Meep Tutorial/Multilevel-atomic susceptibility

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-Meep 1.4 introduced a feature to model saturable absorption/gain media 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 quantum 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 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).
-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 at the edge of the cavity and then compute its Fourier transform in post processing to obtain the cavity spectra.+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 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. 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 discrete simulation model, however, produces peaks with finite width. We therefore need to integrate a fixed number of points around each peak, a 'window', to obtain continuously-varying results.+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.
(set-param! resolution 500) (set-param! resolution 500)
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(lambda (p) (if (= (vector3-z p) (+ (* -0.5 sz) (* 0.5 Lcav))) 1 0))) (lambda (p) (if (= (vector3-z p) (+ (* -0.5 sz) (* 0.5 Lcav))) 1 0)))
(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) Lcav)))) "\n")))+ (real-part (get-field-point Ex (vector3 0 0 (+ (* -0.5 sz) (* 0.5 Lcav))))) "\n")))
- (define-param endt 20000)+ (define-param endt 30000)
(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. The results will therefore be meaningless. 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, this thresholds lie within pumping-rate interval of [0.0053,0.0096]. This is shown in the figure below.+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).
-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 therefore reach an asymptote.+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.

Revision as of 20:34, 16 December 2016

Meep 1.4 introduced a feature to model saturable absorption/gain via a 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 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 angular frequency of the passive cavity modes for this simple structure 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 reference 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 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.

(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 (/ 8 (* 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 30000)
(run-until endt (after-time (- endt 250) print-field))

The choice of the simulation 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 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).

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.

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