Optical Communications
Lecture 1
Fiber Parameter:
\[ \begin{align} NA=\sqrt{n_1^2-n_2^2}=n_1\sqrt{2\Delta}\\\\ \Delta=\frac{n_1^2-n_2^2} {2n_1^2}\approx \frac{n_1-n_2}{n_1} \end{align} \]
Pulse Broadening
\[ \delta T = \frac{Ln_1}{c}\left(\frac{n_1}{n_2}-1\right)\sim \frac{Ln_1\Delta}{c}=\frac{L(NA)^2}{2cn_1} \]
\[ \sigma = \frac{Ln_1\Delta}{2\sqrt3 c} \]
Bit rate for RZ code
\[ B_T(max)=\frac{1}{2\delta T}=\frac{0.2}{\sigma} \]
Lecture 2
Mode index or Effective index
\[ \frac{\beta}{k} \in [n_2,n_1) \]
Normalized Frequency
\[ V=ak(n_1^2-n_2^2)^{1/2}=akNA=akn_1\sqrt{2\Delta} \]
Number of Modes
\[ \frac{V^2}{2} \]
Normalized Propagation Constant
\[ b=\frac{(\frac{\beta}{k})^2-n_2^2}{n_1^2-n_2^2}=\frac{W^2}{V^2} \]
Single-Mode Requirement
\[ V \le 2.405 \]
Lecture 3
graph TD
A(Transmission of Pulses) -->B1("Attenuation(limit dB)")
A-->B2("Dispersion(limit bit rate)")
B1-->C1(Absorption)
B1-->C2(Scattering)
B1-->C3(min->1.55)
B2-->C4(Intermodal)
B2-->C5(Intramodal)
Attenuation
\[ \alpha(dB/km)=-\frac{10}{L}lg(\frac{P_o}{P_i}) \]
\[ P_o=P_i \times 10^{-\frac{\alpha L}{10}} \]
Intermodal Dispersion
Intermodal Broadening
\[ \sigma_n(ps \cdot km^{-1}) = \frac{Ln_1\Delta}{2\sqrt3c}, L= 1km \]
Material Dispersion
\[ \sigma_m(ps \cdot km^{-1}) = \sigma_\lambda(nm)D_M(ps \cdot nm^{-1}\cdot km^{-1}) \]
\(\sigma_\lambda\) : Optical Source
Intramodal Dispersion
Total Pulse Broadening(Test1 Q2)
\[ \sigma_T = \sigma_\lambda \cdot |D_T| \]
\[ D_T=D_M + D_W + D_p \]
\(D_p\) can be ignored.
\(D_W\) can be ignored in multi-mode fiber
Material Dispersion Parameter(Exercise3 Q5,7)
\[ D_M=-\frac{\lambda}{c}\cdot\frac{d^2n}{d\lambda^2} \]
Waveguide Dispersion Parameter(Exercise3 Q3,4 )
\[ D_W = -\frac{n_1-n_2}{\lambda c}V\frac{d^2(bV)}{dV^2} \]
Dispersion Slope(Exercise3 Q6)
\[ D_T(\lambda)=\frac{\lambda S_0}{4}\left[1-\left(\frac{\lambda_0}{\lambda}\right)^4\right] \]
estimate \(D_T\) near the minimum intramodal dispersion point
Output Pulse Width and Power(Test1 Q3)
\[ \sigma_o=\sqrt{\sigma_i^2+\sigma_T^2} \]
\[ P_o=P_i \cdot \frac{\sigma_o}{\sigma_i} \]
Polarization Mode Dispersion(Exercise3 Q8,9)
Modal Birefringence
\[ B=\frac{\beta_x - \beta_y}{k} \]
Beat Length
\[ L_B = \frac{2\pi}{\beta_x - \beta_y}=\frac{\lambda}{B} \]
Lecture 4
Measurement of NA(Exercise4 Q3)
\[ NA = sin\theta_a=\frac{A}{\sqrt{A^2+4D^2}}\approx \frac{A}{2D} \]
Cut-back method(Exercise4 Q2)
\[ \alpha=\frac{10}{L_1-L_2}lg\frac{P_{o2}}{P_{o1}} \]
control Launch Condition, because we need to obtain a equilibrium modal power distribution and high-order modes have high attenuation per unit length which makes power distribution not uniform.
Time Domain Measurement(Exercise4 Q4)
3-dB pulse (full duration at half maximum) broadening
\[ \tau=\frac{\sqrt{\tau_o^2-\tau_i^2}}{L} \]
Bandwidth for Gaussian pulses $$ \sigma = 0.43\tau $$
\[ B = \frac{0.19}{\sigma}=\frac{0.44}{\tau} \]