Biophotonics_Concepts_to_Applications

external force on one or both of the plates varies due to changes in factors such as
pressure, temperature, or stress, the bend radius of the opticalfiber changes. The
result is that the optical power level transmitted through the opticalfiberfluctuates.
For measuring changesΔP in pressure, the operation of the sensor can be
expressed in terms of the change in the transmission coefficientΔt of the light
propagating through the bentfiber as

Dt¼

Dt

DX



Ap kfþ

AsYs

Ls

 1

DP

Dt

DX



Apkf^1 DP ð 7 : 4 Þ

HereΔX is the displacement of the deformer plates, (Δt/ΔX) is the device
sensitivity, Apis the plate area, and kfis the force constant or effective spring
constant of the bentfiber. The parameters As,Ys, and Lsare the cross-sectional
area, Young’s modulus, and the thickness, respectively, of the deformer spacers.
The approximation on the right-hand side of Eq. (7.4) is valid for the design
condition AsYs/Ls≪kf. The effective spring constant kfcan be expressed as

kf^1 ¼

K^3

3 pYd^4 g

ð 7 : 5 Þ

whereΛis the spacing of the deformers, Y is the effective Young’s modulus, d is
the diameter of thefiber, andηis the number of deformation intervals.
Similarly, for changesΔT in temperature, the operation of the sensor can be
expressed as

Dt¼

Dt

DX



AsasYs kfþ

AsYs

Ls

 1

DT

Dt

DX



asLsDT ð 7 : 6 Þ

Hereαsis the thermal expansion coefficient of the spacers and the approximation
on the right-hand side of Eq. (7.5) is valid for the design condition kfLs≪AsYs.

Example 7.9 Consider a microbending pressure sensor that has the fol-

lowing characteristics: Ap=1cm^2 and kf^1 =33× 10 −^8 cm/dyn. If the min-

imum measurable displacement isΔXmin=10−^10 cm, what is the minimum

detectable pressureΔPmin?

Solution:From Eq. (7.4), the minimum detectable pressureΔPminis

DPmin¼

DXmin

Apkf^1

¼

10 ^10 cm

ðÞ1cm^2 ðÞ 33  10 ^8 cm=dyn

¼ 3  10 ^4 dyn=cm^2

¼ 3  10 ^5 Pa (pascals)

7.4 Optical Sensors 215