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Abstract
Microresonator Kerr frequency combs are coherent light sources that emit broadband spectrum of evenly spaced narrow lines in an optical microresonator, which provide breakthroughs in many technological areas, such as spectroscopy, metrology, optical telecommunications, and molecular sensing. The development of mid-infrared (MIR) optical frequency comb (OFC) based on microresonators could pave the way for high performance spectroscopy in the MIR “molecular fingerprint” region. However, the generation of microresonator MIR OFC, especially towards the long-wavelength MIR (>10 µm) region, is prohibited by the transmission window of the commonly used Kerr optical media such as Si and Si3N4, and low nonlinearity at long wavelengths. Here, we seek the possibility to realize an ultra-broadband frequency comb operating in the long-wavelength MIR region based on a cadmium telluride (CdTe) ring microresonator. CdTe features a broad transmission range covering the wavelengths of 1∼25 µm, a flat dispersion profile, and an extraordinary third-order nonlinear refractive index (∼1.4 × 10−17 m2W−1 at 7 µm) which is 2-order greater than that of Si3N4, making it a promising platform to realize MIR Kerr frequency comb. Based on the above excellent optical properties, we design a CdTe/cadmium sulfide (CdS)/Si heterojunction microring resonator to generate an ultra-broadband MIR OFC. Through the numerical simulation, the geometric parameters (width, height, and radius) of the microresonator, polarization, wavelength of the pump, and quality factor are investigated and optimized. As a result, a MIR OFC covering 3.5∼18 µm is numerically demonstrated by using the pump wavelength of 7 µm and a pump power of 500 mW. This is the first simulation demonstration of Kerr OFC with the spectral range extending beyond 10 µm, to the best of our knowledge. This work provides new opportunities for the realization of ultrabroad microresonator frequency combs based on novel Kerr optical medium, which can find important applications ranging from calibration of astronomical spectrographs to high-fidelity molecular spectroscopy.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
Corrections
Siyi Lu, Xin Liu, Yaqi Shi, Hang Yang, Zhe Long, Yang Li, Han Wu, and Houkun Liang, "Mid-infrared ultra-broadband optical Kerr frequency comb based on a CdTe ring microresonator: a theoretical investigation: erratum," Opt. Express 30, 40627-40627 (2022)https://opg.optica.org/oe/abstract.cfm?uri=oe-30-22-40627
1. Introduction
Optical frequency comb (OFC) [1] is a broadband coherent light source consisting of equally spaced sharp spectral lines. The microresonator Kerr frequency comb is a new kind of OFC based on the four-wave mixing (FWM) effect in an optical microresonator with an ultra-high-quality factor (Q factor) [2,3]. The microresonator OFC has a size that can range from a few micrometers to millimeters, and accordingly, the free spectral range (FSR) may vary in the range of gigahertz to terahertz. In the telecom band, materials such as silicon dioxide (SiO2) [4,5] and silicon nitride (Si3N4) [6,7] have been used for microresonators to generate Kerr optical frequency comb, which provides a wide range of practical applications such as optical clocks [8], telecommunications [9] as well as light detection and ranging (LiDAR) [10,11]. The mid-infrared (MIR) wavelength range of 2.5∼25 µm is known as the “molecular fingerprints” because almost all molecules undergo strong and sharp characteristic vibrational transitions in MIR spectral region. Therefore, extending the Kerr frequency combs to MIR is expected to bring new possibilities in tremendous applications from molecular spectroscopy to astronomy science [12]. Recently, microcombs or electro-optic frequency combs based-MIR comb was generated via differential frequency generation (DFG) in nonlinear crystal by using two near-IR combs [13,14]. Compared to the directly pumped MIR Kerr frequency comb, the use of mature near-IR microcombs to generate MIR comb avoids the requirement of highly coherent high-power MIR lasers and MIR waveguides. However, the DFG of near-IR microcombs requires bulky and complicated free-space optical components, and the conversion efficiency of DFG is relatively low. Besides, the wavelength range of MIR comb is limited by both the phase-matching bandwidth and transmission range of the used nonlinear crystal. With the development of high power single-mode quantum cascade lasers (QCL) in long-wavelength MIR with hundreds milliwatts level output power [15,16], it is promising to utilize such laser source to directly pump micro Kerr frequency comb in the long-wavelength MIR region featuring high compactness and relatively high efficiency [17,18].
In the past ten years, several optical media have been explored for developing MIR Kerr frequency combs. The spectral coverage of the generated MIR OFCs and the properties of such materials are summarized in Table 1. To generate a broadband Kerr frequency comb, the ideal materials should have a high nonlinearity, a wide transparent range and a flat dispersion profile. It is conceived from Table 1 that the generation of MIR Kerr OFC has been focused in the wavelength range of 2 to 4 µm, except for two recent works based on germanium (Ge) and magnesium fluoride (MgF2), and MIR Kerr OFC beyond 10 µm has not yet been realized. Notably, although Ge has a large nonlinear coefficient and a transparent window extending up to 15 µm, a recent study indicates that the strong multiphoton absorption and free-carrier absorption set the hurdles to realize high-performance OFC in Ge due to its relatively small band gap [30]. On the contrary, we found that cadmium telluride (CdTe), as a material widely used in the photovoltaic and X-ray detector industries, has an ultrabroad transparency window spanning from 0.8 to 25 µm [26] which can cover the entire MIR fingerprint region, and an outstanding nonlinearity with the third-order nonlinear refractive index (n2) of ∼1.4 × 10−17 m2 W−1 at 7 µm [28,29], which is 2-order greater than that of Si3N4 [25], and 3-order larger than that of MgF2 [27]. In addition, CdTe is a direct band-gap semiconductor with a band-gap energy of ∼1.5 eV [31], thus two-photon or three-photon absorption as well as free-carrier absorption [32] can be neglected in long-wavelength MIR band. Moreover, high-quality CdTe epi-layers could be grown by mature semiconductor deposition technologies, such as sputtering deposition techniques [33]. These excellent properties make CdTe a promising candidate for MIR nonlinear devices, especially at long MIR wavelengths. Ultra-broadband supercontinuum generation pumped by femtosecond pulses has recently been investigated based on CdTe waveguide in our previous work [34].
Table 1. Summary of bandwidths of MIR Kerr frequency combs and optical properties of common MIR materials
In this work, based on the generalized Lugiato-Lefever equation [35], we theoretically analyze the generation of microcombs on an on-chip ring resonator based on CdTe/Cadmium sulfide (CdS)/Si hetero-waveguide. By optimizing the geometric parameters (width, height, and radius) of microresonator, polarization, pump wavelength and quality factor we numerically demonstrate an ultra-broadband Kerr frequency comb spanning from 3.5 µm to 18 µm at a pump wavelength of 7 um and pump power of 500 mW, with a FSR of 110 GHz, which suggests the great potential of CdTe as the Kerr medium for high performance MIR OFC.
2. Waveguide design and dispersion management
The structure of the proposed CdTe/CdS/Si ring microresonator is illustrated in Fig. 1. Si is chosen as the substrate to be adapted by the CMOS-compatible integrated-circuit fabrication technologies. CdTe with the refractive index of n∼2.68 [36] serves as the waveguide core. A layer of CdS with a smaller refractive index (n∼2.3) [37] and good transparency in the wavelength range of 1 to 15 µm [38], is deposited between the Si substrate and CdTe as the lower cladding layer. The CdS cladding layer has a thickness of 10 µm, which serves to confine the optical field in the CdTe waveguide core, and thus reduce the power leakage to the Si substrate [34].
Fig. 1. The three-dimensional structural diagram of the CdTe/CdS/Si hetero-microresonator and the schematic diagram of Kerr frequency comb generation. The insets show the waveguide cross-section and electric-field distribution of the TE0 mode in the CdTe waveguide.
Dispersion management of the microresonator is crucial for the broadband microcomb generation. For the selected material, the waveguide dispersion can be engineered by controlling the geometric parameters of the resonator. We use the commercially available software (COMSOL Multiphysics) as an eigenvalue solver of axisymmetric structures to obtain resonator dispersion values under different geometric parameters (width, height, and radius). The waveguide cross-section used in the simulation is shown in the lower right section of Fig. 1. Besides, 90-degree sidewall angle is used in the simulation as it is found that the deviation of less than 20 degree of the sidewall angle has little effect on the dispersion profile. By considering both the material and waveguide dispersions, the resonator’s total dispersion with different parameters can be obtained by calculating the integrated dispersion Dint defined as the deviation of the resonance frequency from an equidistant FSR [39]:
Fig. 2. The calculated dispersion of the TE (a) and TM (b) modes as a function of the wavelength. Different pump wavelengths are used in the calculation represented by different colors. Inset: electric-field distributions at 7 and 15 µm wavelengths in corresponding polarization.
Fig. 3. Simulated integrated dispersion Dint of CdTe ring microresonator with different geometric parameters: height (a), width (c) and radius (e) in µm, and the corresponding calculated second-order dispersion coefficient β2 curves (b), (d), (f).
The effect of electric-field polarization on tailoring a flat dispersion profile is investigated as the first step. The dispersion curves of the fundamental transverse-electric (TE) and transverse-magnetic (TM) modes are numerically calculated and presented in Fig. 2. It is observed from Dint curves that the dispersion of TE mode is much flatter than that of TM mode. Moreover, by comparing the electric field distributions of TE and TM modes in the waveguide, we can clearly see that the electric field of TE polarization has less leakage into air and CdS claddings, which indicates a better mode confinement. Therefore, the CdTe microresonator with TE polarization is selected for the flatter dispersion and less loss. To determine a suitable pump wavelength, the dispersion profiles of the designed CdTe microresonator are simulated at various pump wavelengths of 6, 7, 8 and 9 µm. As shown in Fig. 2(a), with long pump wavelengths of 8 and 9 µm, the dispersion increases abruptly at wavelengths around 4 µm, while for a short pump wavelength of 6 µm, there exhibits substantially large dispersion at long wavelength side. Thus 7 µm is chosen as the pump wavelength from the flat dispersion point of view, and it needs to be verified by the simulation of OFC generation in later section.
Secondly, the dispersion of the designed microresonator is investigated by varying one while fixing other geometric parameters among height, width and radius of the CdTe microresonator. Figures 3(a) and (b) present the calculated integrated dispersion Dint and second-order dispersion coefficient β2 as a function of wavelength for the CdTe microresonators with different microring heights. In the calculation, the width and the radius of the CdTe microresonator are fixed as w = 15 µm and R = 150 µm to generate flat dispersions with the detailed simulation and discussion in the later part. It is observed that as the height of the CdTe microresonator decreases from 4.5 µm to 3.3 µm, the Dint curve becomes flatter, however, a broad wavelength range in MIR falls into the normal dispersion region when the height of microresonator is below 4 µm, as shown in Fig. 3(b), which is detrimental to generate Kerr OFC with the pump wavelength around 7 µm. Therefore, 4 µm is chosen as the height of the microresonator for a balance of flat and anomalous dispersion. Subsequently, Fig. 3(c) simulates dispersion curves of the CdTe microresonator when the width of microring varies from 11 to 18 µm, while the height and radius are fixed as 4 µm and 150 µm, respectively. It is observed that the dispersion profiles at widths of 15 and 18 µm are very similar and much flatter than cases of 11 and 12 µm. Moreover, a broad spectral band ranging from 6 to 18 µm is located in the anomalous dispersion region for all the values of microresonator widths as shown in Fig. 3(d). Thus, to pursue a smaller mode area that not only leads to higher nonlinearity but also be beneficial for suppressing higher-order modes, the width of microresonator is chosen to be 15 µm. Thirdly, the radius of CdTe microresonator is also taken into investigation as presented in Fig. (e) and (f). Only minor variation of dispersion profiles with different microresonator radii is observed, especially when the radii are equal to 120 and 150 µm. Thus, simulations of OFC generation with the comparison of multiple microresonator radii are to be conducted, which would be presented in the next section.
3. Comb generation
The numerical simulation of the MIR microcomb generation is based on the generalized Lugiato-Lefever equation [41]. The additional slowly-varying field envelope assumption, Fourier transform (FT) and inverse Fourier transform (FT-1) are introduced for fast calculation, with the equation used in the simulation shown below [42]:
OFCs generated in microresonators operate in the dissipative soliton region, which relies on a double balance of nonlinearity and dispersion as well as loss and gain [43]. The balance of nonlinearity and dispersion has been studied in Section 2, while now we discuss gain and loss, which is directly related to the pump power. At a pump wavelength of 7 µm, we use a microresonator with a bending radius of 150 µm (for a balance of flat and small anomalous dispersion), waveguide height of 4 µm and width of 15 µm to simulate the frequency comb generation. Figure 4 shows the dynamics of MIR OFC generation based on the designed CdTe microresonator under different pump powers. Figures 4(a)-(d) present the average intracavity power changing with the scan of pump detuning, as the pump power increases from 30 mW to 500 mW, and Figs. 4(e)-(h) and (i)-(l) simulate the corresponding spectral and temporal characterization, respectively, at different pump levels. At a starting pump power of 30 mW, as shown in Fig. 4(a), the intracavirty power quickly drops after a spike by experiencing modulation instability, which leads to the spectral broadening only with two main combs through FWM [3]. Thus, no trace of soliton but only modulation of the continuous envelope is observed in the time domain as shown in Fig. 4(i). When the pump power is increased to the range of 100-300 mW, the dynamics of OFC generation moves across chaos region and enters the stable multi-soliton region as shown in Figs. 4(b), (c). Multiple stable solitons in one troundtrip are observed in time domain as presented in Figs. 4(j), (k), and spectral modes of the OFC are not locked but exhibit a rough and noisy envelope as shown in Figs. 4(f), (g). Finally, it is noted that when the pump power is increased to 500 mW in Fig. 4(d), the dynamics of the mid-IR Kerr frequency comb go through chaos, stable multi-soliton and reaches the single-soliton region [44], which results in the smooth and ultra-broadband OFC spectrum spanning from 3.5 µm to 18 µm at -60 dB level as shown in Fig. 4(h). In addition, it is worth mentioning that a dispersive wave is formed at short wavelengths, which is associated with the zero-dispersion wavelength at ∼ 4 µm, as simulated in Figs. 3(a), (c), (e). The time-domain character verifies the single-soliton formation in one roundtrip, as presented in Fig. 4(l), and a pulse waveform in the cavity with a temporal width of ∼ 50 fs is generated as shown in the zoom-in inset.
Fig. 4. (a)-(d) Simulated comb power of intracavity field as a function of the laser detuning under different pump powers. The green, blue and yellow shadows correspond to the regions of chaos, multi-soliton and single-soliton. (e)-(h) OFC spectra at a detuning marked by the red arrows shown in (a)-(d). (i)-(l) The corresponding temporal profiles of (e)-(h). The inset of (l) shows a pulse waveform with a pulse width of ∼ 50 fs.
To design the MIR OFC based on the CdTe heterostructure, and maximize the MIR spectral coverage, in the first step, the MIR OFC generation with different radii in the designed CdTe heterostructure is simulated. A pump power of 500 mW is chosen for producing a single-soliton OFC. Figure 5(a) reveals that the broadband OFC is generated at radii of 150 µm and 120 µm, and it is found that increasing the radius to 200 µm substantially narrows the bandwidth of OFC due to its large dispersion on both the short and long wavelength sides. Since the simulated OFCs with radii of 120 µm and 150 µm have similar spectral spanning, the radius can be determined according to the desired FSR in specific applications. Subsequently, the MIR OFC is simulated with three pump wavelengths of 6, 7, and 8 µm. As shown in Fig. 5(b), the pump wavelength of 6 µm produces the narrowest spectrum especially at long wavelengths which agrees well with the calculated large dispersion of the microresonator on the long-wavelength side as shown in Fig. 2(a). At a pump wavelength of 8 µm, bacause the nonlinear coefficient reduces as the wavelength increasing according to γ = n2ω0/(cAeff), the single-soliton OFC can only be generated at a higher pump level of 800 mW. It is observed that at pump power of 500 mW, a pump wavelength of 7 µm provides a broad OFC covering 3.5 to 18 µm at -60 dB. Notably, OFC extending to even longer wavelength could be realized by choosing a longer pump wavelength, but higher pump power is aquired for compensating the reduced nonlinear coefficient.
Fig. 5. Comparison of simulated OFC spectra under different radii (a) and pump wavelengths (b). With longer pump wavelength, higher pump power is aquired for compensating the reduced nonlinear coefficient
The Q factor of the optical microresonator represents the binding ability of the resonant light field in a resonator, and its value is crucial for the spectral extension of OFCs. The reported Q factors of chip-integrated microresonators in the near-infrared wavelengths are in the range of 106–107 [45]. In the MIR wavelength region, the fabrication defects based on the current epitaxial growth and CMOS technologies generate less scattering loss, and CdTe has negligible material absorption loss (the absorption coefficient of CdTe is less than 0.003 cm-1 in the wavelength range from 2.5 to 20 µm [46]). It is thus reasonable and safe to assume a Q factor of 106 in our simulation. In addition, there is still room for further improving the smoothness of the fabricated microresonator by using advanced deposition technique such as metalorganic chemical vapor deposition (MOCVD), which can further increase the Q factor of microresonator.
Although it is still lack of commercially available high power, single-mode QCL in the long-wavelength MIR region, the single-mode QCL at 7 µm region has been reported with the output power more than 300 mW at -20°C [14]. In principle, the output power could be further enhanced to watt-level by using the liquid nitrogen cooling. It is worth mentioning that, even with lower Q value, the power of the available single-mode QCL at 7 µm could still support broadband frequency comb in multi-soliton state regime, as shown in Fig. 4. Besides, with the Q factor of 106, the resonance linewidth of the CdTe microresonator is about 40 MHz, and the typical linewidth of single-mode QCL is several MHz [18], which is narrower than the resonance linewidth. Therefore, we believe with the sustainable development of high power single-mode QCL in long-wavelength region, the QCL pumped CdTe microresonator could provide a promising platform for realizing ultrabroadband long-wavelength MIR frequency comb.
4. Sample fabrication
Here, we briefly describe our fabrication process of CdTe microresonator as shown in Fig. 6. Step 1, CdS film of 10 µm thickness is first grown on silicon substrate by magnetron sputtering. Step 2, CdTe film of 4 µm thickness is grown on top of the CdS layer by the same method. Step 3, the deposited CdTe film is pretreated to remove surface contaminants and prebaked at 110 °C. After cooling to room temperature, the CdTe film is coated with positive photoresist using a spin coater. Step 4, after pre-baking, the ring pattern on the mask is transferred to the photoresist on the sample via deep-UV exposure for 7 seconds with a surface irradiance of about 17 mW/cm2. Subsequently, post-exposure bake is conducted. Step 5, we immerse the sample in the developer for about 20 seconds, and then bake the sample again to make the photoresist film firmly adhere to the surface of the CdTe film as well as to increase the resistance of the photoresist in the subsequent etching process. Step 6, the sample is wet-etched for about 12 minutes using a mixture of phosphoric acid, nitric acid and water with a ratio of 78:2:20. Finally in step 7, the photoresist is removed with acetone. As presented in the micrograph in Fig. 7(a), the fabricated waveguide shows good density and uniformity and the measured X-ray diffraction patterns of CdTe and CdS shown in Figs. 7(b) and (c) are both consistent with those measured from CdTe and CdS single crystals [47,48]. It is worth mentioning that, the gap between the microring and the straight waveguide is still not ideal in the current fabricated CdTe microresonator, which requires more research effort on the fabrication process.
Fig. 6. Schematic diagram of CdTe microresonator fabrication process. (1) and (2): Deposition of CdS and CdTe thin films by magnetron sputtering. (3): Spin coating of a layer of positive photoresist. (4): Exposure under a deep-UV lithographic mask aligner . (5): Development. (6): Wet etching by using phosphoric acid and Nitric acid. (7): Removal of the residual photoresist.
Fig. 7. (a) Micrograph of CdTe microresonator sample. (b) X-ray diffraction pattern of CdTe. (c) X-ray diffraction pattern of CdS
5. Conclusion
In summary, we propose a CdTe/CdS/Si heterostructural microresonator with CMOS-compatible material growth and fabrication technologies. The broad transmission range, engineered flat dispersion profile throughout the entire MIR spectral region, and the outstanding third-order nonlinearity of the designed CdTe microresonator enable an ultra-broadband MIR OFC generation. Based on the generalized Lugiato-Lefever equation with additional slowly-varying field envelope assumption and Fourier transforms, the geometric parameters such as the width, height, and radius of the CdTe ring-shape microresonator, as well as the pump wavelength, polarization and Q factors are systematically investigated to study and design the ultra-broadband MIR OFC generation. As a result, at pump wavelength of 7 µm with 500 mW pump power, an ultra-broadband MIR OFC spanning from 3.5 to 18 µm corresponding to 2.5 octave at -60 dB level is generated with a Q factor of 106. This work provides possibilities to generate integrated OFCs in the long-wavelength MIR region, based on CdTe as a new nonlinear medium platform. In addition, CdTe has been routinely used in the photovoltaic and X-ray detection industries with silicon-compatible fabrication techniques, therefore we foresee the proposed design in this work would find experimental realization soon and thus open new research frontiers of integrated MIR OFCs.
Funding
Outstanding Youth Science and Technology Talents Program of Sichuan (2022JDJQ0031); National Natural Science Foundation of China (62075144).
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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