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Abstract
Thermo-optic actuators based on bulk materials are considered too slow in applications such as laser frequency control. The availability of high-quality optical materials that have extremely fast thermal response times, such as diamond, present an opportunity for increasing performance. Here, diamond thermal actuators are investigated for configurations that use a planar thermal resistive layer applied to a heat-sinked rectangular prism. A general analytical formulation is obtained which simplifies substantially for high thermal conductivity such as diamond. Expressions for modulation depth, bandwidth and power requirements are obtained as functions of modulator dimensions and heat-transfer coefficients. For a 1 mm × 1 mm cross-section diamond at wavelength of 1 μm, around 450 W of applied heat power is needed to achieve a π phase shift at a modulation frequency of 2 kHz.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Single-longitudinal mode (SLM) lasers are fundamental tools in atomic physics, spectroscopy and optical metrology due to their spectral purity [1–4]. Their narrow linewidths are critical for high-precision optical measurement [5–7], and for addressing hyperfine transitions that underpin spectroscopic-based applications spanning areas as diverse as quantum science, sodium-beacon adaptive optics and environmental sensing [8–11]. In most applications, active frequency control is required to keep the centre frequency fixed to either an atomic transition or a reference etalon in the presence of environmental noise, which is achieved using a feedback scheme such as the Pound-Drever-Hall (PDH) locking technique [12,13]. As a result, some form of active frequency control is necessary for stabilized SLM lasers.
Frequency control is commonly implemented via cavity optical-path-length actuation of piezoelectric, thermal, electro-optic effects, or, in the case of diode lasers, free-carrier induced index changes. Modulation schemes are typically evaluated through bandwidth (the frequency range over which noise suppression is achieved), modulation depth (the amount of noise that can be corrected for) and power consumption. There is no universal “best" modulation scheme; the optimum for a given application depend on the specific architecture and environmental noise conditions.
In this paper, we explore the theory of thermo-optic modulator materials that have fast thermal response. To date, thermo-optic modulators have been most prevalent in resonant silicon-on-insulator (SOI) waveguide devices as an form of active phase control [14,15]. This adoption is principally being driven by the need for integrated devices that offer low power consumption and are compatible with scalable manufacturing techniques. Thermo-optic modulation in bulk devices is generally not used due to the slow speed of high heat-capacity devices, which constrains bandwidth and modulation depth. Diamond is a potential exception as its thermal conductivity 2-3 orders of magnitude higher than standard optical media (2200 versus 1-10 W/(m.K)). Furthermore, bulk diamond has also recently shown to be an effective SLM Raman and Brillouin laser gain medium [11,16–22], which opens up the opportunity for an optical element that simultaneously act as both a generator and frequency-controller of high-spectral-purity light beams [23,24].
2. Theory
The output frequency spectrum of a laser is fundamentally dependent on the optical cavity enclosing the laser gain medium. Eigenfrequencies for the lowest-order spatial mode satisfy the condition
where $m$ is the mode number, $c$ is the speed of light in vacuo and $\phi$ represents accumulated Gouy phase and phase shifts at reflecting interfaces. $L_0$ is the optical path length given by where $L_i$ are the physical lengths of cavity elements with refractive indices $n_i$ are the corresponding. These resonant frequencies (longitudinal modes) differ through the index $m$. The frequencies produced by a laser also depend on the gain medium, how it is pumped, the presence of other frequency-selective elements such as gratings or etalons, and nonlinear optical effects.In the presence of environmental noise, $L_0$ has random fluctuations, resulting in commensurate fluctuations in the output laser frequency. Temperature changes, mechanical vibrations and air currents contribute to this environmental noise with typical frequencies up to the kHz range. The ultimate purpose of the modulator and its associated feedback servo is to correct for these fluctuations.
To model modulator characteristics, Newton’s heating equation is applied in a planar (slab) geometry. In Section 2.1, we analyze the general case where the modulator is partitioned into $N$ layers. We then show that this can be reduced to a 2 layer (resistive heater plus modulator) system when the thermal conductivity is sufficiently large, as in the case of bulk diamond modulator. Section 2.2 then analyzes this special case, whereby the associated system of linear differential equations is used to derive modulation depth and bandwidth as a function of modulator dimensions and heat-transfer (boundary) coefficients.
2.1 $N$-layer model
As shown in Fig. 1, the modulator architecture considered here consists of an electrical resistive layer, the transparent modulator layer and a heat sink. To begin, the modulator is partitioned into layers of uniform temperature. The internal energy of the heating layer as a function of time is
where $P(t)$ is the electrical input power, $L$ and $w$ are the length and width of the modulator respectively, $T_r$ is the temperature of the resistive layer, $T_1$ is the temperature of the initial layer and $h_r$ is the heat-transfer coefficient across the boundary between the resistive layer and modulator. Here, $h_r$ is treated as a free parameter with a maximal value of $k/t_r$, where $k$ is the thermal conductivity of the modulator and $t_r$ is the thickness of the resistive layer. Practically, this is justified on the basis that a sputtered or vapor-deposited resistive heating layer would have excellent thermal contact with the modulator. This equation is derived from Newton’s cooling law with an added source term. An alternate form of Eq. (3) is where and $\rho _r$, $c_r$ and $\rho$ are the density, specific heat and resistivity of the resistive heating layer. $V(t)$ is the applied voltage. Similar equations governing the temperature evolution for each modulator layer are similarly derived. For the first layer where and $\rho _m$ and $c_m$ are the density and specific heat of the modulator, $t_m$ is the thickness of the modulator layers and $h_m$ = $k/t_m$ is the heat-transfer coefficient between modulator layers. For the final ($N^{\mathrm {th}}$) modulator layer where $T_0$ is the temperature of the heat sink, which is held fixed. It is noted that all $a$ terms (Eq. 5, 7, 8, 10) have units of s$^{-1}$. Here, $h_s$ is the heat transfer coefficient between the modulator and heat sink and is treated as a free parameter with a maximal value of $k/t_m$ in same vein as $h_r$. Finally, for the intermediate modulator layers,
Fig. 1. The planar architecture of a thermo-optic modulator. The resistive heater is driven electrically at half the modulation frequency, with the heat conducted across the metal/modulator boundary, into the modulator and across the second interface and into the heat sink. The modulation set point can be adjusted via the heat sink temperature. Focusing the beam is typical when diamond is used as a gain medium, but not essential for the operation of the modulator.
This system of equations can be expressed collectively in matrix form as
where andIncreasing the number of layers improves the spatial fidelity with which the temperature profile can be calculated. In the limit $N \rightarrow \infty$, the set of discrete equations describing the modulator layers converge to the heat equation
An example numerical solution is shown in Fig. 2 for a modulation frequency of $\omega = 5000$ rad/s. The difference in thermal conductivities results in a stark effect on the spatial uniformity of the thermal profile.
Fig. 2. Temperature deviation from equilibrium for a 50-layer model modulated at a frequency of $5000/2\pi$ Hz in a) diamond ($k = 2200$ W/(m.K)) and b) silica ($k = 1$ W/(m.K)).
2.2 Single-layer diamond model
When the temperature through the modulator is sufficiently uniform, the system is accurately approximated as a single-layer system. In this case
andTo determine the modulation depth, $V(t)$ is set to $V_0\cos (\frac {1}{2}\omega t)$, where the factor of $\frac {1}{2}$ is included because the modulation in power is naturally twice the frequency of voltage modulation. The modulator temperature as a function of time is found by substituting this expression into Eq. (21) and evaluating the required integrals, giving
In this single-layer model, the modulator behavior is completely defined by the interface parameters $h_r$ and $h_s$, and is not influenced by the thermal conductivity of the modulator. A proposed criterion for the suitability of the single-layer approximation is
where $k$ is the thermal conductivity of the modulator. This criterion is derived from the fact that $k/t_m$ has the same units as $h_r, h_s$ and can thus be thought of as something equivalent to a heat-transfer coefficient. As seen in Eq. (23), combining interfaces requires adding reciprocals (analogous to resistors in parallel), and so if the reciprocal of $k/t_m$ is sufficiently small it can be neglected. By this criterion, the advantage of using diamond versus conventional optical media is that it permits $t_m$ to be large, allowing modulators that are thicker by several orders of magnitude compared to a silica-based modulator.3. Discussion
3.1 Frequency-dependence of $\boldsymbol {M_n}$
The general form of $M_n$ as a function of frequency is shown in Fig. 3. In the low-frequency regime, $M_n$ is constant and independent of modulator thickness. This constant value is determined from the low frequency limit of Eq. (30)
and thus only depends on the heat transfer coefficient $h_s$ and the thermal expansion and conductivity of the modulator. At higher frequencies, $\Xi \gg \Psi$, so that $M_n$ varies inversely with $\omega$. The transition point can be determined by taking the limitFig. 3. $M_n$ as a function of modulation frequency for several modulator thicknesses using Eq. (30) and validated using a numerical finite-difference model with parameters $t_r = 150$ nm, $h_r = k/t_r$, $h_s = 100$ W.m$^{-2}$.K$^{-1}$. Inset is an illustration of how $\omega '$ is derived in Eq. (34).
If we construe the product $M_n \times w'$ as a figure-of-merit for the modulator, then it only depends on $t_m$ (the thinner the modulator, the better). The role of $h_s$ is to exchange modulation depth for bandwidth.
Once $M_n$ is known, the total modulation in the optical path length is
where it is recalled that $P$ is the applied (average) power, $A$ is the surface area of the modulator and $w$ is the modulator width. Note there is no dependence on the modulator length. To increase modulation depth and bandwidth, it is only necessary to reduce $w$ and $t_m$ as much as possible.3.2 Example devices
As a practical example used in single frequency diamond Raman lasers [11], a modulator that has a thickness of $t = 1$ mm would have a $M_n$ of around $1.18 \times 10^{-12}$ m$^2$/W at a modulation frequency of 2 kHz (Fig. 3). For a modulator width of $w = 1$ mm (giving it a square cross-section), this gives a total $\Delta L_0$ of $1.18 \times 10^{-9}$ m per Watt of applied power.
For a wavelength of 1064 nm with a standard cavity length (around 20 cm) the mode number (Eq. (1)) is around $m$ = 375,000, $n \approx$ 2.39, $\frac {dn}{dT} \approx 1.5 \times 10^{-5}$ K$^{-1}$ and $\alpha \approx 1 \times 10^{-6}$ K$^{-1}$. This yields a $df/dL_0$ of around −1.4 $\times$ 10$^{15}$ Hz/m (neglecting secondary effects such as dispersion, thermal Raman shift detuning and mode pulling [26]), hence a modulation in $L_0$ of 1.18 $\times 10^{-9}$ m/W corresponds to a frequency shift of 1.615 MHz/W. The power required to achieve a $\pi$ phase shift (equivalent to spanning the free spectral range of a cavity), is around 450 W. The corresponding bandwidth according to Eq. (34) is 45 mHz, thus the modulation frequency of 2 kHz is many orders of magnitude beyond this figure in this example.
Performance characteristics of a selected range of modulator dimensions under similar conditions is shown in Table 1. From these results we can see that under the conditions tested, the modulation depth is approximately inversely proportional to the modulator thickness, and therefore the cross-sectional area as well. For example, reducing the modulator cross-section to 0.5 mm $\times$ 0.5 mm, the required power drops by a factor of four to 113 W. Extrapolating to order 1 $\times$ 1 $\mu$m device sizes typical of integrated platforms, modulation depths around GHz (or one free spectral range) per mW at 2 kHz are predicted, which is not too dissimilar to the device performance reported in [14,15] (neglecting the influence of radiative heat loss).
Table 1. Predicted performance for selected modulator dimensions using the same parameters as Fig. 3, at a modulation frequency of 2 kHz and assuming a cavity optical length of 20 cm.
4. Conclusion
The performance of thermo-optic modulators heated in a planar geometry has been theoretically investigated. Bulk diamond modulators can be treated as single-layer devices by virtue of their outstanding thermal conductivity. For modulators with dimensions of 1 mm and a heat transfer coefficient between modulator and heat sink ($h_s$) of 100 WK$^{-1}$m$^{-1}$, the computed frequency modulation depth was 1.65 MHz per Watt of applied power for a standard 20 cm cavity and the bandwidth was 45 mHz. Although increasing $h_s$ (by improving thermal contact between diamond modulator and heat sink) improves the bandwidth, it does so at the cost of reducing the modulation depth.
Funding
Australian Research Council (LP200301594); Air Force Office of Scientific Research (FA2386-21-1-4030).
Disclosures
The authors declare no conflicts of interest.
Data availability
No data were generated or analyzed in the presented research. MATLAB code for generating results and figures is available from the authors on request.
Supplemental document
See Supplement 1 for supporting content.
References
1. R. Wyatt and W. J. Devlin, “10 khz linewidth 1.5 μm ingaasp external cavity laser with 55 nm tuning range,” Electron. Lett. 19(3), 110–112 (1983). [CrossRef]
2. T. J. Kane and R. L. Byer, “Monolithic, unidirectional single-mode nd:yag ring laser,” Opt. Lett. 10(2), 65–67 (1985). [CrossRef]
3. C. E. Wieman and L. Hollberg, “Using diode lasers for atomic physics,” Rev. Sci. Instrum. 62(1), 1–20 (1991). [CrossRef]
4. S. Fu, W. Shi, Y. Feng, L. Zhang, Z. Yang, S. Xu, X. Zhu, R. A. Norwood, and N. Peyghambarian, “Review of recent progress on single-frequency fiber lasers,” J. Opt. Soc. Am. B 34(3), A49–A62 (2017). [CrossRef]
5. M. Harris, G. N. Pearson, J. M. Vaughan, D. Letalick, and C. Karlsson, “The role of laser coherence length in continuous-wave coherent laser radar,” J Mod. Opt. 45(8), 1567–1581 (1998). [CrossRef]
6. P. Kwee, C. Bogan, K. Danzmann, M. Frede, H. Kim, P. King, J. Pöld, O. Puncken, R. L. Savage, F. Seifert, P. Wessels, L. Winkelmann, and B. Willke, “Stabilized high-power laser system for the gravitational wave detector advanced ligo,” Opt. Express 20(10), 10617–10634 (2012). [CrossRef]
7. B. S. Sheard, G. Heinzel, K. Danzmann, D. A. Shaddock, W. M. Klipstein, and W. M. Folkner, “Intersatellite laser ranging instrument for the grace follow-on mission,” J Geod. 86(12), 1083–1095 (2012). [CrossRef]
8. M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of bose-einstein condensation in a dilute atomic vapor,” Science 269(5221), 198–201 (1995). [CrossRef]
9. J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011). [CrossRef]
10. Y. Lu, L. Zhang, X. Xu, H. Ren, X. Chen, X. Wei, B. Wei, Y. Liao, J. Gu, F. Liu, L. Xu, J. Wang, T. Chen, M. Wan, W. Zhang, C. Tang, and G. Fan, “208 w all-solid-state sodium guide star laser operated at modulated-longitudinal mode,” Opt. Express 27(15), 20282–20289 (2019). [CrossRef]
11. X. Yang, O. Kitzler, D. J. Spence, Z. Bai, Y. Feng, and R. P. Mildren, “Diamond sodium guide star laser,” Opt. Lett. 45(7), 1898–1901 (2020). [CrossRef]
12. R. V. Pound, “Electronic frequency stabilization of microwave oscillators,” Rev. Sci. Instrum. 17(11), 490–505 (1946). [CrossRef]
13. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31(2), 97–105 (1983). [CrossRef]
14. P. Dong, W. Qian, H. Liang, R. Shafiiha, D. Feng, G. Li, J. E. Cunningham, A. V. Krishnamoorthy, and M. Asghari, “Thermally tunable silicon racetrack resonators with ultralow tuning power,” Opt. Express 18(19), 20298–20304 (2010). [CrossRef]
15. N. C. Harris, Y. Ma, J. Mower, T. Baehr-Jones, D. Englund, M. Hochberg, and C. Galland, “Efficient, compact and low loss thermo-optic phase shifter in silicon,” Opt. Express 22(9), 10487–10493 (2014). [CrossRef]
16. O. Lux, S. Sarang, O. Kitzler, D. J. Spence, and R. P. Mildren, “Intrinsically stable high-power single longitudinal mode laser using spatial hole burning free gain,” Optica 3(8), 876–881 (2016). [CrossRef]
17. O. Lux, S. Sarang, R. J. Williams, A. M. Mckay, and R. P. Mildren, “Single longitudinal mode diamond raman laser in the eye-safe spectral region for water vapor detection,” Opt. Express 24(24), 27812–881 (2016). [CrossRef]
18. S. Sarang, O. Kitzler, O. Lux, Z. Bai, R. J. Williams, D. J. Spence, and R. P. Mildren, “Single-longitudinal-mode diamond laser stabilization using polarization-dependent raman gain,” OSA Continuum 2(4), 1028–1038 (2019). [CrossRef]
19. X. Yang, O. Kitzler, D. J. Spence, R. J. Williams, Z. Bai, S. Sarang, L. Zhang, Y. Feng, and R. Mildren, “Single-frequency 620 nm diamond laser at high power, stabilized via harmonic self-suppression and spatial-hole-burning-free gain,” Opt. Lett. 44(4), 839–842 (2019). [CrossRef]
20. X. Yang, Z. Bai, D. Chen, W. Chen, Y. Feng, and R. P. Mildren, “Widely-tunable single-frequency diamond raman laser,” Opt. Express 29(18), 29449–29457 (2021). [CrossRef]
21. Z. Bai, R. J. Williams, O. Kitzler, S. Sarang, D. J. Spence, Y. Wang, Z. Lu, and R. Mildren, “Diamond brillouin laser in the visible,” APL Photonics 5(3), 031301 (2020). [CrossRef]
22. D. Jin, Z. Bai, Z. Lu, R. Fan, Z. Zhao, X. Yang, Y. Wang, and R. Mildren, “22.5-w narrow-linewidth diamond brillouin laser at 1064 nm,” Opt. Lett. 47(20), 5360–5363 (2022). [CrossRef]
23. E. Granados, G. Stoikos, D. T. Echarri, K. Chrysalidis, V. N. Fedosseev, C. Granados, V. Leask, B. A. Marsh, and R. P. Mildren, “Tunable spectral squeezers based on monolithically integrated diamond raman resonators,” Appl. Phys. Lett. 120(15), 151101 (2022). [CrossRef]
24. E. Granados, C. Granados, R. Ahmed, K. Chrysalidis, V. Fedosseev, B. Marsh, S. Wilkins, R. Mildren, and D. Spence, “Spectral synthesis of multimode lasers to the fourier limit in integrated fabry–perot diamond resonators,” Optica 9(3), 317–324 (2022). [CrossRef]
25. R. P. Mildren, Intrinsic optical properties of diamond (Wiley-VCH, 2013), pp. 1–34.
26. E. Granados and G. Stoikos, “Spectral purification of single-frequency stokes pulses in doubly resonant integrated diamond resonators,” Opt. Lett. 47(16), 3976–3979 (2022). [CrossRef]
