===== Optical Gain =====

In this tutorial we calculate the optical gain upon optical irradiation.
The irradiation parameters are the
  * photon energy of the irradiation and the
  * line width.

==== Physics model ====
The transition rate per volume element can be expressed with the following sum,
\[
R = R_{ab} - R_{ba} = \frac{2}{V} \sum_{k_a} \sum_{k_b} \frac{2 \pi}{ \hbar} |H_{ba}| ^2 \delta(E_b - E_a -\hbar \omega)(f_a-f_b).
\]

In order to evaluate the sum much faster we calculate the $H_{ba}$ matrix element at $k_a = 0; k_b = 0$ (Remark: $k_a = k_b$), and we neglect the $k$ dependence of it. 
Then we can simplify the sum as follows, 
\[
R(E, w) = C_0(E)   \int \gamma(E_a-E, w) \cdot  H(E_a-E) \cdot  [n(E_a) - p(E_b)]  {\rm d}E_a {\rm d}E_b,
\]
where $E$ is the irradiation energy, $w$ is the line width and we assume that the irradiation has the $\gamma(E, w)$ broadening function.

Here $C_0(E)$ is an energy dependent constant,
\[
C_0 = \frac{\pi e^2 \hbar}{n_{\rm r} c \epsilon_0 m_0^2 E}.
\] 

====Input file structure====

A new keyword has been introduced to handle an optical device, ''opticaldevice{}''.

<code>
opticaldevice{
	name = "quantum_region_name"
	line_broadening = 1            # Line broadening model (1: Lorentzian)
	photon_energy   = 1.0          # Photon energy in (eV)
	line_width      = 1.0          # Line width in (eV)
}
</code>

The ''run'' keyword requires ''solve_optical_device{}'' to be included.

==== Results ====
<figure>
;#;
<dataplot xlabel="Position (nm)" ylabel="Gain (1/s/m^3)" ylegends="Gain" title="Optical gain">
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27	1.2537626372951e+18
27.2	1.3687797494775e+18
27.4	1.4906596701e+18
27.6	1.6203036970485e+18
27.8	1.7588245451379e+18
28	1.9097926805291e+18
28.2	2.0636970928785e+18
28.4	2.2290417373803e+18
28.6	2.4067177327278e+18
28.8	2.5976767637574e+18
29	2.8029360983978e+18
29.2	3.0235838842453e+18
29.4	3.2607847549977e+18
29.6	3.5157857777756e+18
29.8	3.7899227733835e+18
30	4.0846270428074e+18
30.2	4.4014325347038e+18
30.4	4.7419834903108e+18
30.6	5.108042604099e+18
30.8	5.5014997405852e+18
31	5.9243812500567e+18
31.2	6.3788599285036e+18
31.4	6.8672656698409e+18
31.6	7.3920968615231e+18
31.8	7.9560325779303e+18
32	8.5619456294403e+18
32.2	9.2129165289105e+18
32.4	9.9122484413969e+18
32.6	1.0663483187338e+19
32.8	1.1470418374159e+19
33	1.2337125736316e+19
33.2	1.3267970769228e+19
33.4	1.4267633748348e+19
33.6	1.5341132230841e+19
33.8	1.6493845143954e+19
34	1.7731538571306e+19
34.2	2.6102791190001e+19
34.4	3.3680495518877e+19
34.6	4.024204778331e+19
34.8	4.6696724981377e+19
35	5.33136060908e+19
35.2	6.0229591063294e+19
35.4	6.7535868495545e+19
35.6	7.5304381313219e+19
35.8	8.3598455211246e+19
36	9.2477871435918e+19
36.2	1.0200162143131e+20
36.4	1.1222957371972e+20
36.6	1.2322358640242e+20
36.8	1.3504832055503e+20
37	1.4777188663989e+20
37.2	1.6146639690226e+20
37.4	1.7620846638476e+20
37.6	1.9207968878743e+20
37.8	2.0916710412647e+20
38	2.2756366970928e+20
38.2	2.4736874266337e+20
38.4	2.6868858023301e+20
38.6	2.9163686279086e+20
38.8	3.1633524371456e+20
39	3.4291392978122e+20
39.2	3.7151229543081e+20
39.4	4.0227953407972e+20
39.6	4.353753495855e+20
39.8	4.7097069094669e+20
40	5.0924853334973e+20
40.2	5.5040470873577e+20
40.4	5.9464878914527e+20
40.6	6.422050262013e+20
40.8	6.9331335020914e+20
41	7.4823043247572e+20
41.2	8.072308145859e+20
41.4	8.7060810850942e+20
41.6	9.3867627155149e+20
41.8	1.0117709602979e+21
42	1.090250967841e+21
42.2	1.1744997487008e+21
42.4	1.2649270359773e+21
42.6	1.3619705553773e+21
42.8	1.4660978408522e+21
43	1.5778081566558e+21
43.2	1.6976345306828e+21
43.4	1.8261459039646e+21
43.6	1.9639494011887e+21
43.8	2.1116927270492e+21
44	2.2700666931334e+21
44.2	2.4398078798886e+21
44.4	2.6217014379889e+21
44.6	2.8165840331219e+21
44.8	3.0253469378291e+21
45	3.2489392735562e+21
45.2	3.4883714054732e+21
45.4	3.7447184919127e+21
45.6	4.0191241894243e+21
45.8	4.3128045134315e+21
46	4.627051853305e+21
46.2	4.9632391393008e+21
46.4	5.3228241572359e+21
46.6	5.7073540049833e+21
46.8	6.1184696828197e+21
47	6.5579108073587e+21
47.2	7.0275204362099e+21
47.4	7.5292499876199e+21
47.6	8.0651642361429e+21
47.8	8.6374463618553e+21
48	9.2484030267551e+21
48.2	9.900469447759e+21
48.4	1.0596214431143e+22
48.6	1.1338345328351e+22
48.8	1.2129712867856e+22
49	1.2973315812189e+22
49.2	1.3872305383423e+22
49.4	1.4829989394338e+22
49.6	1.5849836016214e+22
49.8	1.6935477107901e+22
50	9.0503277327804e+21
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50.4	0
50.6	0
50.8	0
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51.2	0
51.4	0
51.6	0
51.8	0
52	0
52.2	0
52.4	0
52.6	0
52.8	0
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53.2	0
53.4	0
53.6	0
53.8	0
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54.2	0
54.4	0
54.6	0
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55.4	0
55.6	0
55.8	0
56	0
56.2	0
56.4	0
56.6	0
56.8	0
57	0
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57.4	0
57.6	0
57.8	0
58	0
58.2	0
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58.6	0
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59.8	0
60	0
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61	0
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61.4	0
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63	0
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63.4	0
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63.8	0
64	0
64.2	0
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64.6	0
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65	0
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67	0
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68	0
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68.6	0
68.8	0
69	0
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80	0
85	0
90	0

</dataplot>
;#;
</figure>