). In addition, the temperature monitoring of the vapor cell yields a temperature stability of 3.7 × 10−6 at 1 s (see Fig. 4d) of averaging. The variation of the 1470-nm laser frequency caused by temperature fluctuations is estimated to be 228 Hz, corresponding to a fractional frequency stability of 1.1 × 10−12 the average time of 1 s.

Lasers are one of the greatest inventions of the twentieth century and have been the most versatile tool available for scientific, industrial, and medical applications, owing to their directionality, high brightness, monochromaticity, and high degree of coherence1. Ultra-stable lasers with ultranarrow linewidths are highly desired for quantum optics, precision spectroscopy, and fundamental physics measurements. Recently, the ground-based observatory consisting of twin laser interferometers has enabled the detection of ripples in space-time caused by binary black hole mergers2; optical clocks based on ultra-stable lasers have resolved the gravitational redshift at millimeter scale3,4; laser-based gyroscopes allow measurements of Earth’s rotation-induced optical path change of the order of 10−16 cm, which is one-thousandth of the classical electron radius5,6; and optical whispering-gallery sensors are capable of detecting single molecules and even ions7. In all these applications, it is necessary for lasers to be (pre-)stabilized to high-finesse optical reference cavities using the Pound-Drever-Hall (PDH) technique8. Stabilizing the laser frequency to a passive reference cavity with higher stability not only substantially suppresses the laser spectrum broadening but also improves the (short-term) laser frequency stability.

We begin with the mirrorless radiation in the absence of the optical cavity. To avoid any reflection, vapor cell windows are oriented at Brewster’s angle (see “Methods”). Emitting the 459-nm pump laser into thermal atoms, we observe the continuous-wave 1470-nm radiation with the fundamental mode (TEM00) imaged on a high-sensitivity fast PIN photodetector (see Fig. 1c). Analogous to conventional lasers, the 1470-nm radiation displays a threshold behavior characterized by a rapid rising power as the pump power is increased (see Fig. 3a). At a high pump power, more atoms can be populated in 7S1/2, not only enhancing the spontaneous emission input signal but also providing a sufficient optical gain that is essential to trigger the so-called continuous-wave amplified spontaneous emission (mirrorless lasing43).

An optical cavity that bounds the optical wave modes is not always a prerequisite for laser generation. For a medium with sufficient optical gain, incoherent photons generated by spontaneous radiation can be amplified along the medium, eventually outputting a large number of coherent photons, which is called mirrorless lasing. Mirrorless lasers do not require an optical resonant cavity, so their cavity-pulling effects can be completely eliminated, and they are expected to become the absolute frequency standard50.

This research was funded by the National Natural Science Foundation of China (NSFC) (91436210), Innovation Program for Quantum Science and Technology (2021ZD0303200), China Postdoctoral Science Foundation (BX2021020), and Wenzhou Major Science & Technology Innovation Key Project (ZG2020046).

Norcia, M. A. & Thompson, J. K. Cold-strontium laser in the superradiant crossover regime. Phys. Rev. X 6, 011025 (2016).

Laske, T., Winter, H. & Hemmerich, A. Pulse delay time statistics in a superradiant laser with calcium atoms. Phys. Rev. Lett. 123, 103601 (2019).

Young, B., Cruz, F., Itano, W. & Bergquist, J. Visible lasers with subhertz linewidths. Phys. Rev. Lett. 82, 3799 (1999).

The atoms in the vapor cell (length l = 5 cm) are directly pumped by the 459-nm laser in the absence of cavity mirrors. Cell windows are set at Brewster’s angle to prevent any reflection of 1470-nm light. As shown in Fig. 7, the 1470-nm laser power depends strongly on the vapor cell temperature T and is maximized at T = 116.2 °C, where the atomic density n = 5.2 × 1013 cm−3 and the number of atoms reaches N = nLS = 4.39 × 1012. Due to the Doppler broadening, only thermal atoms with the velocity between −vD/2 and vD/2 can be effectively pumped to 7P1/2. According to the Maxwell velocity distribution, the effective number of atoms contributing to the 1470-nm laser is then given by \({N}_{{{\rm{eff}}}}=N\int_{-{v}_{{{\rm{D}}}}/2}^{{v}_{{{\rm{D}}}}/2}\frac{1}{\sqrt{2\pi \Delta v}}{{{\rm{e}}}}^{{(-v/\Delta v)}^{2}}dv=7.55\times 1{0}^{10}\). Here, the velocity distribution width of thermal atoms is computed as \(\Delta v=\sqrt{{k}_{{{\rm{B}}}}T/m}=156\,{{\rm{m}}}\,{{{\rm{s}}}}^{-1}\)(≫vD) with the atomic mass m and Boltzmann constant kB. The mirrorless superradiance spectrum is measured through the beating signal with an extremely bad-cavity laser (R = 0.485%).

Laser cavityfilling

Extremely bad-cavity lasers represent a new exciting field of research that synthesizes laser physics and the science of quantum precision measurement. The physical mechanisms underlying the extremely bad-cavity lasers have never been studied before. Thus, we believe that these findings might open a completely new research area of physics such as active optical clock, cavity QED, and quantum optics. As an important application, the concept of an extremely bad-cavity laser offers a promising opportunity for the implementation of novelly super-stable lasers, which might overcome the practical challenges faced by ultrahigh-finesse optical cavities.

Zhang, J., Shi, T., Miao, J. et al. An extremely bad-cavity laser. npj Quantum Inf 10, 87 (2024). https://doi.org/10.1038/s41534-024-00880-3

Kuppens, S., Van Exter, M. & Woerdman, J. Quantum-limited linewidth of a bad-cavity laser. Phys. Rev. Lett. 72, 3815 (1994).

Optical resonator inlaserpdf

Yu, D., Zhang, J., Zhang, S. & Chen, J. Prospects for an active optical clock based on cavityless lasing. Adv. Quant. Technol. 7, 2300308 (2024).

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Module (Mod) I and II are 459-nm external cavity diode lasers stabilized through modulation transfer spectroscopy. Mod III contains two 1470-nm lasers and the heterodyne measurement is implemented in Mod IV. IF-ECDL interference filter configuration external cavity diode laser, ISO isolator, PBS polarizing beam splitter, HWP half-wave plate, EOM electro-optic modulator, PD photodetector, SIG signal generator, PID proportional-integral-derivative locking system, FC fiber coupler, FA frequency analyzer. M1 is the 459-nm reflective mirror and M2 is a dichroic mirror with the anti-reflection coating at 1470 nm and the high-reflection coating at 459 nm.

Chen, H., Jiang, Y., Fang, S., Bi, Z. & Ma, L. Frequency stabilization of Nd: YAG lasers with a most probable linewidth of 0.6 Hz. J. Opt. Soc. Am. B 30, 1546–1550 (2013).

In this work, an extremely bad-cavity laser with cavity finesse close to the limit of 2 is demonstrated. The obtained laser power is as high as tens of μW and the laser linewidth reaches 1.2 kHz, limited by pump power and vapor cell temperature fluctuations. We also explore the suppression of cavity pulling and find a pulling coefficient of 0.0148, the lowest value ever achieved for a continuous-wave laser. Additionally, the mirrorless lasing is studied for comparison. Although the cavity pulling vanishes completely in the absence of the cavity, the mirrorless superradiance linewidth is strongly broadened by two orders of magnitude, compared to that of the extremely bad-cavity laser. That is, the laser spectrum is extremely sensitive to the weak optical feedback from the mirrorless limit to the extremely bad-cavity regime.

Kessler, T. et al. A sub-40-mHz-linewidth laser based on a silicon single-crystal optical cavity. Nat. Photon. 6, 687–692 (2012).

a Dependence of finesse on cavity reflectance. In the extremely bad-cavity regime, the cavity reflectance R is close to zero. Symbols: logarithmic cavity finesse \({{\rm{Lo{g}}_{10}({{\mathcal{F}}}-\; 2)}}\) and cavity linewidth Γcavity used in experiment. Solid line: Eq. (1). Dashed line: Γcavity derived from Eq. (1) with the free spectral range FSR = 882 MHz. b Sensitivity of the 1470-nm laser frequency ν0 to the detuning δ of the optical cavity from the atomic transition frequency for different cavity reflectances R. For a more intuitive representation, we have vertically shifted the cavity pulling data measured under different cavity reflectivity conditions, as well as so that there is no overlap between the data points. Symbols: central frequency of beat note between two 1470-nm lasers in the same environment. The detuning of one laser is set at zero and the detuning of the other varies. Dash-dotted lines: linear curve fitting. The slope of a curve denotes the corresponding cavity-pulling coefficient at δ. For all measurements, the pump power is set at 5.2 mW and the temperature is 116.2 °C. Each data point in b is the averaged result of 5 measurements, and the error bars represent the standard deviation.

Figure 1b illustrates the general setup of the experiment, where a cylindrical glass cell (diameter of 2 cm and length l = 5 cm) of Cs atoms is placed inside an invar optical cavity (length L = 17 cm and free spectral range FSR = 882 MHz) whose finesse is adjustable. A home-made 459-nm external cavity diode laser (spot area S = 1.69 mm2) with an interference filter configuration is used to drive atoms in the ground 6S1/2 state to the excited 7P1/2 state (see the inset in Fig. 1b). To suppress the frequency drift (so as to provide a continuous pump), the 459-nm pump laser is stabilized to the 6S1/2(F = 4)-\(7{{{\rm{P}}}}_{1/2}({F}^{{\prime} }=3)\) hyperfine transition in Cs through the modulation transfer spectroscopy as shown in Fig. 1c (for more details refer to ref. 44).

J.C. conceived the idea to use an extremely low-finesse cavity to realize the extremely bad-cavity laser as a stable active optical clock. J.Z., T.S., and J.M. performed the experiments. J.Z. and D.Y. carried out the theoretical calculations and wrote the manuscript. All authors participated in the discussion of results and revision of the manuscript.

Laser cavitytypes

In the experiment, we observe only 1470 nm mirrorless lasing in the direction of the 459 nm pump laser. Also, it is found that the laser output power depends strongly on the atomic cell length. This is understandable, considering that the laser action starts with the spontaneous emission of photons at one end of the active sample. Only a sufficiently high spontaneous emission input can trigger the light amplification process along the gain medium atoms. In fact, if all atoms inside the cesium cell are in a state of population inversion, then the emission axis of the mirrorless laser should depend on the shape of the cell. It is expected that the power of the mirrorless laser is greatest in the direction along the maximum optical depth.

by MS Hur · 2023 · Cited by 18 — We propose a new method of compressing laser pulses to ultrahigh powers based on spatially varying dispersion of an inhomogeneous plasma.

Häfner, S. et al. 8 × 10−17 fractional laser frequency instability with a long room-temperature cavity. Opt. Lett. 40, 2112–2115 (2015).

Kolobov, M., Davidovich, L., Giacobino, E. & Fabre, C. Role of pumping statistics and dynamics of atomic polarization in quantum fluctuations of laser sources. Phys. Rev. A 47, 1431 (1993).

a Trend of logarithmic finesse \({{{\rm{Log}}}}_{10}({{\mathcal{F}}}-2)\)45 over time. Red circle symbols correspond to the good-cavity finesse in Pound-Drever-Hall frequency stabilization method10,11,12,13,14,15,16,17,18,19,20 and blue square symbols denote the bad-cavity finesse in active optical clocks (AOCs)25,26,27,28,29,30,31,32,33. b Schematic diagram of extremely bad-cavity laser. A 459-nm laser drives thermal Cs atoms from the ground 6S1/2 to the excited 7P1/2 level. The population inversion is achieved between upper 7S1/2 and lower 6P3/2 levels through the spontaneous emission from 7P1/2 to 7S1/2. An extremely bad cavity with the cavity reflectance R = r1r2 and amplitude reflection coefficients r1,2 of cavity mirrors is used to introduce weak optical feedback. The lasing action at the wavelength of 1470 nm occurs when the pump exceeds the optical loss. The laser spectrum is narrowed as R grows, as shown in the bottom image. c Lasing at 1470 nm. The blue curve denotes the dispersion-shaped modulation transfer spectroscopy (MTS) signal that is used to stabilize the 459-nm pump laser to the 6S1/2(F = 4)-\(7{{{\rm{P}}}}_{1/2}({F}^{{\prime} }=3)\) hyperfine transition in Cs. The red curve shows the corresponding 1470-nm laser power. Inset: Transverse mode of the 1470-nm laser imaged on a high-sensitivity fast photodetector.

Bohnet, J. G., Chen, Z., Weiner, J. M., Cox, K. C. & Thompson, J. K. Relaxation oscillations, stability, and cavity feedback in a superradiant Raman laser. Phys. Rev. Lett. 109, 253602 (2012).

Insert symbols: experimental results with the resolution bandwidth (RBW) of 47 Hz. The Lorentz fit (solid line) results in a spectral linewidth of 482 Hz, indicating a linewidth of 341 Hz for each laser mode.

When measuring the 1470-nm mirrorless superradiance power (red squares), a long-pass interference filter is used to filter out the 1359-nm component in the output light. The total power of the output light (black circles) is measured in the absence of the long-pass interference filter. For all measurements, the 459-nm pump power is set at 5.2 mW.

Laser cavityalignment

Pan, D., Arora, B., Yu, Y., Sahoo, B. & Chen, J. Optical-lattice-based Cs active clock with a continual superradiant lasing signal. Phys. Rev. A 102, 041101 (2020).

Robinson, J. M. et al. Crystalline optical cavity at 4 K with thermal-noise-limited instability and ultralow drift. Optica 6, 240–243 (2019).

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Hotter, C., Plankensteiner, D., Kazakov, G. & Ritsch, H. Continuous multi-step pumping of the optical clock transition in alkaline-earth atoms with minimal perturbation. Opt. Express 30, 5553–5568 (2022).

Miao, J., Shi, T., Zhang, J. & Chen, J. Compact 459-nm Cs cell optical frequency standard with 2.1 × 10−13/ short-term stability. Phys. Rev. Appl. 18, 024034 (2022).

Moreover, within the pump power range, we did not observe the saturation behavior of the 1470-nm laser power, although the 459-nm pump-beam intensity well exceeds the saturation intensity of the atomic 6S1/2-7P1/2 transition. This is because a high pump power may excite more thermal atoms in different velocity groups from 6S1/2 to 7P1/2, thereby always enhancing the population inversion on the laser transition. The velocity distribution of effective atoms contributing to the 1470-nm laser is far narrower than the Doppler velocity distribution.

In fact, the frequency shift due to changes in the cell temperature depends strongly on the cavity length. This is because the thermal expansion of the glass walls at both ends of the vapor cell leads to changes in the equivalent cavity length. In ref. 30, after locking the cavity length, the slope of the 1470 nm laser frequency shift with temperature fluctuations was below 45 ± 1.2 kHz/°C. In this work, the cavity length is free-running. Therefore, assuming that the variation in cavity length caused by changes in cell temperature can be ignored, the linewidth broadening caused by the temperature fluctuations of the atomic vapor cell is about 19.3 Hz. To minimize the effects of temperature fluctuations, we are preparing cold atoms as the gain medium.

The laser frequency shift is measured through the beat note between two independent 1470-nm lasers. a Dependence of laser frequency on pump power. The pump power of one laser is set at 5.2 mW and the pump power of the other varies. The temperature of both lasers is set at 116.2 °C. Symbols: experimental results. Dash-dotted line: linear curve fitting with the resultant slope of −750 ± 20 kHz/mW. b Power stability of the 459-nm pump laser. The shaded area denotes the error band. c Dependence of laser frequency shift on vapor cell temperature T. Symbols: experimental results. Dash-dotted line: linear curve fitting with the resultant slope of −530 ± 30 kHz/°C. The temperature of one laser is set at 116.2 °C and the temperature of the other varies. d Temperature stability at 116.2 °C. The shaded area denotes the error band, indicating 1-σ confidence intervals for each Allan deviation value.

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Norcia, M. A. et al. Frequency measurements of superradiance from the strontium clock transition. Phys. Rev. X 8, 021036 (2018).

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Yu, D. et al. Whispering-gallery-mode sensors for biological and physical sensing. Nat. Rev. Methods Prim. 1, 83 (2021).

We demonstrated the extremely bad-cavity laser under a weak magnetic field and measured the beating linewidth between Zeeman sublevels. By applying a weak magnetic field perpendicular to the direction of the light to the vapor cell, a splitting of the energy levels 7S1/2 and 6P3/2 occurs. As a result, beating signals between different Zeeman levels can be observed, as shown in Fig. 6. Here, we select a typical beat frequency signal for Lorentz fitting, demonstrating a power spectrum with a linewidth of 482 Hz (see the Fig. 6 insert). Assuming equal contribution from each laser mode in the beating linewidth between two Zeeman sublevels, the linewidth of each laser mode is 341 Hz, primarily reflecting the limited linewidth of our extremely bad-cavity laser using thermal atoms. Because the two Zeeman sublevels share the same cavity, which has an elimination of the common-mode noise, such as the cavity-length vibrations, the pumping power fluctuations, and the vapor cell temperature changes, the beating linewidth is narrower than that between two extremely bad-cavity lasers.

Shi, T., Pan, D. & Chen, J. Realization of phase locking in good-bad-cavity active optical clock. Opt. Express 27, 22040–22052 (2019).

To reduce the influence on the spectral linewidth and frequency stability of the 1470-nm extremely bad-cavity laser caused by the Doppler broadening and temperature fluctuations of the thermal atom’s active medium, we are preparing cold atoms as the gain medium. Recently, a one-dimensional 1-m-long sample of cold Rb atoms based on diffuse laser cooling has been demonstrated in experiment48. The resultant temperature of atoms reaches as low as 25 μK and the corresponding Doppler broadening is well below the natural linewidth of the laser (clock) transition. Using this cooling method combined with a four-level system, when pumping light is injected, the cooling light and the repumping light can be turned on simultaneously to cool the atoms, and the surrounding cold atoms can be replenished quickly and benefit from the large volume of cold atoms. This approach holds promise for a cold-atom active optical clock that can operate in a continuous manner and in the extremely bad-cavity regime.

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We examine the cavity pulling based on our laser system. To determine the laser frequency shift ν0 − νa, the heterodyne detection is performed on two identical 1470-nm lasers that are in the same environment (see “Methods”). The cavity detuning for one laser is set to be zero while the detuning νc − νa for the other laser is precisely tuned within the range −60 < νc − νa < 60 MHz by adjusting the piezoelectric ceramic. The central frequency of the beat note, i.e., ν0 − νa, can be extracted using the frequency analyzer (FA, Keysight N9000B). Measurement results for different cavity reflectivities R (finesses \({{\mathcal{F}}}\)) have been summarized in Fig. 2b, which manifests the linear relationship between ν0 − νa and νc − νa. The pulling coefficient P is then derived through the linear curve fitting.

a Dependence of the 1470-nm laser power on the 459-nm pump power. Symbols: experimental results. Solid lines: linear curve fitting above the threshold. Inset: laser threshold as a function of the cavity reflectance R. b Laser spectrum. Symbols: experimental results. The spectral linewidth is derived through the Lorentz fit (solid curves). Inset: detailed laser spectrum with R = 0.485%. c Laser linewidth as a function of R. Each data point for the linewidth of the extremely bad-cavity laser and mirrorless laser is the average of 80 measurements, and the error bars represent the standard deviation.

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State Key Laboratory of Advanced Optical Communication Systems and Networks, Institute of Quantum Electronics, School of Electronics, Peking University, Beijing, 100871, China

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The optical gain bandwidth is related to the pump power broadening and the Doppler broadening of the laser transition. The atomic 6S1/2-7P1/2 transition (pump line) has a natural linewidth of γpump = 1.1 MHz and the corresponding saturation intensity is evaluated as Is = 1.27 mW cm−2 49. The 459-nm pump laser used in the experiment has a spot area of S = 1.69 mm2. The pump light intensity reaches I = 308 mW cm−2 at the typical pump power of 5.2 mW. Thus, the saturation broadening of the pump line is given by \({\Gamma }_{{{\rm{pump}}}}={\gamma }_{{{\rm{pump}}}}\sqrt{(1+I/{I}_{{{\rm{s}}}})}=17.2\,{{\rm{MHz}}}\). According to the velocity-selective mechanism, only atoms with a velocity in the pump-beam direction less than vD = Γpump × (459 nm) = 7.9 m s−1 can be effectively pumped to 7P1/2 and then decay to 7S1/2. Thus, the Doppler broadening of the atomic 6P3/2-7S1/2 line (laser transition) reaches ΓD = vD/(1470 nm) = 5.36 MHz. Combining the Doppler broadening with the natural linewidth of the laser transition γgain = 1.81 MHz, one obtains the optical gain bandwidth Γgain = γgain + ΓD = 7.17 MHz. Using Eq. (1) to obtain the different finesses, the corresponding cavity-mode linewidth \({\Gamma }_{{{\rm{cavity}}}}={{\rm{FSR}}}/{{\mathcal{F}}}\) are all much wider than the gain bandwidth, as shown in Table 1.

Laserresonator Pdf

Ludlow, A. D. et al. Compact, thermal-noise-limited optical cavity for diode laser stabilization at 1 × 10−15. Opt. Lett. 32, 641–643 (2007).

To reveal the intrinsic linewidth of the 1470-nm laser based upon thermal atoms, a weak magnetic field is used to induce energy-level splitting of the laser transition, and the beat note between 1470-nm laser beams based on different Zeeman sublevel transitions is measured. Since these Zeeman sublevel transitions share the same optical cavity, technical and environmental noises can be substantially suppressed in the beat spectrum, leading to a spectral linewidth as narrow as 482 Hz (see “Methods”). Assuming statistical independence and equal contribution of different 1470-nm laser beams, we infer the linewidth of 341 Hz (see Fig. 3c) for each laser, implying a short-time fractional frequency stability of 3.6 × 10−14.

In contrast to the pursuit of high finesse, a bad-cavity laser requires a cavity relaxation rate that exceeds the atomic relaxation rates by several orders of magnitude, which requires a low-finesse cavity. Indeed, it has been recognized that lasing in the bad-cavity regime, where Γcavity ≪ Γgain, strongly reduces the sensitivity of the laser frequency ν0 to cavity-length fluctuations so that ν0 is primarily determined by the atomic transition frequency νa23,24. Such bad-cavity lasers may directly serve as active optical clocks without the need for extra laser frequency stabilization to atomic transitions23,24,25,26,27,28,29,30,31,32,33, and there has been excellent theoretical work34,35,36,37,38,39,40,41 recently demonstrating this point. In addition, the relatively long coherence time of the macroscopic polarization of active atoms substantially suppresses the laser phase noise42, resulting in a laser linewidth that overcomes the usual Schawlow-Townes limit21,22,23. Contrary to the pursuit of high-finesse cavity in PDH frequency stabilization, bad-cavity lasers expect as low-finesse as possible, which results in two diametrically opposed trends of the cavity finesse, as shown in Fig. 1a. Thus far, most laser cavities used in experiments have a reflectance higher than 0.3 and a logarithmic finesse value \({{{\rm{Log}}}}_{10}({{\mathcal{F}}}-2)\) exceeding 0. Here, \({{\mathcal{F}}}\) approaching 2 denotes the mirrorless radiation (i.e., in the absence of the cavity)43. Nonetheless, lasing in the extremely bad-cavity regime, where R is close to zero (i.e., \({{\mathcal{F}}} \sim 2\)) and the laser cavity only provides extremely weak optical feedback, has never been accessed. In this work, we demonstrate for the first time the extremely bad-cavity laser whose mirror reflectance is close to zero, which may lead to new research in laser physics, detection of gravitational waves, quantum information, cavity QED fields, and so on. In addition to the interesting connections with many areas of fundamental science, we also anticipate that new applications and technologies will continue to emerge from the study of the physics of extremely bad-cavity lasers.

Yu, D., Chen, J. & Zhang, S. Active whispering-gallery microclock in pulsed-operation mode. Phys. Rev. A 107, 043712 (2023).

Above the threshold, the resultant laser spectrum is also strongly narrowed down to a width Δν = 281.6 kHz (see Fig. 3b), less than one-fortieth of the optical gain bandwidth Γgain. To the best of our knowledge, this is the largest spectral narrowing of the amplified spontaneous emission ever observed46. In addition, since the cavity effect vanishes in the mirrorless lasing, the laser frequency stability is completely determined by the atoms, and the corresponding Allan deviation potentially takes the form \({\sigma }_{y}(\tau )=\sqrt{\Delta \nu /2\pi {\nu }_{0}^{2}\tau }=1.0\times 1{0}^{-12}/\sqrt{\tau }\) at the averaging time τ47. Here, ν0 denotes the central frequency of the 1470-nm laser.

Overview of Tuffak® OP (Makrolon® OP) Optical Grade Polycarbonate ... Tuffak® OP (formerly Makrolon® OP) polycarbonate sheet is a clear, polished surface, UV ...

Hotter, C., Plankensteiner, D., Ostermann, L. & Ritsch, H. Superradiant cooling, trapping, and lasing of dipole-interacting clock atoms. Opt. Express 27, 31193–31206 (2019).

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Yu, D., Vollmer, F. & Zhang, S. Proposal for an active whispering-gallery microclock. Quant. Sci. Technol. 8, 025005 (2023).

Liu, H. et al. Rugged mHz-linewidth superradiant laser driven by a hot atomic beam. Phys. Rev. Lett. 125, 253602 (2020).

Laser cavityuses

Bychek, A., Hotter, C., Plankensteiner, D. & Ritsch, H. Superradiant lasing in inhomogeneously broadened ensembles with spatially varying coupling. Open Res. Europe 1, https://open-research-europe.ec.europa.eu/articles/1-73/v2 (2021).

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Lasing in the bad-cavity regime has promising applications in quantum precision measurement and frequency metrology due to the reduced sensitivity of the laser frequency to cavity-length fluctuations. Thus far, relevant studies have been mainly focused on conventional cavities whose finesse is high enough that the resonance linewidth is sufficiently narrow compared to the cavity’s free spectral range, though still in the bad-cavity regime. However, lasing output from the cavity whose finesse is close to the limit of 2 has never been experimentally accessed. Here, we demonstrate an extremely bad-cavity laser, analyze the physical mechanisms limiting cavity finesse, and report on the worst-ever laser cavity with finesse reaching 2.01. The optical cavity has a reflectance close to zero and only provides weak optical feedback. The laser power can be as high as tens of μW and the spectral linewidth reaches a few kHz, over one thousand times narrower than the gain bandwidth. In addition, the measurement of cavity pulling reveals a pulling coefficient of 0.0148, the lowest value ever achieved for a continuous-wave laser. Our findings open up an unprecedentedly innovative perspective for future new ultra-stable lasers, which could possibly trigger future discoveries in optical clocks, cavity QED, continuous-wave superradiant laser, and explorations of quantum many-body physics.

Cavitymodes inlaser

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What should you do if the high power objective lens touches or breaks the coverslip? Microscope Part. 1. 2. 3. 4. 5. 6. 7. 8.

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Laser cavitylength formula

Bothwell, T. et al. Resolving the gravitational redshift across a millimetre-scale atomic sample. Nature 602, 420–424 (2022).

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In principle, the population inversion can be also achieved on the 7S1/2-6P1/2 transition (wavelength of 1359 nm). However, the corresponding lasing action is inhibited in practice (see “Methods”). This is because the dipole moment of the 7S1/2-6P3/2 transition (2.4ea0 with elementary charge e and Bohr radius a0) is larger than that of the 7S1/2-6P1/2 transition (1.8ea0). Thus, more atoms in 7S1/2 decay to the 6P3/2 level, and eventually the competition prevents the lasing action upon the 6P1/2-7S1/2 transition. To prove that the lasing action on the 6P1/2-7S1/2 transition (wavelength of 1359 nm) is inhibited, a long-pass interference filter at 1359 nm is used to filter out the 1359-nm component in the output light. It is found that the power of the filtered light is almost equal to that of the unfiltered light (see Fig. 7).

The frequency fluctuations of the free-running 459-nm pump laser result in the large frequency shift and spectral broadening of the 1470-nm laser. To address this issue, the pump laser is stabilized to the 6S1/2(F = 4)-\(7{{{\rm{P}}}}_{1/2}({F}^{{\prime} }=3)\) hyperfine transition in Cs through the modulation transfer spectroscopy and the resultant frequency stability reaches \(2.8\times 1{0}^{-13}\sqrt{\tau }\) 44. However, power fluctuations of the pump laser still strongly influence the 1470-nm laser. As shown in Fig. 4a, the central frequency of the 1470-nm laser depends linearly on the pump power with a slope of −750 ± 20 kHz/mW. The power stability of the pump laser is measured to be 3.9 × 10−5 at 1 s (see Fig. 4b). Thus, at the typical pump power of 5.2 mW used in this work, the variation of 1470-nm laser frequency fluctuations induced by pump power fluctuations is estimated to be 152 Hz, corresponding to fractional frequency stability of 7.5 × 10−13 at 1 s of averaging. We also measure the relative intensity noise of the 459 nm laser, the results show that at 10 kHz, the RIN after frequency stabilization is increased by about 13 dB compared to unstabilized frequency. Thus, the pump power stability can be further enhanced using an acousto-optic modulator for power stabilization in subsequent experiments.

According to ref. 9, the central frequency ν0 of a conventional laser is pulled away from the atomic transition νa by an amount (ν0 − νa) = P(νc − νa) with the pulling coefficient P when the cavity frequency νc is detuned from νa. The cavity-pulling effect directly maps cavity (thermal and mechanical) fluctuations onto the laser frequency, deteriorating the stability of ν0. As has been pointed out in refs. 25,37,42, the laser phase information in the bad-cavity regime is primarily stored in the active medium’s polarization because of the atomic memory effect, and P can be much smaller than unity, making the active optical clock laser frequency robust against cavity fluctuations. Using cavity-mode linewidth Γcavity and atomic gain linewidth Γgain, P can also be expressed as \(P = \frac{\Gamma_{\text{gain}}}{{\Gamma _{\text{cavity}}}\, +\, {\Gamma _{\text{gain}}}}\). For lasers working in bad-cavity region, where Γgain ≪ Γcavity, P ≪ 1, the effect of cavity-mode frequency variations on output laser frequency is greatly suppressed. Nevertheless, the cavity pulling in the extremely bad-cavity limit has never been explored.

Atoms are accumulated in 7S1/2 via the spontaneous emission decay from 7P1/2 to 7S1/2, leading to the population inversion on the laser 7S1/2-6P3/2 transition (wavelength of 1470 nm, frequency νa = 204 THz, and natural linewidth of 1.81 MHz). The transition, although its linewidth is relatively broad, fits into the four-level structure. The energy levels of the pump laser and the clock laser are separated from each other, which helps to reduce the light shift introduced by the pump laser and achieve a truly continuous lasing output. The transition probability corresponding to natural linewidths on the order of MHz increases by nine orders of magnitude over those of mHz, allowing for a more efficient cycling lasing rate between the four energy levels (6S1/2, 7P1/2, 7S1/2, and 6P3/2) and contributing to the realization of laser output power on the order of ten microwatts33. Moreover, atoms can be easily coupled to the cavity, which largely reduces the difficulty of realizing stimulated radiation. The lasing action occurs once the pump rate exceeds the optical loss rate of the system. Taking into account the Doppler broadening, the optical gain bandwidth is estimated to be, for example, Γgain = 7.17 MHz at the pump power of 5.2 mW (see “Methods”). The vapor cell temperature is kept at T = 116.2 °C to maximize the laser output power (see “Methods”).

To measure the frequency shift of the extremely bad-cavity laser, two identical 1470-nm laser systems are built, where one laser plays the reference role. The specific schematic is depicted in Fig. 5, which contains four modules (Mod): Mod I and Mod II are two identical 459-nm pump lasers utilizing the modulation-transfer-spectroscopy frequency stabilization method. Mod III has two 1470-nm extremely bad-cavity lasers, where the finesse of optical cavities is adjustable. In each laser system, the plane mirror of the optical cavity (length L = 17 cm) is high transmission coated at 459 nm and the concave output mirror has a radius of curvature of 18 cm. A piezoelectric ceramic is bonded to the plane mirror so as to precisely adjust the cavity length. Mod IV is for heterodyne detection, where a frequency analyzer is used to evaluate the central frequency and spectral linewidth of the beating signal between two extremely bad-cavity lasers.

Various noises cause the frequency shift and extra broadening of the 1470-nm laser. We focus on two main sources, pump power and vapor cell temperature fluctuations.

Experimentally, P can be as low as 0.0148 (as shown in Fig. 2b and Table 1), i.e., the cavity pulling is suppressed by almost seventy times, the strongest suppression ever achieved for a continuous-wave laser, over a wide range of about 120 MHz adjusting cavity frequency.

Norcia, M. A., Winchester, M. N., Cline, J. R. & Thompson, J. K. Superradiance on the millihertz linewidth strontium clock transition. Sci. Adv. 2, e1601231 (2016).