Optical Coherence Tomography: Technical Aspects - Biomedical Optical Imaging Technologies: Design and Applications - page 165

E D .t / D E S .t L S =c/ C E R .t L R =c/; (5.2)

using the principle of superposition. By applying Eq. 5.1 and putting r and to 0

for the convenience, Eq. 5.2 can be written as

E D .t / D E S e i.tL S =c/ C E R e i.tL R =c/ :

(5.3)

The detector measures the intensity of the output beam, which is proportional to
the square of the electromagnetic field. The irradiance (power per unit area) at the
detector can be expressed in terms of the Poynting vector S D " 0 c 2 E D B D .The
irradiance measured by the detector is the time average of the rapidly varying fields
( 10 14 Hz for near-infrared region), that is, I D

Dhj S ji D " 0 c 2 hj E D B D ji ,since

D " 0 c 2

2

j E jD c j B j . The irradiance can be written as I D hj E D j i .UsingEq. 5.3 ,the

resultant intensity I D can be expressed as
I D D " 0 c D .E S .t / C E R .t C // .E S .t / C E R .t C // E : (5.4)

where D L=c is the time delay corresponding to the roundtrip optical path
length difference between the reference and sample arm, that is,

L D L S L R D 2n.l S l R /; (5.5)

where l S and l R are the geometric lengths of the sample arm and reference arms,
respectively, n Š 1 corresponds to the refractive index of air, and the bar symbol
denotes a complex conjugate operation. The sharp brackets denote the time average:

Z

T

h f i D lim

T !1

f.t/dt: (5.6)

0

Then by omitting the constant term " 0 c, the resulting intensity can be written as
follows:

D

C Re h E S .t / E R .t C / iE :

2 2

I D D j E S .t / j Cj E R .t C / j (5.7)

By assuming that the optical fields of both reference and sample b eams

have the same polarization states, the dot product term D

E

j E S .t / E R .t C / j D

D E . By introducing the correlation function (mutual coherence

j E S .t /E R .t C / j

D E S .t /E R .t C / E , the interference Eq. 5.7

function) of the two fields, SR . / D

can be written as

I D D I S C I R C 2ReΠSR . /; (5.8)

2 ˛ . For convenience, a normalized
correlation function is defined, which gives the degree to which the temporal

˝ j E S .t / j

2 ˛ and I S

˝ j E S .t C / j

where I S D D