3 Sample Calculations

Below are detailed sample calculations that show how mirror surface temperature is calculated using the three temperature rises:

1. Mirror Surface to Bulk Mirror Delta
2. Bulk Mirror to Silicon Delta
3. Silicon to Ceramic Delta

In the first example calculation, t_off is greater than $5\tau$ , therefore the bulk mirror fully cools after each pulse and only one pulse needs to be analyzed.

Example 1

A pulsed laser illuminates a DLP650LNIR DMD with 1064 nm wavelength light and fills the active array with no overfill.

Pulse duration (t_pulse) = 1 $\mu s$

t_off = 999 $\mu s$ (pulsed repetition rate of the optical source is 1 kHz)

Peak incident power during the pulse is 25 kW/cm²

Calculation of the temperature rise of the mirror surface above the DMD ceramic temperature:

1. Mirror Surface to Bulk Mirror Delta: ΔT_{MIRROR_SURFACE-TO-BULK_MIRROR}

   $T\left(t\right)=2q*\frac{{\left(\frac{\alpha t}{\pi }\right)}^{\frac{1}{2}}}{k}+T_i$

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

   $T_i$ = initial mirror temperature

   q = absorbed heat flux on mirror surface [W/m²]

   α = Mirror thermal diffusivity = 6.4667 x 10^-5 m²/s

   k = Mirror thermal conductivity = 160 W/m-ºC

   q = 25 kW/cm² × (1 – 0.94) = 1.50 kW/cm²

   t_pulse = 1 $\mu s$

   T(1 $\mu s$ ) = 2 × 1.50 kW/cm² × {[(6.4667e-5 m²/s × 1.0e-6 s)/π] ^$1/2$ /160 W/m-ºC} + 0 = 0.85 ºC

   Figure 3-1 DMD Mirror Surface Temperature Rise at 25 kW/cm²
2. Bulk Mirror to Silicon Delta: ΔT_{BULK_MIRROR-TO-SILICON}

   $T\left(t\right)=T_f+(T_i-T_f)e^{\left(\frac{-t}{\tau }\right)}$

   $T_f=T_i+Q_{MIRROR}\times R_{MIRROR-TO-SILICON}$

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

   $Q_{INCIDENT\_MIRROR}$ = 25 kW/cm² × (10.8 μm)² = 0.02916 W

   ${FF}_{MIRROR}$ = 0.931 (on-state)

   $MR$ at 1064 nm = 0.94

   $Q_{MIRROR}$ = 0.02916 W × [0.931 × (1 – 0.94)] = 1.629 mW

   $R_{SILICON-TO-CERAMIC}$ = 3.39 x 10⁵ ºC/W

   $T_f$ = 0 + 1.629 mW × (3.39 x 10⁵ ºC/W) = 552.23 ºC

   $\tau$ = 32.27 $\mu s$

   t_off = 999 $\mu s$

   t_pulse = 1 $\mu s$

   $5\tau$ = 5 × 32.27 $\mu s$ = 161.35 $\mu s$

   Since t_off = 999 $\mu s$ (> $5\tau$ ) the bulk mirror fully cools to the initial temperature T_i and analyzing a single pulse cycle is sufficient.

   T(1 $\mu s$ ) = 552.23 ºC + (0 – 552.23 ºC)e^{-(1 μs/32.27 μs)} = 16.85 ºC

   Figure 3-2 DMD Bulk Mirror Temperature Rise 25 kW/cm²
3. Silicon to Ceramic Delta: ΔT_{SILICON-TO-CERAMIC}

   From the DLP650LNIR data sheet:

   ${∆T}_{SILICON-TO-CERAMIC}= Q_{SILICON}\times R_{SILICON-TO-CERAMIC}$

   $R_{SILICON-TO-CERAMIC}$ = 0.5 ºC/W

   $Q_{ELECTRICAL}$ = 1.8 W

   $Q_{SILICON}= Q_{ELECTRICAL}+ Q_{ILLUMINATION}$

   ${FF}_{MIRROR}$ = 0.726 (off-state)

   $MR$ at 1064 nm = 0.94

   ${\alpha }_{Window}$ at 1064 nm = 0.007 (per pass)

   $Overfill$ = 0

   ${\propto }_{DMD}=\left(1-Overfill\right) * \{\left[{FF}_{MIRROR} * (1-MR)] + \left(1-{FF}_{MIRROR}\right)\right\} + \left(2 *{\propto }_{WINDOW}\right)+Overfill$

   ${\alpha }_{DMD}$ = (1 - 0) × {[0.726 × (1-0.94)] + [1-0.726]} + (2 × 0.007) + 0

   ${\alpha }_{DMD}$ = 0.33 (off-state)

   $Q_{INCIDENT}$ = total incident average optical power to DMD

   Active array area = (1280 × 10.8 μm) × (800 × 10.8 μm) = 1.1944 cm²

   t_pulse = 1 $\mu s$ , t_off = 999 $\mu s$

   Therefore pulse duty cycle = 1 $\mu s$ / ( 1 $\mu s$ + 999 $\mu s$ ) x 100% = 0.1%

   Average optical power density = (25 kW/cm²) × 0.1% = 25 W/cm²

   Average optical power = 25 W/cm² × 1.1944 cm² = 29.86 W

   Average absorbed optical power = 29.86 W × 0.33 = 9.85 W

   $∆T_{SILICON-TO-CERAMIC}$ = (1.8 W + 9.85 W) × 0.5 ºC/W = 5.8 ºC

   Mirror Surface to Ceramic Delta: (ΔT_{MIRROR_SURFACE-TO-CERAMIC})

   T_{MIRROR SURFACE} - T_CERAMIC = ΔT_{SILICON-TO-CERAMIC} + ΔT_{BULK_MIRROR-TO-SILICON} + ΔT_{MIRROR_SURFACE-TO-BULK_MIRROR}
   T_{MIRROR_SURFACE} - T_CERAMIC = 5.8 ºC + 16.85 ºC + 0.85 ºC = 23.5 ºC

In the second example calculation, since t_pulse and t_off are both < $5\tau$ , analysis is required over several pulses until the temperature rise stabilizes.

Example 2

A pulsed laser illuminates a DLP650LNIR DMD with 1064 nm wavelength light and fills the active array with no overfill.

Pulse duration (t_pulse) = 10 ps

t_off = 99.99999 $\mu s$ (pulsed repetition rate of the optical source is 10 kHz)

Peak incident power during the pulse is 250 MW/cm²

Calculation of the temperature rise of the mirror surface above the DMD ceramic temperature:

1. Mirror Surface to Bulk Mirror Delta: ΔT_{MIRROR_SURFACE-TO-BULK_MIRROR}

   $T\left(t\right)=2q*\frac{{\left(\frac{\alpha t}{\pi }\right)}^{\frac{1}{2}}}{k}+T_i$

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

   $T_i$ = initial mirror temperature

   q = absorbed heat flux on mirror surface [W/m²]

   α = Mirror thermal diffusivity = 6.4667 x 10^-5 m²/s

   k = Mirror thermal conductivity = 160 W/m-ºC

   q = 250 MW/cm² × (1 – 0.94) = 15 MW/cm²

   t_pulse = 10 ps

   T(10 ps) = 2 × 15 MW/cm² × {[(6.4667e-5 m²/s × 1.0e-11 s)/π] ^$1/2$ /160 W/m-ºC} + 0 = 26.9 ºC

   Figure 3-3 DMD Mirror Surface Temperature Rise at 250 MW/cm²
2. Bulk Mirror to Silicon Delta: ΔT_{BULK_MIRROR-TO-SILICON}

   $T\left(t\right)=T_f+(T_i-T_f)e^{\left(\frac{-t}{\tau }\right)}$

   $T_f=T_i+Q_{MIRROR}\times R_{MIRROR-TO-SILICON}$

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

   $Q_{INCIDENT\_MIRROR}$ = 250 MW/cm² × (10.8 $\mu$ m)² = 291.6 W

   ${FF}_{MIRROR}$ = 0.931 (on-state)

   $MR$ at 1064 nm = 0.94

   $Q_{MIRROR}$ = 291.6 W × [0.931 × (1 – 0.94)] = 16.289 W

   $R_{SILICON-TO-CERAMIC}$ = 3.39 x 10⁵ ºC/W

   $T_f$ = 0 + 16.289 W × (3.39 x 10⁵ ºC/W) = 5.522 x 10⁶ ºC

   $\tau$ = 32.27 $\mu s$

   t_off = 99.99999 $\mu s$

   t_pulse = 10 ps

   $5\tau$ = 5 × 32.27 $\mu s$ = 161.35 $\mu s$

   Bulk mirror only partially heats and partially cools between pulses and we need to iterate and analyze a series of pulses until the mirror temperature no longer changes.

   1st t_pulse heating:

   T(10 ps) = 5.522 x 10⁶ ºC + (0 – 5.522 x 10⁶ ºC)e^{-(10 ps/32.27 μs)} = 1.710 ºC

   1st t_off cooling:

   T(99.99999 μs) = 0 ºC + (1.720 – 0 ºC)e^{-(99.99999 μs/32.27 μs)} = 0.077 ºC

   2nd t_pulse heating:

   T(10 ps) = 5.522 x 10⁶ ºC + (0.077 ºC – 5.522 x 10⁶ ºC)e^{-(10 ps/32.27 μs)} = 1.787 ºC

   2nd t_off cooling:

   T(99.99999 μs) = 0 ºC + (1.787 – 0 ºC)e^{-(99.99999 μs/32.27 μs)} = 0.081 ºC

   3rd t_pulse heating:

   T(10 ps) = 5.522 x 10⁶ ºC + (0.081 ºC – 5.522 x 10⁶ ºC)e^{-(10 ps/32.27 μs)} = 1.791 ºC

   3rd t_off cooling:

   T(99.99999 μs) = 0 ºC + (1.791 – 0 ºC)e^{-(99.99999 μs/32.27 μs)} = 0.081 ºC

   4th t_pulse heating:

   T(10 ps) = 5.522 x 10⁶ ºC + (0.081 ºC – 5.522 x 10⁶ ºC)e^{-(10 ps/32.27 μs)} = 1.791 ºC

   4th t_off cooling:

   T(99.99999 μs) = 0 ºC + (1.791 – 0 ºC)e^{-(99.99999 μs/32.27 μs)} = 0.081 ºC

   Notice the temperature did not change from the 3^rd pulse to the 4^th pulse. When the temperature after each pulse iteration stops changing, it has reached steady-state. The bulk mirror temperature rise above the silicon is 1.8 ºC.

   Figure 3-4 DMD Bulk Mirror Temperature Rise at 250 MW/cm²
3. DMD silicon temperature rise above ceramic: ΔT_{SILICON-TO-CERAMIC}

   From the DLP650LNIR data sheet:

   ${∆T}_{SILICON-TO-CERAMIC}= Q_{SILICON}\times R_{SILICON-TO-CERAMIC}$

   $R_{SILICON-TO-CERAMIC}$ = 0.5 ºC/W

   $Q_{ELECTRICAL}$ = 1.8 W

   $Q_{SILICON}= Q_{ELECTRICAL}+ Q_{ILLUMINATION}$

   ${FF}_{MIRROR}$ = 0.726 (off-state)

   $MR$ at 1064 nm = 0.94

   ${\alpha }_{Window}$ at 1064 nm = 0.007 (per pass)

   $Overfill$ = 0

   ${\propto }_{DMD}=\left(1-Overfill\right) * \{\left[{FF}_{MIRROR} * (1-MR)] + \left(1-{FF}_{MIRROR}\right)\right\} + \left(2 *{\propto }_{WINDOW}\right)+Overfill$

   ${\alpha }_{DMD}$ = (1 - 0) × {[0.726 × (1-0.94)] + [1-0.726]} + (2 × 0.007) + 0

   ${\alpha }_{DMD}$ = 0.33 (off-state)

   $Q_{INCIDENT}$ = total incident average optical power to DMD

   Active array area = (1280 × 10.8 μm) × (800 × 10.8 μm) = 1.1944 cm²

   t_pulse = 10 ps, t_off = 99.99999 $\mu s$

   Therefore pulse duty cycle = 10 ps / ( 10 ps + 99.99999 $\mu s$ ) x 100% = 0.00001%

   Average optical power density = (250 MW/cm²) × 0.00001% = 25 W/cm²

   Average optical power = 25 W/cm² × 1.1944 cm² = 29.86 W

   Average absorbed optical power = 29.86 W × 0.33 = 9.85 W

   $∆T_{SILICON-TO-CERAMIC}$ = (1.8 W + 9.85 W) × 0.5 ºC/W = 5.8 ºC

   Mirror Surface to Ceramic Delta: (ΔT_{MIRROR_SURFACE-TO-CERAMIC})

   T_{MIRROR SURFACE} - T_CERAMIC = ΔT_{SILICON-TO-CERAMIC} + ΔT_{BULK_MIRROR-TO-SILICON} + ΔT_{MIRROR_SURFACE-TO-BULK_MIRROR}
   T_{MIRROR_SURFACE} - T_CERAMIC = 5.8 ºC + 1.8 ºC + 26.9 ºC = 34.5 ºC

   In the second example, the mirror surface temperature rise above the bulk mirror is much larger than the bulk mirror temperature rise above the silicon. This is common with very short pulses of high intensity. Notice the average power is the same between examples 1 and 2 (29.86 W) and therefore the temperature rise of the silicon above the ceramic is equal in the two examples.