{
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"cell_type": "markdown",
"id": "intro",
"metadata": {},
"source": [
"# Linear Absorption — Beer–Lambert Law\n",
"\n",
"When an electromagnetic field is weak enough that it never significantly\n",
"depletes the ground-state population, the medium responds linearly: the\n",
"field amplitude decays exponentially with propagation distance. This is\n",
"the **Beer–Lambert law**,\n",
"\n",
"$$\n",
"I(z, \\omega) = I_0(\\omega)\\,e^{-\\alpha(\\omega)\\,z},\n",
"$$\n",
"\n",
"where the absorption coefficient is $\\alpha(\\omega) := k\\chi_I(\\omega)$\n",
"(the wavenumber times the imaginary part of the susceptibility).\n",
"Since $I \\propto |\\Omega|^2$ this is equivalent to\n",
"\n",
"$$\n",
"|\\Omega(z, \\omega)| = |\\Omega_0(\\omega)|\\,e^{-\\alpha(\\omega)\\,z/2}.\n",
"$$\n",
"\n",
"For the resonant two-level system $\\alpha(0) = 2Ng/\\Gamma$, giving\n",
"$|\\Omega(z)| = |\\Omega_0|\\,e^{-Ng\\,z/\\Gamma}$ at $\\omega = 0$.\n",
"We use $\\Omega_0 = 10^{-3}\\,\\Gamma$ throughout.\n",
"\n",
"This notebook demonstrates:\n",
"\n",
"1. How increasing optical depth attenuates the pulse.\n",
"2. The role of spontaneous decay in reaching a true Beer–Lambert steady state.\n",
"3. That the exponential attenuation is independent of pulse shape (Gaussian, CW)."
]
},
{
"cell_type": "markdown",
"id": "parameters",
"metadata": {},
"source": [
"## Parameters\n",
"\n",
"The `interaction_strengths` parameter $Ng$ sets the coupling between field and\n",
"medium. It is proportional to atomic density $N$ and the dipole matrix element\n",
"squared: $C \\propto N |\\mu|^2 / \\hbar$. In the weak-field limit the resonant\n",
"absorption coefficient is $\\alpha \\propto Ng$, so doubling $Ng$ doubles the\n",
"optical depth of the medium.\n",
"\n",
"We will compare three values:\n",
"\n",
"| $Ng$ | Character |\n",
"|---|---|\n",
"| 0.1 | Optically thin — pulse passes through almost unchanged |\n",
"| 1.0 | Moderate absorption |\n",
"| 10.0 | Optically thick — strong attenuation |"
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "imports",
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"source": [
"import numpy as np\n",
"import plotly.graph_objects as go\n",
"from maxwellbloch import mb_solve, plot"
]
},
{
"cell_type": "markdown",
"id": "low-od-intro",
"metadata": {},
"source": [
"## Optically thin medium — $Ng = 0.1$\n",
"\n",
"With very few atoms the pulse propagates essentially undistorted.\n",
"There is a small reduction in amplitude and a slight group delay, but\n",
"the pulse shape is preserved."
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "solve-low-od",
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"shell.execute_reply": "2026-05-05T15:16:53.298768Z"
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{
"name": "stderr",
"output_type": "stream",
"text": [
"\r",
" 0%| | 0/10 [00:00\n",
" window.PlotlyConfig = {MathJaxConfig: 'local'};\n",
" if (window.MathJax && window.MathJax.Hub && window.MathJax.Hub.Config) {window.MathJax.Hub.Config({SVG: {font: \"STIX-Web\"}});}\n",
" \n",
" \n",
" "
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"source": [
"fig = plot.field_spacetime(mbs_ng01)\n",
"fig.update_layout(title=\"|Ω(z, t)| — Gaussian pulse, Ng = 0.1 (optically thin)\")\n",
"fig.show(renderer='notebook_connected')"
]
},
{
"cell_type": "markdown",
"id": "mod-od-intro",
"metadata": {},
"source": [
"## Moderate optical depth — $Ng = 1$\n",
"\n",
"With $Ng = 1$ the absorption is noticeable: the output amplitude is\n",
"appreciably smaller than the input and the pulse acquires a clear group\n",
"delay as the leading edge drives the atoms before the trailing edge arrives."
]
},
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"name": "stderr",
"output_type": "stream",
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"\r",
" 0%| | 0/20 [00:00\n",
" window.PlotlyConfig = {MathJaxConfig: 'local'};\n",
" if (window.MathJax && window.MathJax.Hub && window.MathJax.Hub.Config) {window.MathJax.Hub.Config({SVG: {font: \"STIX-Web\"}});}\n",
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"source": [
"fig = plot.field_spacetime(mbs_ng1)\n",
"fig.update_layout(title=\"|Ω(z, t)| — Gaussian pulse, Ng = 1\")\n",
"fig.show(renderer='notebook_connected')"
]
},
{
"cell_type": "markdown",
"id": "high-od-intro",
"metadata": {},
"source": [
"## Optically thick medium — $Ng = 10$\n",
"\n",
"With $Ng = 10$ the medium is strongly absorbing. The pulse is heavily\n",
"attenuated and significantly reshaped: the leading spectral components\n",
"are absorbed first, delaying the effective peak and broadening the\n",
"transmitted pulse."
]
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"cell_type": "code",
"execution_count": 6,
"id": "solve-high-od",
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"output_type": "stream",
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"\r",
" 0%| | 0/100 [00:00\n",
" window.PlotlyConfig = {MathJaxConfig: 'local'};\n",
" if (window.MathJax && window.MathJax.Hub && window.MathJax.Hub.Config) {window.MathJax.Hub.Config({SVG: {font: \"STIX-Web\"}});}\n",
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"source": [
"fig = plot.field_spacetime(mbs_ng10)\n",
"fig.update_layout(title=\"|Ω(z, t)| — Gaussian pulse, Ng = 10 (optically thick)\")\n",
"fig.show(renderer='notebook_connected')"
]
},
{
"cell_type": "markdown",
"id": "comparison-intro",
"metadata": {},
"source": [
"## Output envelope comparison\n",
"\n",
"Plotting the output field envelope $|\\Omega(z=1, t)|$ for all three cases\n",
"alongside the common input shows the progressive attenuation. In the\n",
"Beer–Lambert limit the peak amplitude scales as $e^{-\\alpha/2}$ with $\\alpha \\propto Ng$."
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "plot-comparison",
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"source": [
"fig = go.Figure()\n",
"\n",
"# input (same for all runs — use Ng=0.1 z=0 as reference)\n",
"fig.add_trace(go.Scatter(\n",
" x=mbs_ng01.tlist,\n",
" y=np.abs(mbs_ng01.Omegas_zt[0, 0, :]),\n",
" mode=\"lines\",\n",
" name=\"input\",\n",
" line=dict(dash=\"dash\", color=\"black\"),\n",
"))\n",
"\n",
"for mbs, label, color in [\n",
" (mbs_ng01, \"Ng = 0.1\", \"#636EFA\"),\n",
" (mbs_ng1, \"Ng = 1\", \"#EF553B\"),\n",
" (mbs_ng10, \"Ng = 10\", \"#00CC96\"),\n",
"]:\n",
" fig.add_trace(go.Scatter(\n",
" x=mbs.tlist,\n",
" y=np.abs(mbs.Omegas_zt[0, -1, :]),\n",
" mode=\"lines\",\n",
" name=label,\n",
" line=dict(color=color),\n",
" ))\n",
"\n",
"fig.update_layout(\n",
" title=\"Output pulse envelope |Ω(z=1, t)| vs optical depth\",\n",
" xaxis_title=\"t (γ⁻¹)\",\n",
" yaxis_title=\"|Ω| (γ)\",\n",
" template=\"maxwellbloch\",\n",
" width=700,\n",
" height=400,\n",
")\n",
"fig.show(renderer='notebook_connected')"
]
},
{
"cell_type": "markdown",
"id": "cw-intro",
"metadata": {},
"source": [
"## Pulse-shape independence — CW field\n",
"\n",
"Beer–Lambert attenuation depends only on frequency and optical depth,\n",
"not on pulse shape. A continuous-wave (CW) field ramped on at $t = 0$ reaches\n",
"a steady-state transmitted amplitude determined solely by $Ng$ once the\n",
"transient has decayed. This is the same $e^{-\\alpha/2}$ factor seen above.\n",
"\n",
"With decay present the transient ring-on settles in a time $\\sim 1/\\Gamma$."
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "solve-cw",
"metadata": {
"execution": {
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"shell.execute_reply": "2026-05-05T15:16:54.635232Z"
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"source": [
"mb_solve_json_cw = \"\"\"\n",
"{\n",
" \"atom\": {\n",
" \"decays\": [{\"channels\": [[0, 1]], \"rate\": 1.0}],\n",
" \"fields\": [\n",
" {\n",
" \"coupled_levels\": [[0, 1]],\n",
" \"detuning\": 0.0,\n",
" \"rabi_freq\": 1.0e-3,\n",
" \"rabi_freq_t_args\": {\"ampl\": 1.0, \"on\": 0.0, \"off\": 8.0, \"fwhm\": 1.0},\n",
" \"rabi_freq_t_func\": \"ramp_onoff\"\n",
" }\n",
" ],\n",
" \"num_states\": 2\n",
" },\n",
" \"t_min\": -2.0,\n",
" \"t_max\": 10.0,\n",
" \"t_steps\": 100,\n",
" \"z_min\": -0.2,\n",
" \"z_max\": 1.2,\n",
" \"z_steps\": 20,\n",
" \"interaction_strengths\": [10.0],\n",
" \"savefile\": \"mbs-linear-absorption-cw\"\n",
"}\n",
"\"\"\"\n",
"\n",
"mbs_cw = mb_solve.MBSolve().from_json_str(mb_solve_json_cw)\n",
"mbs_cw.mbsolve(recalc=False);"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "plot-cw-spacetime",
"metadata": {
"execution": {
"iopub.execute_input": "2026-05-05T15:16:54.637472Z",
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"source": [
"fig = plot.field_spacetime(mbs_cw)\n",
"fig.update_layout(title=\"|Ω(z, t)| — CW ramp-on field, Ng = 10, Γ = 1\")\n",
"fig.show(renderer='notebook_connected')"
]
},
{
"cell_type": "markdown",
"id": "beer-lambert-intro",
"metadata": {},
"source": [
"## Beer–Lambert comparison\n",
"\n",
"With decay and a CW field, the medium reaches a true steady state: the\n",
"on-resonance ($\\omega = 0$) field amplitude decays exponentially with\n",
"penetration depth. Extracting the spatial profile at a time well inside\n",
"the CW plateau ($t = 5\\,\\gamma^{-1}$, long after the ramp-on transient\n",
"has decayed) and comparing with the Beer–Lambert prediction\n",
"$|\\Omega(z)| = |\\Omega_0|\\,e^{-Ng\\,z/\\Gamma}$ shows excellent agreement."
]
},
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"cell_type": "code",
"execution_count": 11,
"id": "beer-lambert-plot",
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"source": [
"# Time index for plateau (t ≈ 5, well after ramp-on transient)\n",
"i_t5 = np.argmin(np.abs(mbs_cw.tlist - 5.0))\n",
"\n",
"# CW amplitude vs z at steady state\n",
"Omega_cw_z = np.abs(mbs_cw.Omegas_zt[0, :, i_t5])\n",
"\n",
"# Reference amplitude before the medium (z < 0, vacuum region)\n",
"Omega_0 = Omega_cw_z[0] # at z = z_min = -0.2\n",
"\n",
"# Beer-Lambert curve: Ω(z) = Ω₀ exp(−Ng z/Γ) inside medium [0, 1]\n",
"# outside [0, 1] the density is zero so amplitude is constant\n",
"Ng_val = 10.0\n",
"Gamma = 1.0\n",
"z_arr = mbs_cw.zlist\n",
"beer_lambert = Omega_0 * np.exp(-Ng_val / Gamma * np.clip(z_arr, 0.0, 1.0))\n",
"\n",
"fig = go.Figure()\n",
"fig.add_trace(go.Scatter(\n",
" x=z_arr, y=Omega_cw_z,\n",
" mode='lines', name='simulation (CW steady state)',\n",
"))\n",
"fig.add_trace(go.Scatter(\n",
" x=z_arr, y=beer_lambert,\n",
" mode='lines', name='Beer–Lambert: Ω₀ exp(−Ng z/Γ)',\n",
" line=dict(dash='dash'),\n",
"))\n",
"fig.update_layout(\n",
" title='CW steady-state spatial profile vs Beer–Lambert (Ng = 10, Γ = 1)',\n",
" xaxis_title='z',\n",
" yaxis_title='|Ω| (γ)',\n",
" yaxis_type='log',\n",
" template='maxwellbloch',\n",
" width=700,\n",
" height=400,\n",
")\n",
"fig.show(renderer='notebook_connected')"
]
},
{
"cell_type": "markdown",
"id": "summary",
"metadata": {},
"source": [
"## Summary\n",
"\n",
"- In the **weak-field limit** ($\\Omega_0 \\ll \\Gamma$) the Maxwell–Bloch equations\n",
" reduce to the Beer–Lambert law: amplitude decays exponentially with $Ng$.\n",
"- Without spontaneous decay the atoms respond coherently; the medium acts as a\n",
" resonant absorber but also re-emits energy, producing a **free induction decay**\n",
" tail after the pulse.\n",
"- Adding decay at rate $\\Gamma$ damps the coherence on timescale $T_2 = 2/\\Gamma$.\n",
" Once $T_2 \\lesssim \\tau_\\text{pulse}$ the transmission matches the steady-state\n",
" Beer–Lambert prediction.\n",
"- The steady-state attenuation is **pulse-shape independent**: both Gaussian and\n",
" CW fields with the same $Ng$ reach the same transmitted amplitude once transients\n",
" have decayed.\n",
"\n",
"The nonlinear regime — where the field is strong enough to saturate the transition\n",
"and the area theorem governs propagation — is covered in the\n",
"Area Theorem and SIT Solitons notebooks."
]
}
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