13 Optical Properties, Lecture-2

K Asokan

Learning Objectives:

 

From this module students may get to know about the following:

  1. Relation between amplitude reflection and Transmission coefficients
  2. Relation between Intensity, Reflectance, and Transmittance

FRESNEL’S EQUATIONS

 

Amplitude Reflection and Transmission Coefficients:

 

Although the ray picture with constant phase wave fronts is useful in understanding refraction and reflection, to obtain the magnitude of the reflected and refracted waves and their relative phases, we need to consider the electric field in the light wave. The electric field in the wave must be perpendicular to the direction of propagation as shown in Figure 1. We can resolve the field of the incident wave into two components, one in the plane of incidence   ∥ and the other perpendicular to the plane of incidence   ,⊥. The plane of incidence is defined as the plane containing the incident and the reflected rays which in Figure 1 corresponds to the plane of the paper. Similarly for both the reflected and transmitted waves, we will have field components parallel and perpendicular to the plane of incidence, i.e.,   ∥,   ⊥ and   ∥,   ⊥.

 

As apparent from Figure 1, the incident, transmitted, and reflected waves all have a wave vector component along the z direction; that is, they have an effective velocity along z. The fields   ,⊥,   ,∥and   ,⊥are all perpendicular to the z direction. These waves are called transverse electric field (TE) waves. On the other hand, waves with   ,∥,   ,∥ and   ,∥ only have their magnetic field components perpendicular to the z direction and these are called transverse magnetic field (TM) waves.

 

We  will   describe   the  incident,  reflected,   and  refracted  waves  by   the  exponential

 

representation of a traveling wave, i.e.,
=    0 exp  (     −  .   ) (1)
=    0 exp  (     −  .   ) (2)
=    0 exp  (     −  .   ) (3)

 

Where r is the position vector; the wave vectors , and describe, respectively, the directions of the incident, reflected, and transmitted waves; and   0,   0and are the respective amplitudes. Any phase changes such as and in the reflected and transmitted waves with  respect to the phase of the incident wave are incorporated into the complex amplitudes   0 and   0. Our objective is to find and with respect to .

 

 

We should note that similar equations can be stated for the magnetic field components in the incident, reflected, and transmitted waves, but these will be perpendicular to the corresponding electric fields. The electric and magnetic fields anywhere on the wave must be perpendicular to each other as a requirement of electromagnetic wave theory. This means that with ∥ in the EM wave we have a magnetic field ⊥ associated with it such that ⊥ = ∥ , Similarly ⊥ will have a magnetic field ∥ associated with it such that ∥ = ( ) ⊥.

 

There are two useful fundamental rules in electromagnetism that govern the behavior of the electric and magnetic fields at a boundary between two dielectric media which we can arbitrarily label as 1 and 2. These rules are called boundary conditions. The first states that the electric field that is tangential to the boundary surface must be continuous across the boundary from medium 1 to 2, i.e., at the boundary y = 0 in Figure 1,

The plane of incidence is the plane of the paper and is perpendicular to the flat interface between the two media. The electric field is normal to the direction of propagation. It can be resolved into perpendicular (⊥) and parallel (∥) components.

(1) =(2) (4)

The second rule is that the tangential component of the magnetic field to the boundary must be likewise continuous from medium 1 to 2 provided that the two media are nonmagnetic (relative permeability = 1),

(1) =(2) (5)

 

Using these boundary conditions for the fields at y = 0, and the relationship between the electric and magnetic fields, we can find the reflected and transmitted waves in terms of the incident wave. The boundary conditions can only be satisfied if the reflection and incidence angles are equal, = , and the angles for the transmitted and incident waves obey Snell’s law,

1 sin                =  2 sin                  .

 

Applying the boundary conditions to the EM wave going from medium 1 to 2, the amplitudes of the reflected and transmitted waves can be readily obtained in terms of 1, 2 and the incidence angle alone. These relationships are called Fresnel’s equations. If we define n = 2, as the relative refractive index of medium 2 to that of 1, then the reflection and transmission 1 coefficients for    ⊥ are

There are corresponding coefficients for the ∥ fields with corresponding reflection and transmission coefficients ∥ and ∥: Further, the reflection and transmission coefficients are related by

∥ +  ∥ = 1⊥ + 1 = ⊥ (10)

 

The significance of these equations is that they allow the amplitudes and phases of the reflected and transmitted waves to be determined from the coefficients ⊥, ∥, ∥ and ⊥. For convenience we take to be a real number so that the phase angles of ⊥ and ⊥ correspond to the phase changes measured with respect to the incident wave. For example, if ⊥ is a complex quantity, then we can write this as ⊥ = | ⊥|   (− ∅⊥) where | ⊥| and ∅⊥ represent the relative amplitude and phase of the reflected wave with respect to the incident wave for the field perpendicular to the plane of incidence. Of course, when| ⊥| is a real quantity, then a positive number represents no phase shift and a negative number is a phase shift of 180° (or ). As with all waves, a negative sign corresponds to a 180° phase shift. Complex coefficients can only be obtained from Fresnel’s equations if the terms under the square roots become negative, and this can only happen when < 1 (    1 > 2), and also when > , the critical angle. Thus, phase changes other than 0 or 180° occur only when there is total internal reflection.

 

Figure 2a shows how the magnitudes of the reflection coefficients| ⊥| and |  ∥| vary with the incidence angle for a light wave traveling from a more dense medium, 1 = 1.44, to a less dense medium, 2 = 1.00, as predicted by Fresnel’s equations. Figure 2b shows the changes in the phase of the reflected wave,∅⊥ and∅∥, with ; the critical angle as determined from

sin             = 2 in this case is 44°. It is clear that for incidence close to normal (small  ), there is no

1

 

phase change in the reflected wave. For example, putting normal incidence ( = 0) into Fresnel’s equations, we find

=   = 1 − 2 (11)

Internal reflection.

 

(a) Magnitude of the reflection coefficients ∥ and ⊥ versus the angle of incidence , for 1 = 1.44 and 2 = 1.00. The critical angle is 44°.

(b) The corresponding phase changes ∅∥and ∅⊥versus incidence angle.

This is a positive quantity for 1 > 2 which means that the reflected wave suffers no phase change. This is confirmed by ∅⊥ and ∅∥ in Figure 2b. As the incidence angle increases, eventually ∥ becomes zero at an angle of about 35°. We can find this special incidence angle, labeled as , by solving the Fresnel equation, Equation 2 , for ∥ = 0 =0. The field in the reflected wave is then always perpendicular to the plane of incidence and hence well-defined. This special angle is called the polarization angle or Brewster’s angle and from Equation 8 is given by

 

The reflected wave is then said to be linearly polarized because it contains electric field oscillations that are contained within a well-defined plane which is perpendicular to the plane of incidence and also to the direction of propagation. Electric field oscillations in un polarized light, on the other hand, can be in any one of an infinite number of directions that are perpendicular to the direction of propagation. In linearly polarized light, however, the field oscillations arecontained within a well-defined plane. Light emitted from many light sources such as a tungsten light bulb or an LED diode is un polarized and the field is randomly oriented in a direction that is perpendicular to the direction of propagation.

 

For incidence angles greater than but smaller than , Fresnel’s equation, Equation 8, gives a negative number for ∥ which indicates a phase shift of 180° as shown in ∅∥ in Figure 2b. The magnitudes of both ∥ and ⊥ increase with as apparent in Figure 2a. At the critical angle and beyond (past 44°), i.e., when > , the magnitudes of both ∥ and ⊥ go to unity, so the reflected wave has the same amplitude as the incident wave. The incident wave has suffered total internal reflection (TIR). When > , in the presence of TIR, the Equations 6 to 9 are complex quantities because then sin > and the terms under the square roots become negative. The reflection coefficients become complex quantities of the type ⊥ = 1. exp(− ∅⊥) and ∥ = 1. exp(− ∅∥) with the phase angles ∅⊥and ∅∥ being other than 0 or 180°. The reflected wave therefore suffers phase changes ∅⊥ and ∅∥ in the components ⊥ and ∥. These phase changes depend on the incidence angle, as apparent in Figure 2b, and on 1and 2.

 

The reflection coefficients in Figure 3 considered the case in which 1 > 2. When light approaches the boundary from the higher index side, that is, 1 > 2, the reflection is said to be internal reflection and at normal incidence there is no phase change. On the other hand, if light approaches the boundary from the lower index side, that is, 1 < 2then it is called external reflection. Thus in external reflection light becomes reflected by the surface of an optically denser (higher refractive index) medium. There is an important difference between the two. Figure 3 shows how the reflection coefficients ⊥ and ∥ depend on the incidence angle for external reflection ( 1 = 1 and 2 = 1.44). At normal incidence, both coefficients are negative, which means that in external reflection at normal incidence there is a phase shift of 180°. Further, ∥ goes through zero at the Brewster angle given by Equation 12. At this angle of incidence, the reflected wave is polarized in the ⊥ component only. Transmitted light in both internal reflection (when < ) and external reflection does not experience a phase shift.

 

Figure4: When > for a plane wave that is reflected, there is an evanescent wave at the boundary propagating along z.

 

What happens to the transmitted wave when > ? According to the boundary conditions, there must still be an electric field in medium 2; otherwise, the boundary conditions cannot be satisfied. When > , the field in medium 2 is a wave that travels near the surface of the boundary along the z direction as depicted in Figure 4. The wave is called an evanescent wave and advances along z with its field decreasing as we move into medium 2, i.e.,

( ,  ,   ) ∝  −∝2 exp  (     −    ) (13)
  ,⊥

 

Where = sin the wave vector of the incident is wave along the z axis, and ∝2 is an attenuation coefficient for the electric field penetrating into medium 2,

2 2 1 2
∝ = [( ) sin2   − 1] (14)
2 2

 

Where is the free-space wavelength. According to Equation 13, the evanescent wave travels along z and has an amplitude that decays exponentially as we move from the boundary into medium 2 (along y) as shown in Figure 1b. The field of the evanescent wave is e-1 in

medium 2 when       = 1 which is called the penetration depth. It is not difficult to show that the evanescent wave is correctly predicted by Snell’s law when > . The evanescent wave propagates along the boundary (along z) with the same speed as the z component velocity of the incident and reflected waves. In past we had assumed that the incident and reflected waves were plane waves, that is, of infinite extent. If we were to extend the plane wavefronts on the reflected wave, these would cut the boundary as shown in Figure 4. The evanescent wave traveling along z can be thought of as arising from these plane wavefronts at the boundary as in Figure 4. (The evanescent wave is important in light propagation in optical waveguides such as in optical fibers.) If the incident wave is a narrow beam of light {e.g., from a laser pointer), then the reflected beam would have the same cross section. There would still be an evanescent wave at the boundary, but it would exist only within the cross-sectional area of the reflected beam at the boundary.

 

Intensity, Reflectance, and Transmittance:

 

It is frequently necessary to calculate the intensity or irradiance of the reflected and transmitted waves when light traveling in a medium of index 1 is incident at a boundary where the refractive index changes to 2. In some cases we are simply interested in normal incidence where = 0 . For example, in laser diodes light is reflected from the ends of an optical cavity where there is a change in the refractive index.

 

Reflectance R measures the intensity of the reflected light with respect to that of the incident light and can be defined separately for electric field components parallel and perpendicular to the plane of incidence. The reflectances ⊥ and ∥ are defined by

= |   0,⊥|2 = (   )2 and = |   0,∥|2 = (   )2 (15)
2 2
⊥ ⊥ ∥ |   0,∥| ∥
|   0,⊥|
From Equations 9.37 to 9.40 with normal incidence, these are simply given by
− 2
= = = ( 1 2 ) (16)
⊥ ∥ 1+ 2

 

Since a glass medium has a refractive index of around 1.5, this means that typically 4 percent of the incident radiation on an air-glass surface will be reflected back.

 

Transmittance T relates the intensity of the transmitted wave to that of the incident wave in a similar fashion to the reflectance. We must, however, consider that the transmitted wave is in a different medium and further its direction with respect to the boundary is also different by virtue of refraction. For normal incidence, the incident and transmitted beams are normal and the transmittances are defined and given by

=  ⊥ = ∥ = 4 1  2 (17)

Further, the fraction of light reflected and fraction transmitted must add to unity. Thus R + T= 1.

 

Summary:

 

References:

 

  1. A.J. Dekker (1957). Solid state physics, Prentce-Hall, Inc .
  2. Leonid A. Azaroff (1960), Introduction to solids, McGRAW-HILL BOOK COMPANY INC.
  3.    3 Kasap, S. (2006). Principles of electronic materials and devices (3rd ed.). Boston: McGraw-Hill
  4. Frederick Wooten (1972) ACADEMIC PRESS INC
  5. Harald Ibach, Hans Luth (2009) Springer series on material science
  6. Ashcroft & Mermin (1976) Harcourt College Publisher.
  7. Philip Hofmann (2008) , Solid State Physics An Introduction, WILEY-VCH Verlag GmbH & KGaA.

References and Suggestive Readings

 

  1. James Patterson, Bernard Bailey (2010) Solid State Physics, Second Edition , Springier series on material science.
  2. S.L. Kakani (2004), Material Science , New age international (P) Limited, publishers
  3. Harald lbach & Hans Luth (1995) , Solid state physics- An introduction to material science , Springier
  4. Donald A. Neamen (2003), Semiconductor physics and devices, Third edition, Mc-Graw Hill Higher Education

 

Web Links

 

  1. https://en.wikipedia.org/wiki/Fresnel_equations
  2. http://physics.stackexchange.com/questions/74046/electric-field-of-unpolarized-light-after-reflect
  3. http://graphics.stanford.edu/courses/cs148-10-summer/docs/2006–degrevereflection_refraction.pdf
  4. http://iqst.ca/quantech/pubs/2013/fresnel-eoe.pdf