% pmd_vs_length.m
% ------------------------------------------------------------
% Plots the Foschini-Poole (1991) analytical variance formula for PMD:
%
%   sigma_T^2(z) = 2*(Delta_beta1)^2 * lc^2
%                  * [ exp(-z/lc) + z/lc - 1 ]
%
% and overlays the two asymptotic regimes:
%   - Short-reach  (z << lc) : sigma_T(z) ~ (Delta_beta1) * z
%   - Long-haul    (z >> lc) : sigma_T(z) ~ Dp * sqrt(z)
%
% Parameters follow typical SMF values from the literature.
% ------------------------------------------------------------

clear; clc;

%% ── Parameters ──────────────────────────────────────────────────────────────
% Intrinsic local birefringence  [ps/km]
Delta_beta1 = 1e-1;           % typical value, adjust as needed

% Correlation length [km]
lc          = 0.01;           % ~50 m, typical for G.652 SMF

% PMD parameter  [ps / sqrt(km)]  —  derived from the two above
Dp          = Delta_beta1 * sqrt(2 * lc);

% Distance axis [km]
z_max       = 10;            % maximum distance
z           = linspace(0.001, z_max, 10000);   % avoid z = 0 in log plot

%% ── Exact Foschini-Poole formula (sigma_T in ps) ────────────────────────────
sigma_T_sq  = 2 .* Delta_beta1.^2 .* lc.^2 ...
    .* (exp(-z ./ lc) + z ./ lc - 1);
sigma_T     = sqrt(sigma_T_sq);          % RMS DGD  [ps]

%% ── Asymptotic regimes ───────────────────────────────────────────────────────
% Short-reach: linear growth  (z << lc)
sigma_T_short = Delta_beta1 .* z;        % [ps]

% Long-haul: square-root growth  (z >> lc)
sigma_T_long  = Dp .* sqrt(z);          % [ps]

%% ── Plot ─────────────────────────────────────────────────────────────────────
figure('Color','w','Position',[100 100 760 480]);
hold on;

% Color palette (matching dissertation style)
c_exact = [0.1216, 0.4706, 0.7059];      % blue  – exact
c_short = [0.8392, 0.1529, 0.1569];      % red   – short-reach asymptote
c_long  = [0.1961, 0.6314, 0.1725];      % green – long-haul asymptote

% Exact solution
h_exact = plot(z, sigma_T, ...
    'Color', c_exact, 'LineWidth', 2.0, ...
    'DisplayName', 'Exact (Foschini \& Poole)');

% Short-reach asymptote  σ_T ≈ Δβ₁ · z
h_short = plot(z, sigma_T_short, ...
    'Color', c_short, 'LineWidth', 1.4, 'LineStyle', '--', ...
    'DisplayName', '$\sigma_T \approx \Delta\beta_1 \cdot z$  \quad ($z \ll l_c$)');

% Long-haul asymptote  σ_T ≈ D_p √z
h_long  = plot(z, sigma_T_long, ...
    'Color', c_long,  'LineWidth', 1.4, 'LineStyle', ':', ...
    'DisplayName', '$\sigma_T \approx D_p \sqrt{z}$  \quad ($z \gg l_c$)');

%% ── Axes & decoration ────────────────────────────────────────────────────────
ax = gca;
set(ax, 'XScale', 'log', 'YScale', 'log');

% ── X-axis: linear-style tick labels on log scale ────────────────────────
x_ticks = [1e-3, 1e-2, 1e-1, 1, 10];
ax.XTick = x_ticks;
ax.XTickLabel = arrayfun(@(v) sprintf('%g km', v), x_ticks, 'UniformOutput', false);

% ── Y-axis: linear-style tick labels on log scale ────────────────────────
y_ticks = [1e-3, 1e-2, 1e-1, 1, 10];
ax.YTick = y_ticks;
ax.YTickLabel = arrayfun(@(v) sprintf('%g ps', v), y_ticks, 'UniformOutput', false);

xlabel('Fiber length $z$ [km]',              'Interpreter', 'latex');
ylabel('RMS DGD $\sigma_T$ [ps]',            'Interpreter', 'latex');

grid on; box on;
xlim([min(z) z_max]);

legend([h_exact, h_short, h_long], ...
    'Location', 'northwest', 'Interpreter', 'latex', 'FontSize', 9);

% Parameter annotation
anno_str = sprintf( ...
    ['$\\Delta\\beta_1 = %.3g$ ps/km\n' ...
    '$l_c = %.0f$ m\n' ...
    '$D_p = \\Delta\\beta_1\\sqrt{2l_c} = %.4g$ ps/$\\sqrt{\\mathrm{km}}$'], ...
    Delta_beta1, lc*1e3, Dp);

annotation('textbox', [0.57 0.14 0.38 0.22], ...
    'String',        anno_str, ...
    'Interpreter',   'latex', ...
    'FontSize',      8.5, ...
    'BackgroundColor','w', ...
    'EdgeColor',     [0.5 0.5 0.5], ...
    'LineWidth',     0.8, ...
    'FitBoxToText',  'on');

%% ── Regime transition marker ─────────────────────────────────────────────────
% Mark the crossover region around z = lc
xline(lc, '--', ...
    'Color', [0.5 0.5 0.5], 'LineWidth', 0.8, ...
    'HandleVisibility', 'off');
text(lc * 1.15, min(sigma_T)*3, '$l_c$', ...
    'Interpreter', 'latex', 'Color', [0.4 0.4 0.4], 'FontSize', 9);

%% ── Export (uncomment to use) ────────────────────────────────────────────────
% mat2tikz_improved('C:\...\tikz\pmd\pmd_vs_length.tikz')