Absorbed heat must be transferred from the DMD mirror to the underlying silicon. This is true in a CW application as well as a pulsed application. Normally the temperature rise of the bulk mirror above the silicon in CW applications is small, but in pulsed applications, the peak optical power can be much higher than the average power. This results in a rise and fall of the bulk mirror temperature with each pulse. The bulk mirror temperature rise is defined by Equation 3.

Where:

$T\left(t\right)$ = mirror temperature at time = t

$T_f$ = final mirror temperature at t = ∞ (steady-state)

$T_i$ = initial mirror temperature at t = 0

$\tau$ = thermal time constant of the mirror (R × C) [Table 2-1]

$T_f=T_i+Q_{MIRROR}\times R_{MIRROR-TO-SILICON}$ [Table 2-1]

$Q_{MIRROR}=Q_{INCIDENT\_MIRROR}\times [{FF}_{MIRROR}\times (1-MR\left)\right]$

Where:

$Q_{INCIDENT\_MIRROR}$ = total incident power per mirror [incident power density × (mirror pitch)²]

${FF}_{MIRROR}$ = fill factor of mirror array (on-state calculates highest temperature) [Table 2-2]

$MR$ = mirror reflectivity [Figure 2-1, Figure 2-2, Figure 2-3]

The thermal time constant of the mirror, $\tau$ , is defined as R_MIRROR ✕ C_MIRROR

Where:

R_MIRROR is the thermal resistance from the mirror to the silicon

C_MIRROR is the thermal capacitance of the mirror

In Table 2-1, R was calculated using a finite element model of the pixel superstructure and distance between the mirror and the silicon. C was calculated as ρVC_p where;

ρ = density of the aluminum mirror

V = volume of the aluminum mirror

C_p = specific heat of the aluminum mirror

Therefore, $\tau$ is different for each mirror pitch

There are several possibilities of the resulting transient response of the mirror depending on the duration of t_pulse and t_off relative to the thermal time constant of the mirror. These possibilities are shown in Table 2-3.