R. John Koshel
A general goal in fiberoptic communications is to carry many distinct channels at once so as to increase the transmission data rate. Wavelength-division multiplexing (WDM) achieves this by using many separate diode-laser sources, each operating at distinct wavelengths. Currently, most WDM systems operate at a channel spacing of 50-100 GHz (0.4-0.8 nm), while each individual laser has an instantaneous linewidth an order of magnitude smaller. Novel systems are approaching 25-GHz channel spacing and 2.5-GHz linewidth. Clearly, there is a sizable amount of wasted bandwidth. The reason for this waste comes primarily from the need to maintain distinct channels over a long period of time, taking into account such things as drift of the diode-laser line centers due to thermal and mechanical effects.
Of course, the desire is to continue decreasing the channel spacing while at the same time decreasing the individual source linewidths. A number of solutions have been proposed for WDM; some of them involve optical feedback into a diode laser with such elements as diffraction gratings, etalons, and prisms.1 These optical elements act as the spectral tuning object through coupled resonators. Spectral frequencies away from the tuned line center show a poor coupling of the return signal into the waveguide structure of the laser. The result is a narrowing of the linewidth of the ultimate output of the laser.
In the research-laboratory environment, such geometries have produced an instantaneous linewidth of less than 1 Hz and a long-term (one-hour) linewidth of 1 kHz. Much care has to be given to the thermal, vibrational, electrical, and other associated fluctuations in these systems in order to obtain such results. Forms of these systems are being considered for WDM applications, albeit not to such a narrow linewidth. Modeling of such external optical stabilization has primarily addressed approximate cases, especially in consideration of the spatial beam profile and the effects from the various intracavity elements.
A WDM source that uses a diffraction grating and flat mirror can be modeled to provide feedback. The optical layout is called the Littman-Metcalf configuration.2 The diffraction grating is designed such that the zero-order beam is ejected from the cavity, while the minus first order is diffracted to the tuning mirror. Tuning is accomplished by rotating the mirror around the center point of the grating. The line-narrowing is then affected by the grating period, and to a lesser extent by the system geometry. The modeling is performed using coherent propagation with Gaussian beam decomposition.3,4
THE LITTMAN-METCALF CONFIGURATION
In the modeled system, the radiation from a diode laser is first incident on a collimating lens. The collimated beam is then incident on a grating at an angle of 82° to the normal (see Fig. 1). The grating ejects the zero diffraction order out of the external cavity, while the -1 order is directed to a planar mirror. This mirror reflects the -1 order back toward the grating, where it is diffracted into new zero- and -1 order components once again. The latter zero order is lost in another direction, while the -1 order is diffracted back toward the lens. Due to the grazing incidence and the small grating period, very few orders can propagate.
In this model, only two orders (0 and -1) can be present. Wavelengths not reflected along the original path will walk off, resulting in a poor coupling back into the laser and high loss for frequencies away from the tuned frequency. Rotation of the planar mirror selects the tuned frequency. Because of the high gain of the laser, the feedback seed is strong enough to pull the spectral distribution toward the tuned wavelength, thus giving a narrowed emission linewidth at the tuned frequency (that is, the operating line center).
The initial conditions for the diode laser include a Gaussian spectral distribution that has a full-width-half-maximum (FWHM) linewidth of 8.7 GHz and a Gaussian angular-emission width of 25° at the e-1 points. Eleven distinct wavelengths are chosen for the model, with the initial line center being 830 nm (the choice of the initial wavelength is purely for the sake of using a known lens system; similar systems at a wavelength of 1550 nm can be designed with no loss of generality). The spacing between neighboring wavelengths is 0.0025 nm, with the amplitude of each wavelength chosen from the Gaussian spectral distribution. The return signal is compared to the original and the coupling coefficient into the waveguide structure of the laser is calculated.
ITERATION PRODUCES FINAL VALUES
The return signals allow computation of the overall spatial and angular distributions of light coupled back into the diode laser, as well as of its spectral narrowing. The process is then repeated starting with an amplified version of this seed signal. After several iterations of this algorithm, values will approach equilibrium, allowing the final signal to be determined. The first return of the laser signal is highlighted, showing the pulling of line center and the narrowing of linewidth as a function of grating period and planar mirror rotation. In this study, the planar mirror and grating are separated by approximately 5 cm. Because the model simulates a bulk system and beams of a finite size, diffraction effects, geometrical issues, and the like are included.
Initially, the mirror is arranged to retroreflect 830 nm, which is the original line center from the laser. A grating with a period of 0.6927 µm is chosen. The model produces a spectral profile (as a function of wavelength away from line center) for the initial Gaussian distribution of the laser, the new profile after coupling back into the laser diode, and the coupling coefficients back into the diode laser (see Fig. 2). The new spectral distribution is obtained by finding the product of the original distribution with the coupling coefficients. The narrowing of the linewidth for a grating with 0.6927-µm period spacing gives approximately a 2.7 times narrowing after one round-trip.
VARYING THE GRATING PERIOD
Three grating periods (0.6927, 1.0, and 1.5 µm) were chosen to investigate how the grating period affects the linewidth narrowing. As expected, decreasing the grating period results in narrowing of the linewidth (see Fig. 3). The FWHM linewidths after one round-trip are 3.3 GHz for the 0.6927-µm grating period, 5.0 GHz for the 1.0-µm grating period, and 7.3 GHz for the 1.5-µm grating period. The change in grating period does nothing to the selection of the central wavelength.
Selecting a different line center requires rotation of the planar mirror. In one example, the wavelength is tuned to three different values using the configuration with the grating period of 0.6927 µm (see Fig. 4). The first case is the standard when tuned to the nominal 830-nm line center. In the second case, the mirror is rotated to choose the neighboring wavelength at -0.0025 nm away from nominal line center. The spectral distribution for this case is skewed to the chosen wavelength, while there is a slight asymmetry in the right-to-left comparison of this distribution. The last case is for detuning between the third (-0.0075 nm) and second (-0.005 nm) wavelengths. The result is an increased asymmetry in the spectral distribution. The FWHM linewidth is largely unaffected by the particular frequency to which the mirror is tuned. The linewidth for all three cases upon return to the diode laser is approximately 3.2 GHz. Slight discrepancies are caused by the asymmetry in the spectral distribution.
EXTENDING THE MODEL
This model, developed with Breault`s ASAP software program, includes effects such as diffraction, interference, and the temporal dynamics of the field as it propagates, as well as the coupling efficiency of light returning to the laser cavity for each of the various wavelengths. The results provide a preliminary understanding of the bulk dynamics in such systems. The model is very effective for modeling source dynamics for applications that require narrow linewidth sources, such as WDM and spectroscopy. Many other effects can be modeled with this system, including the effects of placement errors of the components, polarization effects, scattering, and ghosting. This preliminary model could also be extended to include thermal and mechanical effects.
The near-term plan is to extend the model to directly include the waveguide structure amplification within the diode laser and coupling into an external fiber. Optimization of the entire model can then be performed, with the figure of merit being the coupling into the fiber. This technique will provide a powerful means to know a priori the stability of the line center, linewidth, and spatial beam profiles of real-life bulk systems in which optical phenomena ranging from diffraction to polarization play a natural part.
The authors would like to thank Ken Anderson, University of Colorado (Boulder, CO), and John Sweetser, Templex Technology (Eugene, OR), for helpful discussions.
1. D. R. Hjelme, A. R. Mickelson, and R. G. Beausoleil, IEEE J. Quantum Electron. 27, 352 (1991).
2. M. G. Littman and H. J. Metcalf, Appl. Opt. 17, 2224 (1978).
3. A. Greynolds, "Propagation of generally astigmatic Gaussian beams along skew ray paths," in Diffractive Phenomena in Optical Engineering Applications, D. M. Byrne and J. E. Harvey, eds., Proc. SPIE 560, 33 (1985).
4. A. Greynolds, "Vector formulation of ray-equivalent method for general Gaussian beam propagation," in Current Developments in Optical Engineering and Diffractive Phenomena, R. E. Fischer, J. E. Harvey, and W. J. Smith, eds., Proc. SPIE 679, 129 (1986).
Joe Shiefman is an optical engineer and R. John Koshel is technical director for illumination at Breault Research Organization, Ste. 350, 6400 East Grant Rd., Tucson, AZ 85715; www.breault.com. For more information, contact the authors at 520-721-0500 or by e-mail at email@example.com and firstname.lastname@example.org.
FIGURE 1. In the Littman- Metcalf configuration, collimated light from a diode laser strikes a grating at a large angle to the normal. The -1 diffracted order exits the grating close to normal and is reflected back on itself by a mirror. The grating directs the -1 diffracted order of the reflected beam back into the laser. The grating dispersion results in only a very narrow band of wavelengths re-entering the laser close enough to the original path to achieve good coupling.
FIGURE 2. A comparison of spectral profiles (as a function of wavelength away from line center) for the initial Gaussian distribution of the laser (blue) and the new profile after coupling back into the laser (red) shows a line-narrowing of 2.7 times. The model also produces coupling coefficients of the beam re-entering the laser (green). All profiles are for nominal line center.
FIGURE 3. As the grating period in the Littman-Metcalf configuration is decreased, the laser linewidth narrows. Note that as the grating period approaches the dimension of the tuned wavelength, the rate of reduction in linewidth lessens. Resulting peak flux ratios (the ratio of feedback flux to initial flux) are 0.72 for a zero offset, 0.69 for a -0.0025-nm offset, and 0.52 for a -0.006875-nm offset.
FIGURE 4. Model illustrates effects on line shape of wavelength tuning via mirror rotation. Optical configuration contains a grating with a 0.6927-µm period. As the wavelength is tuned away from nominal, the spectral distribution becomes asymmetrical. The relative fluxes in the coupled signal decrease as the external cavity is tuned away from line center. This loss is due to the lower initial signal in frequencies away from the nominal line center.