Biophotonics_Concepts_to_Applications

ID¼

n

2 l 0 c

EtotalEtotal¼

n

2 l 0 c

E^2 s0þE^2 r0þ2Es0Er0cos2ðkzrkzsÞ



ð 10 : 5 Þ

Here n is the refractive index of the transmission medium (e.g., n≈1.5 for
glass),μ 0 is the magnetic permeability, and the asterisk (*) denotes the complex
conjugate. Letting the intensity from the sample be

Is0¼

n

2 l 0 c

E^2 so ð 10 : 6 Þ

and letting the intensity from the reference mirror be

Ir0¼

n

2 l 0 c

E^2 ro ð 10 : 7 Þ

then the intensity at the detector can be expressed as

ID¼Is0þIr0þ 2

ffiffiffiffiffiffiffiffiffiffi

Is0Ir0

p

cos 2

2 p

k

ðzrzsÞ



ð 10 : 8 Þ

whereλ=2π/k is the wavelength of the optical source. The factor zr−zsis the
path length difference between the reference and the sample arms.
Alternatively, Eq. (10.8) can be written in terms of the optical power received at
the detector as

PD¼Ps0þPr0þ 2

ffiffiffiffiffiffiffiffiffiffiffiffi

Ps0Pr0

p

cos 2

2 p

k

ðzrzsÞ



ð 10 : 9 Þ

where Pr0and Ps0are the monochromatic powers from the reference and sample
arms, respectively. In Eqs. (10.8) and (10.9) thefirst two terms are constants for
particular reflection values, whereas the third term represents the wave interference.
When the factor in square brackets in the cosine term is 0 or a multiple of± 2 πthe
cosine value is a maximum of 1, thereby yielding a maximum optical power at the
detector.
For an actual OCT setup, the source is not monochromatic. In fact, as described
below, to increase the axial resolution it is desirable to have a coherent light source
with a wide spectral width. In this case the total optical power falling on the detector
is found by integrating over the entire spectrum of the light source, thereby yielding

PD¼PsþPrþ 2

ffiffiffiffiffiffiffiffiffi

PsPr

p

sinc

pðzrzsÞ

lc



cos 2

2 p

k

ðzrzsÞ



ð 10 : 10 Þ

Here Prand Psare the wideband spectral optical powers from the reference and
sample arms, respectively,λ 0 is the center wavelength of the optical source,lcis the
source coherence length, and thesinc functionis defined by sinc x = (sin x)/x.

10.1 Optical Coherence Tomography 297