DATOS GENERALES:

The reflection and transmission coefficients (or Fresnel coefficients) [1, 2, 4] provide information on the amplitude and phase of the reflected and transmitted waves relative to the incident wave's amplitude and phase. There are two reflection and two transmission coefficients (one for each plane of polarisation) and are simply stated here (a brief derivation can be found in appendix 1). The coefficients for a TE polarised wave are:

:TE =

y TE

_ P2 ^ 1 C0S8j -|Ltik2 COS0t

jijki COS + Pikj cos 0 t

(4.4)

tjE —" it " v E i,

2 p2k, cos8 j

TE p^ki cos8 j +|Xik2 cos8 (

(4.5)

where, Ej, Ej. and E^ are the complex amplitudes of the incident, reflected and tramsmitted wave electric field strengths respectively. The coefficients for a TM polarised wave are:

^TM -

V ^ i / T M

_

Optics o f multilayer thin film systems___________________________________ Chapter 4

where, Hj, Hj. and are the complex amplitudes of the incident, reflected and transmitted wave magnetic field strengths respectively.

Using the relation (see appendix 1):

the ratios of the reflected and transmitted electric field strengths to the incident electric field strength can be obtained for a TM polarised wave:

V^i 7tM TM Et v E i ,_TM

= ^ t ™

(4.9)

4.2.3

Total internal reflection

It is clear from equation 4.2 that when a plane wave in medium 1 is incident on an interface with an optically less dense medium 2, (as in fig. 4.1) there exists a range of incident angles such that:

sine, = ^ s i n e i > l

(4.10)

^ 2

The angle of incidence (GJ at which equation 4.10 becomes true is known as the critical angle. At angles of incidence greater than the critical angle, the angle 0^ becomes complex and |rTE| = kTM| = l» ie. all incident light energy is reflected back into medium 1 (total internal reflection).

Optics o f multilayer thin film systems Chapter 4

Although all the light energy is reflected, the electromagnetic field in medium 2 does not disappear and it can be shown (appendix 1) that the field decays exponentially

as the perpendicular distance from the interface increases (see fig. 4.2) [1 ,2 ]. The wave in medium 2 is described as an evanescent wave and it is also non-uniform - ie. the

planes of constant phase and constant amplitude are not co-incident. In this case, the planes o f constant phase are perpendicular to the interface and the planes o f constant amplitude are parallel to the interface. An important figure when considering evanescent field optical biosensors is the evanescent field penetration depth - the distance from the interface at which the field strength o f the evanescent wave in medium 2 falls to 1/e of its value at the interface (fig. 4.2). The penetration depth is given by [1, 2 ]: ^pen - k ,^ s in ^ e i- ( e2/e ,) (4.11)

I

I

Distance from interface

Fig, 4.2 The evanescent field o f a totally internally reflected plane wave. The penetration depth is given by dpen.

Optics o f multilayer thin film systems Chapter 4

When designing an evanescent field optical biosensor, it is important to minimise the penetration depth and maximise the transmitted field strength. Equation 4.11 indicates that to minimise the penetration depth, it is required that the ratio 6 / 6 2 is

large and that the angle of incidence is also large.

It can be shown [2] that, for a TE wave, the transmitted electric field vector remains polarised perpendicular to the plane of incidence and the magnetic field vector rotates in the plane of incidence with its axis of rotation parallel to the electric field vector (fig. 4.3). The converse is true for a TM wave. This phenomenon can be useful in an evanescent field biosensor because the absorption o f many fluorescent molecules (when immobilised) is highly dependent on the polarisation o f the exciting light. Thus, because electron transition is mainly dependent on the electric field strength, it is useful to use TM polarised exciting light as the rotating electric vector in the evanescent field will excite a greater number of fluorescent molecules bound to the tagged antibodies. In a simple total internal reflection sensor, it would be possible to use unpolarised light

Medium 2 Medium 1

E

Fig. 4.3 Total internal reflection o f a TE polarised plane wave. The transmitted wave remains TE polarised but the magnetic field vector (H) rotates with respect to time in the plane o f incidence. The transmitted k vector breaks down to a real x- component (p) and an imaginary z-component (a) indicating that there is a time averaged energy flow in the x-direction but not in the z-direction.____________________

Optics o f multilayer thin film systems____________________________________Chapter 4

but, in the resonant multilayer thin film devices relevent to this thesis, this is not possible as the resonance is dependent on the polarisation of the incident light.

4.2.4

Power flow across the interface

It is useful to know the power that is transferred from medium 1 into medium 2 and to calculate this it is required to know the Foynting vectors [1, 2 ] for the incident,

reflected and transmitted waves, and to calculate the power flow perpendicular to the interface. The ratios of the reflected and transmitted power flowing normal to the interface relative to the incident normal power flow are given, respectively, for a TE polarised wave by (see appendix 1):

9ÎTC=|rTEf (4.12)

_ H ik;C os9,| , 2

p^k|Cos8 ,

Similarly, for a TM wave:

^TM — kyMl (4-14)

q — ^1 ^ 2 COSQt L 2 14 15)

82k l C 0 S 8 ;

91 and 3 are known as the reflectivity and the transmittivity respectively. The law of