**Rise Time Budget:**

- A rise-time budget analysis is a convenient method for determining the dispersion limitation of an optical fiber link, useful for digital systems.
- The total rise time $t_{sys}$ of the link is the root sum square of the rise times from each contribution t_i, to the pulse rise-time degradation. $$t_{sys} = \bigg(∑_{i=1}^Nt^2_i\bigg) ^{1/2}$$ The four basic elements that limit system speed are:

- Transmitter rise time $t_{tx}$
- Group-velocity dispersion (GVD) rise time $t_{GVD}$ of the fiber
- Modal dispersion rise time $t_{mod}$ of the fiber
- Receiver rise time $t_{tx}$

- Single-mode fibers do not experience modal dispersion.
- The transmitter rise time is attributable primarily to the light source and its drive circuitry.
- Receiver rise time results from the photodetector response and 3dB electrical bandwidth of the receiver front end.

To find $t_{tx}$:

The response of the receiver front end can be modeled by a first order lowpass filter having a step response.

$$g(t) = [1 - exp (-2π B_{rx t} u(t)]$$

$B_{rx} = 3dB$ electrical bandwidth of the receiver

U (t) = step function which is 1 for t >= 0 & 0 for t < 0

$t_{rx}$ = rise time of receiver

g(t) = 0.9 (10 to 90% rise time)

If $B_{rx}$ is given in MHz then $t_{rx}$ is in ns

$$t_{rx} = \frac{350}{B_{rx}}$$

To find $t_{GVD}:$

The fiber rise time $t_{GVD}$ resulting from group velocity dispersion over length L

$$t_{GVD} = |D|Lσ_λ$$

D = dispersion

$σ_λ$ = Half power band width of source

To find $t_{mod}:$

Empirical relation for bandwidth $B_M$ of link length L:

$$B_M (L) = \frac{B_0}{Lq}$$

Where 0.5 < q < 1

$B_0$ = Bandwidth of 1km length of fiber cable

- 3dB bandwidth is defined as modulation frequency $f_{3dB}$ at which received optical power has fallen to 0.5 of zero frequency value.

$$f_{3dB} = B_{3dB} = \frac{0.44}{t_{FWHM}}$$

- Letting $t_{FWHM}$ be the rise time resulting from modal dispersion.

$$t_{mod} = \frac{0.44}{B_M} = \frac{0.44 L^q}{B_0}$$

- If $t_{mod}$ is in ns & $B_M$ is in MHz

$$t_{mod} = \frac{440}{B_M} = \frac{440 L^q}{B_0}$$

- Total rise time of a fiber link is the root - sum - square of rise time of each contributor to the pulse rise time degradation.

$$t_{sys} = \sqrt{t_{tx}^2+t_{mod}^2+t_{GVD}^2+t_{rx}^2}\\ t_{sys} = \sqrt{t_{tx}^2+\bigg(\frac{440 L^q}{B_0}\bigg)^2+(DLσ_λ )^2+\bigg(\frac{350}{B_{rx}} \bigg)^2}$$

All the times are in nanoseconds.