Optics Series Lecture, Lecture – VII.

“Conditions of interference, Interference of two plane harmonic waves.” This lecture was delivered on 7th February in a lecture session of 1 and 1/2 hours. This lecture was delivered to Physics elective students but intended as a lecture towards Honors students at a later date.

Electromagnetic Waves.

Light is an electromagnetic wave. In-fact its a transverse electromagnetic wave which means the oscillation of E and B fields produces light which propagates in a direction that is perpendicular to the plane that contains the E and B fields. In other words E, B and k the vector that denotes the direction of light propagation, are mutually perpendicular vectors. We will study these details in a later intended lecture. EM waves are not only transverse waves but also vector waves, that is; E and B are vector fields whose undulation is summarized as light.

Light is a general name for all EM waves but visible light is that particular part of EM waves which has frequency of wave such that the wavelength varies from approximately 400 – 700 nm. In vacuum — only in vacuum, light always moves at a fixed speed: namely c = 3×108 m/s. Therefore light whose wavelength lies between 400 – 700 nm is called as visible light: we can write in vacuum c = νλ.

Light as a transverse wave phenomenon of vector fields is comprehensively described by four equations known as Maxwell’s Equations. The Maxwell’s Equations are a summary of important and fundamental laws of electricity and magnetism — together called as electromagnetism, such as Gauss Law and Ampere’s Law. These equations produce the wave equation of motion, a linear, homogeneous, 2nd order differential equation that we will study a few lectures afterwards. If you are quite serious and technically well equipped though, you can have a glimpse of it all — and may be work out to your satisfaction, by following the link to my slideshare presentations. There are many other important Physics concepts that are worked out in great detail, in those slideshare presentations by me. eg check: Electromagnetic Waves.

Let us therefore write the wave equation of motion, where the 3 components of E field — such as Ex, Ey or Ez or the 3 components of B field such as Bx, By, Bz, are denoted as ψ chosen anyone at one time. eg we can chose Ex = ψ. In general we have: $\inline&space;\dpi{120}&space;\fn_jvn&space;\frac{\partial^2&space;\psi}{\partial&space;x^2}+\frac{\partial^2&space;\psi}{\partial&space;y^2}+\frac{\partial^2&space;\psi}{\partial&space;z^2}=\epsilon&space;\mu&space;\frac{\partial^2&space;\psi}{\partial&space;t^2}\\&space;\psi&space;=&space;E_x,\hspace{3pt}&space;E_y,\hspace{3pt}&space;E_z,&space;\hspace{3pt}B_x,&space;\hspace{3pt}B_y,\hspace{3pt}&space;B_z$ so ψ is any one of the scalar component of any vector field E or B. Also ε, μ are the permittivity and the permeability respectively of any medium so that εμ = 1/v2 with the recognition that v is the speed of wave propagation, entailed by the above wave equation. Needless to say, if the wave travels in free-space — or vacuum, we replace the above equation with the proper subscripts that represents the quantities in free space. We have for free space: ε0μ0 = 1/c2.

Plane Harmonic Waves.

Let us now consider two monochromatic point sources S1 and S2 in a homogeneous medium. A monochromatic source would mean a single frequency in a homogeneous medium. Let the sources be a distance a apart from each other so that the wave length of light is negligible compared to this distance. That is; a >> λ, where λ is the wavelength of the light emitted from S1 or S2. Let the light emitted from these sources meet at point P. Lets consider only linearly polarized light. That is there is a fixed plane instead of arbitrarily many planes on which the E field — and correspondingly the B field, oscillate. Let us now draw a suitable diagram for our situation.

Let us consider what happens when the two plane harmonic waves from monochromatic sources S1 and S2 reach the point P. When two waves reach the same space-point they can be together described by the principle of superposition. According to this principle two different vecors or scalars are added to determine the combined effect. Here we are considering the electric field vector oscillation of the plane wave-fronts — given as E1(t) and E2(t) in the diagram whose amplitude or maximum oscillation value is independent of time and space points and given as E01 and E02 in the diagram. Accordingly we write the effective electric field vector E at the point P as the sum of the individual electric field vectors E1(t) and E2(t) and explicitly write their harmonic monochromatic form. Harmonic means sine or cosine or both and monochromatic means single wavelength for both sources — or the same frequency value for both waves, ω as shown below. $\inline&space;\fn_jvn&space;\vec{E}(t)=\vec{E}_{1}(t)+\vec{E}_{2}(t)\\&space;\vec{E_1}(\vec{r}_1,t)=\vec{E}_{01}\cos{\,}&space;(\vec{k}_{1}\cdot\vec{r}-\omega&space;t+&space;\epsilon_{1})\\&space;\vec{E_2}(\vec{r}_2,t)=\vec{E}_{02}\cos{\,}&space;(\vec{k}_{2}\cdot\vec{r}-\omega&space;t+&space;\epsilon_{2})$

Interference.

Interference of optical waves can be defined in terms of irradiance I which is the amount of optical energy falling on an unit surface area in an unit time. If the energy is emitted out of the surface its called as exitance. But radiant flux density is the general name for both falling on or emission out of the surfaces. This irradiance can be written as I = εv〈E2T where T is time elapsed, such that the time period of light wave τ is far smaller than this elapsed time T and we can write: T >> τ. The factor in front of the expression: εv is constant for a homogeneous medium. We will disregard this factor, as its same for all waves for calculating irradiance. Thus for our analysis irradiance would be given as: I = 〈E2T .

Let us define optical interference as interaction of two or more light waves such that resultant irradiance is different from sum of component irradiance. From superposition of the waves in terms of electric field vectors: E(t) = E1(t) + E2(t) hence $\inline&space;\fn_jvn&space;\vec{E}^{2}=&space;\vec{E}\cdot\vec{E}=(E_{1})^{2}+(E_{2})^{2}+&space;2\vec{E}_1\cdot\vec{E}_2$ so we can write the irradiance of source S1, that is I1 and source S2, that is Iand the irradiance of the combined waves from the sources S1 and S2, that is I. I = I1 + I2 + I12 where: $\inline&space;\fn_jvn&space;I_1=\langle&space;E_{1}^{2}\rangle&space;_T{,}\hspace{4pt}I_2=\langle&space;E_{2}^{2}\rangle&space;_T&space;{,}\hspace{4pt}&space;I_{12}=&space;2\langle&space;\vec{E}_1\cdot\vec{E}_2\rangle&space;_T$But we see that: $\inline&space;\fn_jvn&space;\vec{E}_1\cdot\vec{E}_2=\\&space;\vec{E}_{01}\cdot\vec{E}_{02}\cos{\,}(\vec{k}_{1}\cdot\vec{r}-\omega&space;t+&space;\epsilon_{1})\cos{\,}&space;(\vec{k}_{2}\cdot\vec{r}-\omega&space;t+&space;\epsilon_{2})\\&space;\\&space;=\vec{E}_{01}\cdot\vec{E}_{02}\\&space;\times&space;\Big[cos{\,}&space;(\vec{k}_{1}\cdot\vec{r}+&space;\epsilon_{1})\,cos\,\omega&space;t&space;+sin{\,}&space;(\vec{k}_{1}\cdot\vec{r}+&space;\epsilon_{1})\,sin\,\omega&space;t\Big]\\&space;\times&space;\Big[cos{\,}&space;(\vec{k}_{2}\cdot\vec{r}+&space;\epsilon_{2})\,cos\,\omega&space;t&space;+sin{\,}&space;(\vec{k}_{2}\cdot\vec{r}+&space;\epsilon_{2})\,sin\,\omega&space;t\Big]$From elementary calculus we know that average of sine and cosine functions over complete periods, when the functions are squared, are one-half. Also the average over complete periods of sine and cosine functions are zero. So we have: $\inline&space;\fn_jvn&space;\langle\vec{E}_1\cdot\vec{E}_2\rangle_T&space;\,\\=\frac{1}{2}\vec{E}_{01}\cdot\vec{E}_{02}\Big[cos{\,}&space;(\vec{k}_{1}\cdot\vec{r}+&space;\epsilon_{1}-\vec{k}_{2}\cdot\vec{r}-\epsilon_{2})\Big]\\&space;I_{12}=\vec{E}_{01}\cdot\vec{E}_{02}\,cos\,\delta&space;,&space;\hspace{6pt}&space;\delta&space;=&space;\vec{k}_{1}\cdot\vec{r}+&space;\epsilon_{1}-\vec{k}_{2}\cdot\vec{r}-\epsilon_{2}$δ is the phase difference arising from combined path length difference and initial phase difference. If the waves emanated from the sources are such that the electric field vectors oscillate in the same direction: $\fn_jvn&space;\vec{E}_{01}\hspace{5pt}||\hspace{5pt}\vec{E}_{02}&space;,&space;\,I_{12}=E_{01}E_{02}\,cos\,&space;\delta$ in this case: $\fn_jvn&space;I_1=\langle&space;E_{1}^2\rangle&space;_T=\frac{E_{01}^2}{2},\\&space;and&space;\,I_2=\langle&space;E_{2}^2\rangle_T=\frac{E_{02}^2}{2}\\&space;\\&space;also\\&space;I_{12}=2\sqrt{I_1&space;I_2}\,&space;cos&space;\,&space;\delta$ so total irradiance equals to individual irradiances and the interference term; $\inline&space;\fn_cs&space;\boxed{&space;I=I_1+I_2+2\sqrt{I_1&space;I_2}\,&space;cos&space;\,&space;\delta}$.

Constructive and Destructive interference.

1. In the above equation we see that maximum irradiance occurs when cos δ is 1;
$\inline&space;\fn_cs&space;I_{max}=I_1&space;+&space;I_2&space;+&space;2\sqrt{I_1I_2}$ This represents the condition of Total Constructive Interference and is possible when phase difference δ equals angles that are integral multiples of . $\inline&space;\fn_cs&space;\delta&space;=&space;0,\pm&space;\,2\pi&space;,&space;\pm&space;\,4\pi&space;\,...$  In this situation the waves from different sources are said to be in phase. The troughs overlap with troughs and crests overlap with crests.
2. If cos δ is between 0 and 1 waves are still partially in phase. — that is not completely out of phase;
Such a situation is known as constructive interference. In this case: I1 + I2 < I < Imax.
3.  When δ = π/2cos π = 0;
That is optical disturbances are 900 out of phase. I = I1 + I2.
4. When cos δ lies between 0 and -1, that is: 0 > cos δ > -1 waves are partially out of phase.
This is known as condition of destructive interference. We have I1 + I2 > I > Imin
5. Minimum amount of irradiance occurs when waves are 1800 out of phase.
The troughs overlap with crests and vice-a-versa. cos δ is -1. $\inline&space;\fn_cs&space;I_{min}=I_1&space;+&space;I_2&space;-&space;2\sqrt{I_1I_2}$ This occurs when the phase difference is equal to odd integral multiples of π$\inline&space;\fn_cs&space;\delta&space;=&space;\pm&space;\,\pi&space;,&space;\pm&space;\,3\pi&space;\,&space;\pm&space;\,5\pi&space;\,...$ This is known as Total Destructive Interference.
6. If amplitudes of S1 and S2 are same that is if: E01 = E02;
Then I1=I2=I0, in this case total irradiance, minimum irradiance and maximum irradiance are as follows: $\inline&space;\fn_cs&space;\boxed{I=2I_0(1+cos\,\delta)=4I_0cos^2{\frac{\delta}{2}}}$ $\inline&space;\fn_cs&space;\boxed&space;{&space;I_{min}=0,\,&space;I_{max}=4I_0}$.