Steady-state ab initio theory of lasers with injected signals

We present an ab initio treatment of the steady-state of lasers with injected signals that describes a regime, valid for microlasers, in which the locking transition is dominated by cross saturation and spatial hole burning. The theory goes beyond standard approaches and treats multimode lasing with injected signals and finds the possibility of partially locked states and as well as repulsion of the free-running frequencies from the injected signal. The theory agrees well with exact integration of the full wave and matter equations for the system. It can also describe accurately complex modern lasers structures and is applied to the example of deformed disk lasers. We show that in the case of a one-dimensional cavity in the locked or regenerative amplifier regime the theory reduces to an improved version of the Adler equations in the appropriate limit.

Figure 1

(Top) Simulations of single-mode injection locking in a one-sided dielectric slab cavity with $n=1.5$ (schematic) with a perfect mirror at one end and an index step to vacuum at the other. First, the pump is increased above the threshold for lasing at ${\omega }_{1,\mathrm{free}}=40.714$, $D_0=0.0603$ to $D_0=0.08$, and then held at a fixed value (vertical black line) while the input signal amplitude is ramped from $B_{\mathrm{in}}=0$ until the free-running signal is quenched and the system “locks” (vertical orange line) to the injected frequency, ${\omega }_{\mathrm{in}}=40.4$ at $B_{\mathrm{in}}=0.176$. Finally, the simulation is continued in the locked regime to $B_{\mathrm{in}}=0.4$. Solid lines are output intensities calculated from I-SALT; blue is lasing output, red is amplified output at signal frequency, dot-dashed black is total output. Triangles are the same quantities from FDTD for the same dielectric slab laser with ${\omega }_a=40$, the width of the gain curve, ${\gamma }_{\perp }=4$, and ${\gamma }_{\parallel }=0.001$. The green curve in the locked regime is the prediction of our generalized Adler equations, (36) and (37). The top inset shows gain curve and ${\omega }_{\mathrm{in}}$ (red), ${\omega }_1$ (blue). (Bottom) Frequency variation of the first lasing mode. Tlue line is from I-SALT and blue triangles from FDTD. The green line shows the prediction of the Adler theory. The red line is the injected signal frequency. Again, the orange dashed line shows the locking threshold from I-SALT; frequencies beyond this point are taken as the real part of the location of the pole of the scattering matrix. As the locking transition is approached, the lasing frequency moves away from the injected frequency, due to spatial hole burning instead of being attracted toward it, as expected from the Adler equation [5]. Blue dashed lines showing negligible frequency shift are I-SALT calculation with uniform gain saturation and no spatial hole burning. The inset shows a plot of the phase shift between input and output signals of an injection-locked dielectric slab cavity at a fixed input intensity. I-SALT (red curve) and FDTD simulations (red triangles) are seen to have a better quantitative and qualitative agreement than the Adler prediction (green curve). For comparison with the Adler theory, the horizontal axis is plotted in terms of the free-running lasing frequency in the absence of an injected signal at threshold, ${\omega }_{1,\mathrm{free}}$. Frequencies and rates are given in units of $c/L$, while the atomic inversion and modal intensities are given in SALT units of $E_c$ and $D_c$.

Figure 2

(a) Motion of the pole as described in the text corresponding to the free-running lasing mode in the locking scenario of Fig. 1. As the pump is increased below threshold the pole of the scattering matrix is pulled upwards towards the real axis and “in” towards ${\omega }_a=40$ (blue dashed line, recall there is no signal yet at ${\omega }_{\mathrm{in}}=40.4)$. Free-running lasing occurs when the pole reaches the real axis at ${\omega }_{1,\mathrm{free}}=40.714$ and continues as the pump is increased further above threshold with negligible further frequency shift. Then the pump is fixed and the input signal is ramped, causing the pole (solid blue line) to move to higher frequency, away from the input frequency, and eventually off the real axis as the effects of gain saturation cause the lasing mode to go below threshold. The inset shows a magnification on the motion of the pole of the lasing mode inside the dotted box. (b) Frequency spectrum from FDTD simulations across the locking transition, showing no additional lines appearing, indicating that the effect is purely due to gain cross saturation. Frequencies and rates are given in units of $c/L$, while the atomic inversion and modal intensities are given in SALT units of $E_c$ and $D_c$.

Figure 3

Partial locking transition as described in the text for a laser with two free-running modes and an injected signal (schematic) using a similar pumping and input ramping scheme as Fig. 1, starting at the first lasing threshold $D_0=0.101$ and pumping until $D_0=0.13$, then increasing the input signal from $B_{\mathrm{in}}=0$ to $B_{\mathrm{in}}=0.4$. Solid lines are output intensities calculated from I-SALT; blue and cyan lines are lasing output, the red line is amplified output at signal frequency, ${\omega }_{\mathrm{in}}=20.3$, the dot-dashed black line is total output. Triangles are the same quantities from FDTD for a similar dielectric slab laser with $n=3$, ${\omega }_a=20.5,{\gamma }_{\perp }=3,{\gamma }_{\parallel }=0.001$. The inset shows the relationship of the three frequencies. As expected, the lasing mode nearest to the injected signal locks to the injected signal (orange line), then the more distant lasing mode locks (purple line). Frequencies and rates are given in units of $c/L$, while the atomic inversion and modal intensities are given in SALT units of $E_c$ and $D_c$.

Figure 4

Comparison of the predictions of I-SALT (blue line), the SPA of I-SALT, given by Eqs. (27) and (28) (green line), the Adler model, as given by Eqs. (36) and (37) (red line), and FDTD (blue triangles) for both the output intensity (top) and the phase offset (bottom). The input intensity is negligible $(|B_{\mathrm{in}}{|}^2={10}^{-4}$ in normalized SALT units) compared to the output, while the pump (gain) is increased from $D_0=0$ to $D_0=0.067$, thus placing the simulations in the regime of validity for the Adler approximations. The vertical orange dashed line denotes the first lasing threshold in the absence of an incident signal, whereas the vertical purple dashed line shows where I-SALT predicts the unlocking transition to occur. Simulations are shown for a single-sided dielectric slab cavity with $n=1.5$, ${\omega }_a=40$, ${\omega }_{\mathrm{in}}=40.7$, ${\omega }_{1,\mathrm{free}}=40.714$, ${\gamma }_{\perp }=4$, and ${\gamma }_{\parallel }=0.001$. Frequencies and rates are given in units of $c/L$, while the atomic inversion and modal intensities are given in SALT units of $E_c$ and $D_c$.

Figure 5

Simulations of a uniform index quadrupole cavity laser amplifier with $n=1.5$ (boundary indicated in white). The parameters chosen are $R_0=1.72\mu \mathrm{m}$, ${\lambda }_a=1\mu \mathrm{m}$, ${\gamma }_{\perp }=0.03\mu \mathrm{m}$, and $\epsilon =0.16$. The injected wavelength for all three simulations is the same as the that for the first free-running mode, ${\lambda }_{\mathrm{in}}={\lambda }_1=0.989\mu \mathrm{m}$. The pump value was increased from $D_0=0$ to the first free-running lasing threshold, $D_0=0.065$, vertical orange dashed line. The three solid curves show the amplifier output for three different injection conditions: blue, injection with the TCF corresponding to the first lasing mode; green, injection with the TCF corresponding to the second lasing mode; red, injection with the TCF corresponding to the third lasing mode. Lower plots show in color scale the normalized mode amplitude profiles: (a) The first free-running lasing mode at threshold. (b) The amplified mode with the first lasing mode's incoming TCF as input. (c) The amplified mode with the second lasing mode's incoming TCF as input. (d) The amplified mode with the third lasing mode's incoming TCF as input. The full disk shown in blue is the simulation region used; only TM modes were simulated. The atomic inversion and modal intensities are given in SALT units of $E_c$ and $D_c$.