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70 2 Extremely Short-External-Cavity Laser Diode
Center plane of
each layer

d

r1
r2 P 1
r3
P 2
P 3

Fig. 2.47. Deflection of a bimorph MC and internal stress due to temperature
change

3
where M i = E i I i /r i (I i = bt /12) is the moment of inertia of i th layer, h i
i
is the distance between the center plane of the MC and that of the i th layer
and r i is the radius of curvature of the i th layer of the MC, and h 1 + h 2 =
(t 1 + t 2 )/2, −h 2 + h 3 =(t 2 + t 3 )/2,h 1 + h 3 =(t 1 +2t 2 + t 3 )/2.
At the interface between the two layers, the normal strain of the materials
must be the same. Therefore
P 1 t 1 P 2 t 2
α 1 ∆T − − = α 2 ∆T − + , (2.31)
bE 1 t 1 2r 1 bE 2 t 2 2r 2
P 2 t 2 P 3 t 3
α 2 ∆T − − = α 3 ∆T + + . (2.32)
bE 2 t 2 2r 2 bE 3 t 3 2r 3
Here, r 1 = r 2 = r 3 = r (very thin compared to length) and we derive the
curvature k =1/r by eliminating P 1 ,P 2 ,P 3 from (2.29) to (2.32). Note that
the deflection d at the free end of the MC from the curvature k is [2.30]
kl 2
d = (2.33)
2
for l r.
Finally, the tip deflection of the MC by thermal strain due to the mismatch
between the thermal coefficient of the expansion is:
A
d = , (2.34)
B
where
2
A =3∆Tl [E 1 E 2 t 1 t 2 (α 1 − α 2 )(t 1 + t 2 )+ E 2 E 3 t 2 t 3 (α 2 − α 3 )(t 2 + t 3 )
+E 1 E 3 t 1 t 3 (α 1 − α 3 )(t 1 +2t 2 + t 3 )]
2
2
2
2
B =2E 1 E 2 t 1 t 2 (2t +3t 1 t 2 +2t )+2E 2 E 3 t 2 t 3 (2t +3t 2 t 3 +2t )
2
2
3
1
2 4
2 4
2 4
2
2
2
+2E 1 E 3 t 1 t 3 (2t +6t +2t +6t 1 t 2 +6t 2 t 3 +3t 1 t 3 )+ E t +E t +E t .
3
3 3
2 2
1 1
1
2