117 lines
5.1 KiB
Matlab
117 lines
5.1 KiB
Matlab
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% pmd_vs_length.m
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% ------------------------------------------------------------
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% Plots the Foschini-Poole (1991) analytical variance formula for PMD:
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%
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% sigma_T^2(z) = 2*(Delta_beta1)^2 * lc^2
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% * [ exp(-z/lc) + z/lc - 1 ]
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%
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% and overlays the two asymptotic regimes:
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% - Short-reach (z << lc) : sigma_T(z) ~ (Delta_beta1) * z
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% - Long-haul (z >> lc) : sigma_T(z) ~ Dp * sqrt(z)
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%
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% Parameters follow typical SMF values from the literature.
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% ------------------------------------------------------------
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clear; clc;
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%% ── Parameters ──────────────────────────────────────────────────────────────
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% Intrinsic local birefringence [ps/km]
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Delta_beta1 = 1e-1; % typical value, adjust as needed
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% Correlation length [km]
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lc = 0.01; % ~50 m, typical for G.652 SMF
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% PMD parameter [ps / sqrt(km)] — derived from the two above
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Dp = Delta_beta1 * sqrt(2 * lc);
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% Distance axis [km]
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z_max = 10; % maximum distance
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z = linspace(0.001, z_max, 10000); % avoid z = 0 in log plot
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%% ── Exact Foschini-Poole formula (sigma_T in ps) ────────────────────────────
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sigma_T_sq = 2 .* Delta_beta1.^2 .* lc.^2 ...
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.* (exp(-z ./ lc) + z ./ lc - 1);
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sigma_T = sqrt(sigma_T_sq); % RMS DGD [ps]
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%% ── Asymptotic regimes ───────────────────────────────────────────────────────
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% Short-reach: linear growth (z << lc)
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sigma_T_short = Delta_beta1 .* z; % [ps]
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% Long-haul: square-root growth (z >> lc)
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sigma_T_long = Dp .* sqrt(z); % [ps]
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%% ── Plot ─────────────────────────────────────────────────────────────────────
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figure('Color','w','Position',[100 100 760 480]);
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hold on;
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% Color palette (matching dissertation style)
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c_exact = [0.1216, 0.4706, 0.7059]; % blue – exact
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c_short = [0.8392, 0.1529, 0.1569]; % red – short-reach asymptote
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c_long = [0.1961, 0.6314, 0.1725]; % green – long-haul asymptote
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% Exact solution
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h_exact = plot(z, sigma_T, ...
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'Color', c_exact, 'LineWidth', 2.0, ...
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'DisplayName', 'Exact (Foschini \& Poole)');
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% Short-reach asymptote σ_T ≈ Δβ₁ · z
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h_short = plot(z, sigma_T_short, ...
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'Color', c_short, 'LineWidth', 1.4, 'LineStyle', '--', ...
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'DisplayName', '$\sigma_T \approx \Delta\beta_1 \cdot z$ \quad ($z \ll l_c$)');
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% Long-haul asymptote σ_T ≈ D_p √z
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h_long = plot(z, sigma_T_long, ...
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'Color', c_long, 'LineWidth', 1.4, 'LineStyle', ':', ...
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'DisplayName', '$\sigma_T \approx D_p \sqrt{z}$ \quad ($z \gg l_c$)');
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%% ── Axes & decoration ────────────────────────────────────────────────────────
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ax = gca;
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set(ax, 'XScale', 'log', 'YScale', 'log');
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% ── X-axis: linear-style tick labels on log scale ────────────────────────
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x_ticks = [1e-3, 1e-2, 1e-1, 1, 10];
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ax.XTick = x_ticks;
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ax.XTickLabel = arrayfun(@(v) sprintf('%g km', v), x_ticks, 'UniformOutput', false);
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% ── Y-axis: linear-style tick labels on log scale ────────────────────────
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y_ticks = [1e-3, 1e-2, 1e-1, 1, 10];
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ax.YTick = y_ticks;
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ax.YTickLabel = arrayfun(@(v) sprintf('%g ps', v), y_ticks, 'UniformOutput', false);
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xlabel('Fiber length $z$ [km]', 'Interpreter', 'latex');
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ylabel('RMS DGD $\sigma_T$ [ps]', 'Interpreter', 'latex');
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grid on; box on;
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xlim([min(z) z_max]);
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legend([h_exact, h_short, h_long], ...
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'Location', 'northwest', 'Interpreter', 'latex', 'FontSize', 9);
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% Parameter annotation
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anno_str = sprintf( ...
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['$\\Delta\\beta_1 = %.3g$ ps/km\n' ...
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'$l_c = %.0f$ m\n' ...
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'$D_p = \\Delta\\beta_1\\sqrt{2l_c} = %.4g$ ps/$\\sqrt{\\mathrm{km}}$'], ...
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Delta_beta1, lc*1e3, Dp);
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annotation('textbox', [0.57 0.14 0.38 0.22], ...
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'String', anno_str, ...
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'Interpreter', 'latex', ...
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'FontSize', 8.5, ...
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'BackgroundColor','w', ...
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'EdgeColor', [0.5 0.5 0.5], ...
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'LineWidth', 0.8, ...
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'FitBoxToText', 'on');
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%% ── Regime transition marker ─────────────────────────────────────────────────
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% Mark the crossover region around z = lc
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xline(lc, '--', ...
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'Color', [0.5 0.5 0.5], 'LineWidth', 0.8, ...
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'HandleVisibility', 'off');
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text(lc * 1.15, min(sigma_T)*3, '$l_c$', ...
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'Interpreter', 'latex', 'Color', [0.4 0.4 0.4], 'FontSize', 9);
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%% ── Export (uncomment to use) ────────────────────────────────────────────────
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% mat2tikz_improved('C:\...\tikz\pmd\pmd_vs_length.tikz')
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