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Abstract
We propose a dynamic-spectral-broadness Littman/Metcalf external cavity diode laser, which replaces the flat end mirror of the external cavity with a curved one with a tunable radius of curvature (RoC). The concept was verified via simulation; first, the frequency selectivity of the cavity was calculated for each RoC using Gaussian-beam optics combined with ray tracing, and second, laser oscillation and amplified spontaneous-emission (ASE) spectra were obtained using the transmission-line laser model. The simulation revealed a tuning range with spectral broadness: 250 kHz for single-mode operation, 1.2–47 GHz for multi-mode operation, and 50 GHz–3.9 THz for ASE.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Linewidth control, which also implies temporal-coherence control [1], is useful for various applications in optics and photonics, including speckle-noise reduction in coherent imaging [2,3], evaluation of optical coherent communication systems [4,5], suppression of stimulated Brillouin scattering in fiber amplifiers for high-power single-frequency fiber lasers [6], enhanced magneto-optical trapping [7], efficient optical pumping of helium-3 [8], and rainbow refractometry for the measurement of droplets [9]. Therefore, techniques to control the broadness of a spectrum purely, without changing the output power, are desired.
Several methods have thus far been developed to tune, or especially broaden, the linewidth dynamically [2,6,8–15]. These include modulating the injection current into a distributed-feedback diode laser [10], changing the pump power of a fiber laser [8], and shifting the frequency of light fed back from the external cavity using an acousto-optic modulator (AOM) [13,14]. However, these methods may increase intensity noise or cause intensity changes when tuning spectral broadness owing to the modulation of the injection current, change in pump power, and change in diffraction efficiency of an AOM. A more typical method to broaden a spectrum is the phase modulation of an output beam from a single-frequency laser [6,11,12]. However, such phase modulation can cause an artificial intensity fluctuation with a given frequency, particularly when a large phase modulation is applied, which is known as residual amplitude modulation [16–18]. Another method used to tune spectral broadness is the addition of only an external mirror and a mode-matching lens to a diode laser to form an external cavity [2,9,15]. However, narrow linewidth in the kHz scale is difficult to achieve by such an external cavity because it does not have any dispersive elements.
A Littman/Metcalf external cavity diode laser (ECDL) can be used to control spectral broadness statically [19,20]. It consists of a Fabry–Perot laser diode (LD), mode-matching lens, diffraction grating, and flat end mirror, and it limits wavelengths that can travel inside the cavity by spatially dispersing different wavelengths using the diffraction grating [19]. Using a curved end mirror with a radius of curvature (RoC) equal to the distance between the grating and end mirror instead of a flat mirror enables a broad spectrum [20]. However, to the best of our knowledge, the use of a curved end mirror with other RoCs and dynamic control have not been previously suggested.
This study proposes a dynamic spectral-broadness-tunable Littman/Metcalf ECDL [21]. The proposed ECDL uses a curved end mirror with a dynamically controllable RoC. Some implementations of a dynamic tunable curved mirror include a dynamic mirror with tunable RoC [22], segmented or continuous deformable mirror [23], and varifocal lens [24], which is well-known for its changeable focal length and can function as a tunable curved mirror by being placed in front of a flat end mirror. Another possible implementation is a liquid crystal on silicon spatial light modulator (LCOS-SLM), which can modulate the spatial phase distribution of light and change the RoC of light by displaying a Fresnel lens pattern with a particular focal length [25,26]; thus, it may be substituted for a tunable curved mirror. Unlike the conventional phase-modulation method for spectral-broadness control, our proposed ECDL is not accompanied by any temporal modulation or artificial intensity fluctuation because this method does not include external temporal modulation to control spectral broadness. The concept was verified via laser simulation, which combined a beam-propagation calculation with the ABCD matrix and the transmission-line laser model (TLLM). The results exhibited a transition from single longitudinal-mode operation to multi-mode operation as well as changes in the spectral broadness on the multi-mode operation and amplified spontaneous-emission (ASE) range. The laser oscillation exhibited consistent average output powers across various spectral spreads, whereas the average output powers of the ASE varied. Our proposed ECDL has potential applications such as reducing speckle noise in coherent imaging, where how low temporal coherence can sufficiently reduce speckle noise depends on its underlying causes [3]. To achieve the optimal temporal coherence, a technique that allows for changing spectral broadness from single- to multi-mode operation and ASE is required.
The rest of this paper is organized as follows. Section 2 describes the working principle of our proposed ECDL. Section 3 explains methods for designing an external cavity and results of simulated frequency selectivity of the cavity. Section 4 illustrates the simulated laser oscillation and ASE spectra as well as the details of the TLLM. In Section 5, we discuss a few problems and challenges in our proposed ECDL as well as possible solutions. Finally, Section 6 presents the conclusions of this study.
2. Principle of dynamic spectral-broadness-tunable Littman/Metcalf ECDL
Figure 1 shows our proposed dynamic spectral-broadness-tunable Littman/Metcalf ECDL, which has several similarities to the conventional Littman/Metcalf ECDLs. Specifically, the lens converges the highly diverged beam from the LD toward the end mirror, the grating disperses the incident light spatially using the first-order light, the angle of the end mirror with respect to the grating determines the center wavelength that exactly traces the outward trip on the return trip and couples back into the LD, and the output can be extracted from the zeroth-order light of the grating or an LD facet that does not face the external cavity. In this study, we assumed that the LD had a single transverse mode so that we could obtain single longitudinal-mode operation when the end mirror was flat. The selectivity of the optical frequency that can exist inside the cavity is determined by the curved mirror as well as the diffraction grating, ECDL geometry, and LD aperture. Thus, the spectral broadness changes according to the RoC of the end mirror. The center wavelength can be tuned by tilting the end mirror in the same way as the conventional Littman/Metcalf ECDL.
Fig. 1. Dynamic spectral-broadness-tunable Littman/Metcalf ECDL. A curved mirror with a tunable RoC is used as the end mirror instead of the flat end mirror that is used in the conventional Littman/Metcalf ECDL. Each color of the beam represents a different wavelength (${\lambda _1},\; {\lambda _2},\; {\lambda _3}$). The output is extracted from the zeroth-order light of the grating. The z axis is defined as parallel to the ray of the center wavelength after diffraction at the grating. LD: laser diode. ${d_1}$: distance between the LD facet and mode-matching-lens flat surface. ${d_2}$: distance between the mode-matching lens curved-surface center and beam-incidence point on the grating. ${d_3}$: distance between the grating and end mirror. ${t_{\textrm{lens}}}$: mode-matching-lens center thickness.
The detailed description of the working principle is as follows. When the RoC of the end mirror is infinity (i.e., flat end mirror), the ECDL is the same as the conventional Littman/Metcalf one. Thus, a narrow spectrum (i.e., single-mode operation) is achieved, where only a fraction of the wavelengths that are spatially dispersed by the grating can couple back into the LD. In contrast, an RoC equal to ${d_3}$ (the distance between the grating and end mirror) cancels the diffraction from the grating, which allows all the wavelengths diffracted to exactly trace the outward trip on the return trip after being reflected at the end mirror. This cancellation of the diffraction enables all the wavelength components to couple back into the LD, and a broad spectrum (i.e., multi-mode operation) is obtained [20]. When the RoC is between ${d_3}$ and infinity, a mid-broad spectrum is obtained, where a transition occurs from single- to multi-mode operation. Furthermore, an injection current that is lower than the laser-oscillation threshold provides a considerably broader spectrum by suppressing the high nonlinearity of the laser oscillation. Note that an RoC shorter than ${d_3}$ does not cancel the diffraction from the grating completely and will result in a narrower spectrum than the case of an RoC that is equal to ${d_3}$.
3. Cavity design with Gaussian-beam optics and ray tracing
3.1 Methods for designing cavity arrangement independent of RoC and calculating optical-frequency selectivity
A cavity geometry that is optimized independently of the RoC of the end mirror is desired, that is, re-alignment of the optical components is not required for different RoCs. The coupling efficiency of a beam that departed from the LD facet, traveled inside the external cavity, and returned to the LD facet was considered as an index of the optimization of a cavity for a particular RoC (see Fig. 2(a)); an optimized cavity has a high coupling efficiency. For simplicity, a trip inside the LD gain medium was ignored. In addition, two independent RoCs for the orthogonal directions ($\textrm{Ro}{\textrm{C}_x}$ and $\textrm{Ro}{\textrm{C}_y}$) were considered for higher coupling efficiencies. Such a curved end mirror can be realized with, for example, a deformable mirror or an LCOS-SLM by displaying two superimposed cylindrical lens patterns. Note that the diffraction at the grating occurs on the $xz$ plane (see Fig. 1) and that changing only $\textrm{Ro}{\textrm{C}_x}$ is sufficient to control the spectral broadness.
Fig. 2. A beam that departed from the LD facet and one that returned for (a) the center wavelength and (b) a wavelength other than the center one. The red arrows represent the rays of the beam centers, the gradated red ellipsoids represent beam-intensity distributions, and the black bidirectional arrows indicate beam width ${w_x}$. A wavelength other than the center one does not return along the same path as that of the outward trip.
The coupling efficiency was defined as a mode matching or spatial-overlapping integral of two electric fields, ${E_0}$ and ${E_1}$, where the former just departed the LD facet, and the latter traveled inside the external cavity and just returned to the LD facet:
Then, the q parameters of the beam that returned to the LD facet ${q_{1x}}({{d_1},{\; \textrm{Ro}}{\textrm{C}_y};{\; \textrm{Ro}}{\textrm{C}_x};{\; }{\lambda_{\textrm{center}}}} )\; $ and ${q_{1y}}({{d_1},{\; \textrm{Ro}}{\textrm{C}_y};{\; \textrm{Ro}}{\textrm{C}_x};{\; }{\lambda_{\textrm{center}}}} )$ were obtained:
Table 1. Some of the fixed parameters used to design the cavity geometry.
Table 2. Obtained parameters after the optimization of the cavity geometry.
Table 3. Some of the laser parameters used for the TLLM.
Using these q parameters, the coupling efficiency formulated in Eq. (1) was reduced as follows:
To obtain a cavity geometry that was optimized independently of the $\textrm{Ro}{\textrm{C}_x}$, the coupling efficiency for the center wavelength was optimized for all $\textrm{Ro}{\textrm{C}_x}$ values with the mode-matching-lens position ${d_1}$ and $\textrm{Ro}{\textrm{C}_y}$ free parameters, which was formulated as follows:
After the optimization of the cavity geometry for the center wavelength, the optical-frequency selectivity or bandwidth of the external cavity (BWEC) for each $\textrm{Ro}{\textrm{C}_x}$ was obtained as follows. Coupling efficiencies for wavelengths that slightly deviated from the center wavelength were calculated using q parameters and the ABCD matrices, as explained above. However, when considering wavelengths other than the center one, optical paths after the grating were different for each wavelength. Therefore, the optical paths were also calculated using ray tracing (see Fig. 2(b)). The optical-path lengths, incidence angles on the LD facet, and beam offsets from the LD facet center that were obtained from ray tracing were used to calculate the beam propagation and mode matching for a wavelength other than the center wavelength (see Supplement 1), where mode matching between two noncollinear beams with a lateral offset was considered [32]. Note that in our case, the incidence angles and beam offset were only negligible, and most of the contributions to the mode matching were from the difference in the beam widths at the LD facet owing to different optical-path lengths. Using the calculated optical-frequency dependence of the coupling efficiency for a particular $\textrm{Ro}{\textrm{C}_x}$, the BWEC value for the $\textrm{Ro}{\textrm{C}_x}$ was determined as its full width at half maximum (FWHM).
3.2 Results: optical-frequency selectivity of the optimized cavity
Figure 3(a) shows optical-frequency dependence of the coupling efficiency for $\textrm{Ro}{\textrm{C}_x} = \infty $, where the narrowest frequency dependence of the coupling efficiency was obtained. The coupling efficiency at the center wavelength was approximately unity, which indicated that most of the power of a beam returning from the external cavity coupled into the LD. The coupling efficiency decreased as the optical frequency deviated further from the center frequency. The optical-frequency selectivity or bandwidth of the external cavity (BWEC) was 50 GHz (FWHM).
Fig. 3. (a) Optical-frequency dependence of the coupling efficiency for $\textrm{Ro}{\textrm{C}_x} = \infty $. (b) $\textrm{Ro}{\textrm{C}_x}$ dependence of the bandwidth of the external cavity (BWEC, blue circles) and the coupling efficiency at the center wavelength (red triangles) with the ordinate-axis range 0.999999 to 1. The solid vertical line shows $\textrm{Ro}{\textrm{C}_x} = {d_3}$.
The $\textrm{Ro}{\textrm{C}_x}$ dependence of BWEC is shown in Fig. 3(b) (see Fig. S1 in Supplement 1 for a wide range of $\textrm{Ro}{\textrm{C}_x}$). BWEC increased monotonically as $\textrm{Ro}{\textrm{C}_x}$ decreased. However, it changed steeply for $\textrm{Ro}{\textrm{C}_x} < $ 90 mm and plateaued for $\textrm{Ro}{\textrm{C}_x} > $ 95 mm. BWEC remained almost constant for $\textrm{Ro}{\textrm{C}_x} > $ 95 mm because the focal length comparable to or longer than the external cavity length (100 mm) has only minor influence on a beam divergence angle. At $\textrm{Ro}{\textrm{C}_x} = {d_3}$, a BWEC value of 6.1 THz was obtained. Note that BWEC becomes narrower for $\textrm{Ro}{\textrm{C}_x} < {d_3}$ from the working principle. The coupling efficiencies at the center wavelength for different $\textrm{Ro}{\textrm{C}_x}$ values are also plotted in Fig. 3(b). These values were almost constant and close to unity, which indicated that the cavity geometry was optimized for a wide range of $\textrm{Ro}{\textrm{C}_x}$ values.
4. Laser simulation
4.1 Methods used for laser simulation
In the above description of the cavity design, the range of possible BWEC values was obtained. For each BWEC within this range, laser oscillation and ASE inside the entire cavity were simulated using the TLLM to calculate a spectrum [30,33]. The TLLM can obtain broad and continuous spectra, whereas other laser models, such as the rate equation model, cannot. This characteristic enables the TLLM to simulate not only laser oscillation but also ASE. In addition, implementing an external cavity in the TLLM is simple.
The TLLM utilized the equivalence between an electric/magnetic field of a transverse electric mode inside a waveguide and a voltage/current on a transmission line. It then modeled a laser on a transmission line, where spontaneous emission and amplification by the stimulated emission were expressed by the current noise and voltage amplification at each section on a transmission line, respectively [33,34]. The TLLM assumed a uniform cross section for a laser cavity and discretized it along only the optical axis, which enabled the TLLM to easily model an external cavity [31]. The optical-frequency dependences of the spontaneous emission, stimulated emission, and coupling efficiency of an external cavity were set using a combination of open and short stubs; the characteristic impedances of the stubs were determined to produce specified FWHMs and the same center wavelength. Time evolution was calculated using the same method that was used for the modeling of a transmission line in combination with the rate equation of the excited carrier density. An electric-field waveform of the laser oscillation was sampled at the external cavity end mirror from an incident electric field, and that of ASE was sampled from an electric field reflected at the mirror. The reflected electric field was extracted to obtain an electric field just after the stub filter corresponding to the frequency selectivity of the cavity geometry. Whether an electric field was sampled just after the stub filter or not did not affect the laser oscillation; however, this was critical for ASE. Finally, Fourier-transforming the simulated electric-field waveform produced a laser spectrum.
The total simulated time length was 8 µs, and the last 5.369 µs ($= {2^{27}} \times 40$ fs) was Fourier-transformed to calculate a spectrum because the laser oscillation and ASE were almost stationary within the time range (see Fig. 4 and its description). A time of 5.369 µs was sufficiently long to provide an adequate frequency resolution of the Fourier transformation to resolve longitudinal modes. As introduced in Ref. [30,33], temporal waveforms were “undersampled” with respect to time to reduce computational loads, but without loss of information. Some of the laser parameters used for the simulation are listed in Table 3 (see Table S2 in Supplement 1 for the full list of the parameters). Our simulated ECDL corresponded to one with strong feedback, rather than those with weak feedback or in a coherence-collapsed state. The reason was that the product of the coupling efficiency, the diffraction efficiency at the grating, and the reflectivity at the end mirror was relatively high for the center wavelength. The product for the center wavelength was approximately $r_{\textrm{EC}}^2{({1 - r_{\textrm{front}}^2} )^2} = $ 0.5 (see Table 3) because the coupling efficiency for the center wavelength was almost unity (see Fig. 3). For ECDLs with weak feedback or in a coherence-collapsed state, the product was considerably lower than our case [35–37].
Fig. 4. Simulated laser-oscillation waveform for $\textrm{Ro}{\textrm{C}_x} = {d_3}$ (BWEC = 6.1 THz). The red graph indicates instantaneous power that is proportional to the squared electric field, and the blue graph indicates the averaged power over 7 ns, which was approximately 10 times the beating period (i.e., the reciprocal of the Fabry–Perot FSR). The vertical black line is at 8 µs – 5.369 µs = 2.631 µs.
4.2 Results: laser oscillation/ASE spectra, output power, and tunability of spectral broadness
Figure 4 shows the simulated laser-oscillation waveform for $\textrm{Ro}{\textrm{C}_x} = {d_3}$ (BWEC = 6.1 THz). The averaged power with an averaging time window of 7 ns, which was approximately 10 times the beating period (i.e., the reciprocal of the Fabry–Perot free spectral range (FSR)), increased rapidly at the beginning of the simulation and reached its stationary value well before 1 µs. The instantaneous power reflected beating between the Fabry–Perot modes, and its waveform exhibited some information about the dynamics and relaxation of the oscillation spectrum. Before 2.631 µs, the averaged power was almost constant; however, the instantaneous power increased gradually. This behavior was understood as follows: at first, the power was distributed over a number of Fabry–Perot modes, and beating between the modes was cancelled to some extent. However, as time progressed, the laser oscillation approached a stationary state, and the power was concentrated on fewer Fabry–Perot modes, which enabled large beating spikes to occur frequently and made the instantaneous power waveform appear higher after 2.631 µs. This also explains why the fluctuation of the averaged power after 2.631 µs was larger than that before 2.631 µs. Therefore, we concluded that the laser oscillation became stationary after 2.631 µs. For other $\textrm{Ro}{\textrm{C}_x}$ values and ASE, the laser oscillation or ASE were also stationary after 2.631 µs.
Figure 5 shows the simulated laser-oscillation spectra. For the case of $\textrm{Ro}{\textrm{C}_x} = \infty $, a single longitudinal-mode operation was obtained, which was similar to that obtained by the conventional Littman/Metcalf ECDL. Here, the single longitudinal-mode operation was defined as a side-mode suppression ratio (SMSR) higher than 30 dB. Fitting the spectrum using a Gaussian fit resulted in an FWHM of 250 kHz; however, the spread was limited by the frequency resolution of the Fourier transformation (Fig. 5(a)). Next, as shown in Fig. 5(c), when the $\textrm{Ro}{\textrm{C}_x}$ was equal to the distance between the grating and curved end mirror ($\textrm{Ro}{\textrm{C}_x} = {d_3}$), a wide spectrum was obtained, which consisted of many Fabry–Perot cavity modes with an FSR of 1.5 GHz (corresponding to the external cavity length of 100 mm). Each of the Fabry–Perot cavity modes had a shape and width that were similar to those of Fig. 5(a). The FWHM of the spectrum was 47 GHz, which was calculated using the standard deviation of the spectrum, assuming a Gaussian envelope, with a spectral-intensity threshold of 0.1. The Gaussian envelope with an FWHM of 47 GHz and a peak value of unity is also plotted in Fig. 5(c). For the case of $\textrm{Ro}{\textrm{C}_x}$ shorter than $\infty $ and longer than ${d_3}$, a spectrum of the multi-mode operation with a spread between $\textrm{Ro}{\textrm{C}_x} = \infty $ and $\textrm{Ro}{\textrm{C}_x} = {d_3}$ was obtained, as shown in Fig. 5(b). The FWHM of the spectrum was calculated to be 5.1 GHz, using the same method as that in Fig. 5(c).
Fig. 5. Simulated laser-oscillation spectra. The abscissa axis of (a) is the optical-frequency difference from the spectral peak, and those of (b) and (c) are optical-frequency differences from the center frequency corresponding to ${\lambda _{\textrm{center}}}$ = 842 nm. The red circles and lines represent the simulated data. The blue solid line in (a) represents the Gaussian fitting, which provided the FWHM of the spectrum. The blue broken lines represent Gaussian envelopes with FWHMs that were calculated simply from the standard deviations of the spectra with a spectral-intensity threshold of 0.1. (a) $\textrm{Ro}{\textrm{C}_x} = \infty $ and BWEC = 50 GHz. (b) $\textrm{Ro}{\textrm{C}_x}$ = 92.3 mm $> {d_3}$ and BWEC = 500 GHz. (c) $\textrm{Ro}{\textrm{C}_x} = {d_3}$ and BWEC = 6.1 THz. The inset of (a) is the spectrum of ASE with an injection current below the laser-oscillation threshold for $\textrm{Ro}{\textrm{C}_x} = \infty $. The abscissa and ordinate axes are the optical-frequency difference from the center frequency and spectral intensity in an arbitrary unit, respectively. The blue solid line in the inset is an envelope for the ASE spectrum obtained by fitting a smoothed ASE spectrum with a TLLM stub filter-response function.
Two characteristics of the laser-oscillation spectra are as follows. The obtained spectral spreads were a few orders of magnitude narrower than BWEC. This may be ascribed to the nonlinearity of the laser oscillation. In addition, multi-mode operation resulted in a spectrum consisting of Fabry–Perot cavity modes; thus, a continuous wide spectrum was not obtained.
The spectrum of the ASE for $\textrm{Ro}{\textrm{C}_x} = \infty $ is shown in the inset of Fig. 5(a). The FWHM of the spectrum was calculated to be 50 GHz by fitting a smoothed ASE spectrum with a TLLM stub-filter response function, which is also plotted with the peak value adjusted. The FWHM for the ASE was the same as BWEC; however, a discrepancy was observed between the spectral spread and BWEC for the case of the laser oscillation. This may be explained by the weaker nonlinearity of ASE compared to that of the laser oscillation.
The $\textrm{Ro}{\textrm{C}_x}$ dependence of the average output power is shown in Fig. 6(a). The averaged time segment was 1–8 µs of the simulation time length of 8 µs, where stationary power appeared to be obtained. For the case of the laser oscillation, the average powers of single-mode operation and those of multi-mode operation were almost the same. However, the average output powers of the ASE were significantly dependent on $\textrm{Ro}{\textrm{C}_x}$, as shown in the inset of Fig. 6(a). The difference in the behavior may be understood as follows. When laser oscillation was well established, the oscillating Fabry–Perot modes consumed most of the gain of the medium. In contrast, for ASE, the gain was distributed over many Fabry–Perot modes, and only a portion of them coupled back into the LD from the external cavity. This led to considerably higher loss that was dependent on the frequency selectivity of the external cavity. Therefore, a significant dependence of the output power on $\textrm{Ro}{\textrm{C}_x}$ was observed, and higher power was obtained for $\textrm{Ro}{\textrm{C}_x}$ closer to ${d_3}$ or wider BWEC in the case of ASE.
Fig. 6. $\textrm{Ro}{\textrm{C}_x}$ dependence of the (a) average output power and (b) spectral spreads (FWHM). The solid vertical lines correspond to $\textrm{Ro}{\textrm{C}_x} = {d_3}$. The red circles represent multi-mode laser oscillation, green triangles indicate single longitudinal-mode laser oscillation, and blue diamonds indicate amplified spontaneous-emission with an injection current below the threshold. The inset in (a) is a vertically enlarged view. The inset in (b) shows the log-scaled spectrum for $\textrm{Ro}{\textrm{C}_x} = $ 143 mm, which is at the boundary of the “gap” in the graph. The abscissa and ordinate axes of the inset are the optical-frequency difference in THz and spectral intensity in an arbitrary unit, respectively.
The spectral spreads for different $\textrm{Ro}{\textrm{C}_x}$ values with the injection current above and below the laser-oscillation threshold are shown in Fig. 6(b). For a spectrum in single longitudinal-mode operation, a spectral spread was calculated from the Gaussian fitting of only the main peak, as in Fig. 5(a). For a spectrum in multi-mode operation, a spectral spread was calculated simply from the standard deviation of the spectrum with a spectral-intensity threshold of 0.1, except for the case where only two side peaks had intensities comparable to 0.1. In such a case, a spectral spread was obtained from the Gaussian fitting of the (three) maximum values of the three peaks, with only the standard deviation of the Gaussian as a free parameter. For both the laser oscillation and ASE, the spectral spreads increased as the $\textrm{Ro}{\textrm{C}_x}$ approached ${d_3}$ from infinity, and a range of controlling spectral broadness was obtained, from 250 kHz for the single longitudinal-mode operation and 1.2–47 GHz for the multi-mode operation to 50 GHz–3.9 THz for the ASE. However, a gap between the single- and multi-mode operation was observed because single-mode operation was defined with an SMSR threshold of 30 dB. The inset shows a log-scaled spectrum for an $\textrm{Ro}{\textrm{C}_x}$ value at the boundary of the gap. Defining a spectral spread for such a comb-like spectrum with side modes that were sufficiently weaker than the main peak was difficult. Therefore, we concluded that the gap was artificial and that it did not have the intrinsic nature of the proposed ECDL.
5. Discussion
Spectral broadness was indeed controlled by $\textrm{Ro}{\textrm{C}_x}$; however, the gap was observed between the single- and multi-mode operation. In reality, the transition between single- and multi-mode operation is continuous. Furthermore, the gap may be bridged by technical noises, such as the instability of an external cavity, fluctuation of the injection current, and temperature variation.
For multi-mode operations, comb-like spectra caused by the Fabry–Perot resonator hindered smooth spectral shapes, whereas the spreads of the envelopes were controlled. The FSR is inversely proportional to cavity length, and longer external cavities result in a narrower FSR and smoother spectra. A longer external cavity may be obtained using two methods: a long free-space external cavity and a fiber external cavity. A long free-space external cavity simply provides long distances between the LD, mode-matching lens, grating, and end mirror. However, a longer free-space cavity is generally more unstable, and single longitudinal-mode operation is difficult to achieve. In contrast, a fiber external cavity appears to be a more reasonable option. If a single-mode fiber is inserted between the LD and mode-matching lens with adequate coupling between the LD and fiber, the frequency selectivity of the cavity is determined by the fiber cross section instead of the LD aperture. Such a modification is not accompanied by longer free-space propagation, and stability can be maintained. In general, a narrower FSR owing to a longer cavity length makes achieving single-mode operation difficult under the possible narrowest frequency selectivity of the cavity geometry (∼ 50 GHz). However, the stability of a fiber external cavity may overcome this challenge.
For ASE, whether an electric field was sampled just before or just after the stub filter at the end mirror was a critical factor, as mentioned in Section 4.1. This may imply that ASE dominated the filtering effect of the external cavity after propagating inside the LD. In real experiments, light is filtered just when it is coupled back into the LD from the external cavity. Therefore, some measures will be essential to extracting the filtered light before amplification inside the LD gain medium. A possible solution is to insert a single-mode fiber between the LD and mode-matching lens, as explained previously, and extract a portion of light from the fiber using a fiber coupler.
Around the gap in spectral-broadness tuning, the presence of homogeneous broadening tends to concentrate energy into the primary single mode while suppressing the other modes. This phenomenon gives rise to a competition between single- and multi-mode operation, which can result in a significant level of intensity noise. The evaluation of laser intensity noise will be undertaken as part of our future research tasks.
In this study, the ECDL end mirror was assumed to be flat or concave; however, it can also be convex. The frequency selectivity in such a case became narrower than that in the case of the flat end mirror. For example, in the case of $\textrm{Ro}{\textrm{C}_x} ={-} {d_3}$, a BWEC value of 25 GHz was obtained by considering the optimized cavity geometry used throughout this study with a coupling efficiency at the center wavelength higher than 0.999999. Given that the flat end mirror provided the single longitudinal-mode operation and that further narrowing of BWEC will not lead to a significant reduction in the linewidth, the use of the convex operation of the proposed ECDL will not be effective. However, such further narrowing of BWEC may be useful to achieve single-mode operation in experiments because the technical noises in experiments often hinder single mode operation.
The relationships between the focal length of the mode-matching lens and other parameters such as $\textrm{Ro}{\textrm{C}_x}$, $\textrm{Ro}{\textrm{C}_y}$, ${d_1}$, and ${d_3}$ can be understood based on the fact that the lens focuses the beam in x direction at the end mirror, as shown in Figs. 1 and 2. We fix the external cavity length ${L_{\textrm{EC}}}$. From the lens formula, the distance between the LD facet and the principal plane of the lens a, the distance between the principal plane and the end mirror $b = {L_{\textrm{EC}}} - a$, and the focal length f should approximately satisfy the following equation (if the effects of the grating are ignored):
By solving this equation for a, we obtain:
Here, ${d_1}$ is close to a, and ${d_3}$ is close to $b = {L_{\textrm{EC}}} - a$. Therefore, for a shorter focal length, ${d_1}$ tends to be shorter and ${d_3}$ tends to be longer. Correspondingly, the minimum $\textrm{Ro}{\textrm{C}_x}$ ($= {d_3}$) tends to be longer. On the other hand, $\textrm{Ro}{\textrm{C}_y}$ focuses a divergent beam in y direction. The difference in beam divergence between x and y direction results from the difference in divergence angles from the LD facet and the effects of the grating. Based on beam-width behavior in x direction, we can infer that longer ${d_3}$ means a smaller divergence angle and that $\textrm{Ro}{\textrm{C}_y}$ tends to increase (weaker focusing) with a decreasing focal length.
Desired tuning speed of a curved mirror with a dynamically controllable RoC varies depending on the application and dynamic range of spectral-broadness tuning. Thus, there is no general answer for the desired tuning speed. Instead, we can estimate the upper limit of the necessary tuning speed. The tuning speed of spectral broadness is ultimately limited by the response of a laser diode. The response is also related to round-trip time and storage time of the ECDL. We may estimate the response time by analyzing the transient behavior of laser oscillation from the initial state, as shown in Fig. 4. The laser oscillation appeared to be stationary after 2–3 µs, as discussed in Section 4.2, and the response time may be estimated to be approximately 2–3 µs. This indicates that in our setup, the upper limit of desired tuning speed or response time of a dynamic tunable curved mirror is approximately 2–3 µs.
6. Conclusion
This study proposed a dynamic spectral-broadness-tunable Littman/Metcalf ECDL. This design replaces the end mirror of the conventional Littman/Metcalf ECDL with a curved end mirror. The RoC of the mirror can be changed, and controlling the RoC and injection current enables spectral-broadness tuning. The concept was verified via simulations of the frequency selectivity of the external cavity and of the laser oscillation and ASE using the TLLM. The tuning range for controlling spectral broadness was 250 kHz for the single longitudinal-mode operation, 1.2–47 GHz for the multi-mode operation, and 50 GHz–3.9 THz for the ASE. However, its spectral shape was comb-like and not smooth. The average output power of the laser oscillation was independent of the RoC of the curved end mirror, in contrast to that of the ASE. The proposed ECDL is not accompanied by any artificial intensity fluctuation because it does not include external temporal modulation. It is applicable to speckle-noise reduction in coherent imaging and the evaluation of optical coherent communication systems. Experimental verification will be conducted in future work.
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.
Supplemental document
See Supplement 1 for supporting content.
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