Meep Tutorial/Multilevel-atomic susceptibility
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| - | Meep 1.4 introduced a feature to model saturable absorption/gain via a multilevel-atomic susceptibility. This is based on a generalization of the [[w:Maxwell-Bloch equations|Maxwell-Bloch equations]] which involves the interaction of a quantized system having an arbitrary number of levels with the electromagnetic fields. We will demonstrate this feature by computing the lasing thresholds of a two-level, multimode cavity in 1d. This is similar to the example used in [https://www.osapublishing.org/oe/abstract.cfm?uri=oe-20-1-474 A. Cerjan et al] (Fig. 2). | + | Meep 1.4 introduced a feature to model saturable absorption/gain via multilevel-atomic susceptibility. This is based on a generalization of the [[w:Maxwell-Bloch equations|Maxwell-Bloch equations]] which involves the interaction of a quantized system having an arbitrary number of levels with the electromagnetic fields. We will demonstrate this feature by computing the lasing thresholds of a two-level, multimode cavity in 1d. This is similar to the example used in [https://www.osapublishing.org/oe/abstract.cfm?uri=oe-20-1-474 A. Cerjan et al] (Fig. 2). |
| - | The cavity consists of a high-index active medium with a perfect-metallic mirror on one end and an abrupt termination in air on the other. We will specify an initial population density for the ground state. The cavity field is initialized to an arbitrary profile and the two-level system is then driven using a fictitious external source at a fixed pumping rate. The fields are time stepped until steady state. Near the end of the time stepping, we will output the electric field in the middle of the cavity and then, in post processing, compute its Fourier transform to obtain the spectra. | + | The cavity consists of a high-index medium with a perfect-metallic mirror on one end and an abrupt termination in air on the other. We will specify an initial population density for the ground state of the two-level system. The field within the cavity is initialized to an arbitrary profile and a fictitious source is used to pump the cavity at a fixed rate. The fields are time stepped until reaching steady state. Near the end of the time stepping, we will output the electric field in the middle of the cavity and then, in post processing, compute its Fourier transform to obtain the spectra. |
| - | The angular frequency of the passive cavity modes for this simple structure can be computed analytically using the equation <math>\omega_{cav}=(m+0.5)\pi/(n_{cav}L_{cav})</math> where <math>n_{cav}</math> and <math>L_{cav}</math> are the cavity index and length and <math>m</math> is an integer. Similar to the paper reference above, we focus on the angular-frequency range of [36,44] which contains four modes corresponding to <math>m</math>=17-20. In the continuum limit, these modes would appear in the cavity spectra as Dirac delta functions. The discretized model, however, produces peaks with finite width. We therefore need to integrate a fixed number of points around each peak to obtain continuously-varying results. | + | The angular frequency of the passive cavity modes can be computed analytically using the equation <math>\omega_{cav}=(m+0.5)\pi/(n_{cav}L_{cav})</math> where <math>n_{cav}</math> and <math>L_{cav}</math> are the cavity index and length, and <math>m</math> is an integer. Similar to the paper referenced above, we focus on the angular-frequency range of [36,44] which contains four modes corresponding to <math>m</math>=17-20. In the continuum limit, these modes would appear as Dirac delta functions in the cavity spectra. The discretized model, however, produces peaks with finite width. We therefore need to integrate a fixed number of points around each peak to smoothen out the mode intensity. |
| (set-param! resolution 500) | (set-param! resolution 500) | ||
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| (set! pml-layers (list (make pml (thickness dpml) (side High)))) | (set! pml-layers (list (make pml (thickness dpml) (side High)))) | ||
| (define-param freq-21 (/ 40 (* 2 pi))) ; emission frequency (units of 2\pia/c) | (define-param freq-21 (/ 40 (* 2 pi))) ; emission frequency (units of 2\pia/c) | ||
| - | (define-param gamma-21 (/ 8 (* 2 pi))) ; emission gamma (units of 2\pia/c) | + | (define-param gamma-21 (/ 4 (* 2 pi))) ; emission gamma (units of 2\pia/c) |
| (define-param sigma-21 8e-23) ; emission sigma | (define-param sigma-21 8e-23) ; emission sigma | ||
| (set! sigma-21 (/ sigma-21 (sqr freq-21))) | (set! sigma-21 (/ sigma-21 (sqr freq-21))) | ||
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| (define print-field (lambda () (print "field:, " (meep-time) ", " | (define print-field (lambda () (print "field:, " (meep-time) ", " | ||
| (real-part (get-field-point Ex (vector3 0 0 (+ (* -0.5 sz) (* 0.5 Lcav))))) "\n"))) | (real-part (get-field-point Ex (vector3 0 0 (+ (* -0.5 sz) (* 0.5 Lcav))))) "\n"))) | ||
| - | (define-param endt 30000) | + | (define-param endt 50000) |
| (run-until endt (after-time (- endt 250) print-field)) | (run-until endt (after-time (- endt 250) print-field)) | ||
| - | The choice of the simulation parameters requires some care. For example, choosing a pumping rate <code>Rp</code> that lies far beyond threshold will cause large inversion which is not physical and produce meaningless results. For these types of problems, it is also important to check that the output fields do not contain relaxation oscillations. This may indicate that the system has yet to reach steady state. The choice of the simulation time is also important when operating near the lasing threshold of a particular mode as the modal intensity is typically weak. For this particular cavity, we can compute the lasing thresholds analytically using the steady-state ab-initio laser theory (SALT) developed by Prof. A. Douglas Stone and his group at Yale. Based on the default parameters of the simulation script above, the thresholds lie within pumping-rate interval of [0.0053,0.0096]. This is shown in the figure below. We also choose a fairly large resolution which is necessary to ensure numerical stability due to the presence of the polarization field (this is also a feature of linear absorbers). | + | The choice of the atomic media parameters requires some care. For example, choosing a pumping rate <code>Rp</code> that lies far beyond threshold will cause large inversion which is not physical and thus produces meaningless results. For these types of problems involving steady-state phenomena, it is important to check that the fields used in the analysis do not contain relaxation oscillations. The choice of the simulation time is also important when operating near the lasing threshold of a particular mode as there the modal intensity is typically weak. For this particular cavity, we can compute the lasing thresholds analytically using the steady-state ab-initio laser theory (SALT) developed by Prof. A. Douglas Stone and his group at Yale. Based on the default parameters of the simulation script above, two modes, <math>m</math>=18 and 19, should begin to lase within the pumping-rate interval of [0.0050,0.0096]. For comparison, note that this is just above the relaxation rate of <code>rate-21</code>=0.005. We also need to choose a large resolution to ensure numerical stability of the polarization fields which is also a feature of linear absorbers. A plot of the variation of the modal intensity with pumping rate is shown below. |
| - | Note that the slopes of each curve is decreasing with increasing pumping rate. This occurs because the onset of lasing from additional modes means there is less gain available for the other modes. The modal intensities reach an asymptote in the limit of large pumping rates. | + | Note that the slopes of each curve is decreasing with increasing pumping rate. This occurs because the onset of lasing from additional modes means there is less gain available to the other modes. The modal intensities reach an asymptote in the limit of large pumping rates. We can convert Meep's dimensionless parameters into real units by specifying the units of the cavity length <math>L_{cav}</math> and then multiplying the rate terms by <math>L_{cav}/c</math>. |
Revision as of 03:12, 17 December 2016
Meep 1.4 introduced a feature to model saturable absorption/gain via multilevel-atomic susceptibility. This is based on a generalization of the Maxwell-Bloch equations which involves the interaction of a quantized system having an arbitrary number of levels with the electromagnetic fields. We will demonstrate this feature by computing the lasing thresholds of a two-level, multimode cavity in 1d. This is similar to the example used in A. Cerjan et al (Fig. 2).
The cavity consists of a high-index medium with a perfect-metallic mirror on one end and an abrupt termination in air on the other. We will specify an initial population density for the ground state of the two-level system. The field within the cavity is initialized to an arbitrary profile and a fictitious source is used to pump the cavity at a fixed rate. The fields are time stepped until reaching steady state. Near the end of the time stepping, we will output the electric field in the middle of the cavity and then, in post processing, compute its Fourier transform to obtain the spectra.
The angular frequency of the passive cavity modes can be computed analytically using the equation ωcav = (m + 0.5)π / (ncavLcav) where ncav and Lcav are the cavity index and length, and m is an integer. Similar to the paper referenced above, we focus on the angular-frequency range of [36,44] which contains four modes corresponding to m=17-20. In the continuum limit, these modes would appear as Dirac delta functions in the cavity spectra. The discretized model, however, produces peaks with finite width. We therefore need to integrate a fixed number of points around each peak to smoothen out the mode intensity.
(set-param! resolution 500)
(define-param ncav 1.5) ; cavity refractive index
(define-param Lcav 1) ; cavity length
(define-param dpad 1) ; padding thickness
(define-param dpml 1) ; PML thickness
(define-param sz (+ Lcav dpad dpml))
(set! geometry-lattice (make lattice (size no-size no-size sz)))
(set! dimensions 1)
(set! pml-layers (list (make pml (thickness dpml) (side High))))
(define-param freq-21 (/ 40 (* 2 pi))) ; emission frequency (units of 2\pia/c)
(define-param gamma-21 (/ 4 (* 2 pi))) ; emission gamma (units of 2\pia/c)
(define-param sigma-21 8e-23) ; emission sigma
(set! sigma-21 (/ sigma-21 (sqr freq-21)))
(define-param rate-21 0.005) ; non-radiative rate (units of c/a)
(define-param N0 1e23) ; initial population of ground state
(define-param Rp 0.015) ; pumping rate of ground to highest state
(define two-level (make medium (index ncav)
(E-polarizations (make multilevel-atom (sigma 1)
(transitions (make transition (from-level 1) (to-level 2) (pumping-rate Rp)
(frequency freq-21) (gamma gamma-21) (sigma sigma-21))
(make transition (from-level 2) (to-level 1) (transition-rate rate-21)))
(initial-populations N0)))))
(set! geometry (list (make block (center 0 0 (+ (* -0.5 sz) (* 0.5 Lcav)))
(size infinity infinity Lcav) (material two-level))))
(init-fields)
(meep-fields-initialize-field fields Ex
(lambda (p) (if (= (vector3-z p) (+ (* -0.5 sz) (* 0.5 Lcav))) 1 0)))
(define print-field (lambda () (print "field:, " (meep-time) ", "
(real-part (get-field-point Ex (vector3 0 0 (+ (* -0.5 sz) (* 0.5 Lcav))))) "\n")))
(define-param endt 50000)
(run-until endt (after-time (- endt 250) print-field))
The choice of the atomic media parameters requires some care. For example, choosing a pumping rate Rp that lies far beyond threshold will cause large inversion which is not physical and thus produces meaningless results. For these types of problems involving steady-state phenomena, it is important to check that the fields used in the analysis do not contain relaxation oscillations. The choice of the simulation time is also important when operating near the lasing threshold of a particular mode as there the modal intensity is typically weak. For this particular cavity, we can compute the lasing thresholds analytically using the steady-state ab-initio laser theory (SALT) developed by Prof. A. Douglas Stone and his group at Yale. Based on the default parameters of the simulation script above, two modes, m=18 and 19, should begin to lase within the pumping-rate interval of [0.0050,0.0096]. For comparison, note that this is just above the relaxation rate of rate-21=0.005. We also need to choose a large resolution to ensure numerical stability of the polarization fields which is also a feature of linear absorbers. A plot of the variation of the modal intensity with pumping rate is shown below.
Note that the slopes of each curve is decreasing with increasing pumping rate. This occurs because the onset of lasing from additional modes means there is less gain available to the other modes. The modal intensities reach an asymptote in the limit of large pumping rates. We can convert Meep's dimensionless parameters into real units by specifying the units of the cavity length Lcav and then multiplying the rate terms by Lcav / c.
