BER and Demultiplexers Apr. 12, 2018

• Noise equivalent power (NEP): minimum optical power per square root of bandwidth to achieve SNR=1.

$\text{ }NEP\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{P}_{in}{|}_{SNR=1}/\sqrt{\Delta f}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\left(\overline{{i}_{T}^{2}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\overline{{i}_{S}^{2}}{\right)}^{1/2}/\left(R\sqrt{\Delta f}\right)$

• Digital communication: The signal quality is controlled by bit error rate (BER), proportional to SNR.
Acceptable quality,$\text{ }BER\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1{0}^{-9}$
• BER (Sec 4.4.6 and handout): It depends on threshold level and coding, .e.g return to zero (RZ) and non-return to zero (NRZ).
For thermal noise limited system,$\text{ }BER\text{\hspace{0.17em}}=\text{\hspace{0.17em}}0.5\text{\hspace{0.17em}}-\text{\hspace{0.17em}}0.5erf\left(0.354\sqrt{SNR}\right)$ where erf is the error function. BER improves a lot when SNR is doubled.
Note:$\text{ }1\text{\hspace{0.17em}}-\text{\hspace{0.17em}}erf\left(x\right)\text{\hspace{0.17em}}\text{≈}\text{\hspace{0.17em}}{e}^{-{x}^{2}}/\left(x\sqrt{\pi }\right)$
• Shot-noise-limited system (Sec. 4.4.1 and handout): Emitted electrons may not be above the threshold even if the optical signal is above the threshold for "1" bit. Another source for error in thresholding is the dark current.
Optimum detection threshold relates to$\text{ }{k}_{T}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{n}_{s}/\text{ln}\left(1\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{n}_{s}/{n}_{n}\right)$ where$\text{ }{n}_{s}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{I}_{ph}\tau /q$ is average number of electrons generated by signal,$\text{ }{n}_{n}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{I}_{d}\tau /q$ is average number of electrons generated by dark current and$\text{ }\tau$ is the bit interval.
The optimum threshold is$\text{ }{k}_{d}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}round\left({k}_{T}\right)$

Handout for noise calculations for digital systems.

• Demultiplexers: Prism separate light with different wavelengths to different spatial locations with dispersion. A novel material called photonic crystal can have the same effect as a prism.
Diffraction gratings can diffract various wavelengths with different angles.

$\text{ }a\left(\text{sin}{\theta }_{i}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\text{sin}{\theta }_{d}\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}m\lambda$

and$\text{ }d{\theta }_{d}/d\lambda \text{\hspace{0.17em}}=\text{\hspace{0.17em}}m/\left(a\text{cos}{\theta }_{d}\right)$ where$\text{ }{\theta }_{i}$ is the incidence angle,$\text{ }{\theta }_{d}$ is diffraction angle and$\text{ }a$ is grating line spacing. (see Sec 3.3.1)
Arrayed waveguide grating (AWG) is similar to WGR, except that it is a Nx1 device.
Characteristics - polarization dep., temp. dep., flat spectral response, insertion loss -3dB and -35dB cross talk.
Operation is based on MZ interferometer.
Spectral filter can be design to transmit only 1 wavelength and reject others. (Fig. 3.20)
Acousto-optic filter plus polarizing beam splitter can be used to extract a certain wavelength.
• Multiplexers: Demultiplexers are bilateral$\text{ }\to$ they can function a multiplexers by reversing signal paths.