#### Coherence, Physics tutorial

What is Coherence?

If you are inquire what coherence is, you might state that it is the condition essential to generate observable interference of the light. And if you are asked what interference is, you might state it is joined by the interaction of waves which are coherent. Well, not anything definite follows from such circular arguments! However, coherence is the property of light while interference is the effect of interaction of the light waves. The important consideration in interference phenomenon is the relative phase of waves occurring at a given point from two or more sources. That is, in order to view interference fringes, there should exist a definite phase relationship between the light waves from the two sources. Therefore, we might state that the necessity of having coherent sources for observing the interference fringes fundamentally implies that the waves from the two sources should encompass a constant and predictable phase relationship. This is the absence of a definite phase relationship between the light waves from ordinary sources which we don't get any observable interference fringe pattern.

Now, the question arises: Why is there no definite phase relationship between the light waves from two ordinary light sources? Well, the fundamental method of emission of light comprises atoms radiating electromagnetic waves in the form of photons. Each and every atom radiates for a small time (that is, of the order of 10-9s). In the meantime, other atoms start to radiate. The phases of such emitted electromagnetic waves are, thus, random; whenever there are two such sources, there can be no definite phase relationship between the light waves emitted from them.

In common, sources, and the waves they emit, are stated to be coherent if they:

• Have equivalent frequencies.
• Maintain a phase difference which is constant in time.

When either of such properties is lacking, the sources are incoherent and the waves don't generate any observable interference.

Let us take a moment and ask ourselves: Why it is a requirement for observing the interference fringe pattern? To answer this question, let us assume that the origin of the bright and dark fringes in the Young's experiment (the figure is shown below). Let E1 and E2 be the electric fields related by the light waves emanating from the slits S1 and S2. These waves superpose and the joint electric field at any point on the screen is represented by:

E = E1 + E2

We are familiar that in the interference pattern, we observe the intensity of light, not the electric field. As the average intensity of light is proportional to the time-averaged value of the related electric field, we have:

I ∝ E2

Therefore, we have:

I = <E12> + <E22> + 2 <E1E2>

I = I1 + I2 + 2 <E1E2>

The above equation represents that the resultant intensity on the screen is the sum of intensities I1 and I2 (that is, due to the individual slit sources) and an interference term 2 <E1E2>. The interference word is crucial as it finds out whether the resulting intensity is a uniform illumination or a fringe pattern on the screen. The contribution of the interference term to the resultant intensity mainly based on the phase relationship between the light waves emanating from the two slits.

Let us first assume the case when the light waves are in phase at one instance and are out of the phase at other instance. In such a condition, the product E1E2 will be positive at one instance and negative at the other. As an outcome, the time average of E1E2 will be zero, that is,

<E1E2> = 0

Waves having this type of phase relationship (that is, varying with time) are stated to be incoherent and the resulting intensity will be:

I = I1 + I2

Therefore, when the light waves from two incoherent sources interfere, the resulting intensity will be the sum of individual intensities and the screen will be uniformly illuminated. To give you a simple illustration, if the headlights of a car illuminate the similar region, their combined intensity is simply the sum of two separate intensities. The headlights are incoherent sources and there is no contribution of the interference term.

Now, what will occur if the light waves from the two slits encompass a definite phase relationship that is, a phase relationship that is constant in time. The source of light emitting such waves is coherent. Whenever light sources are coherent, the resulting intensity is not simply the sum of individual intensities. It is so because in that condition, the interference term in equation is non-zero. Let us observe what the form of the interference term is if the two coherent light waves superposed. The two cases are as follows:

1) If E1 = E2, that is, the two waves encompass the similar amplitude, frequency and phase. Therefore:

I1 = <E12> = <E22> = I2

And 2 <E1E2> = 2 <E12> = 2 I1

The resultant intensity:

I = I1 + I2 + 2 <E1E2>

I = I1 + I1 + 2 I1

I = 4 I1

Therefore, at the points on screen where the two interfering waves are in phase, the resulting intensity is four times that due to the individual source. Such points will, thus, appear bright on the screen.

2) E1 = - E2, that is, the two waves encompass similar amplitude and frequency however their phases differ by 180° that remains constant in time. In that situation, the two waves are fully out of phase and the resulting wave amplitude and intensity will be zero.

E = E1 + E2 = 0

=> I = 0

The points on the screen where the interfering light waves satisfy the above state will encompass zero intensity and therefore they will come out dark.

Therefore, the constant phase relationship between the superposing light waves, that is, coherence, is an essential condition for getting the interference fringe pattern. When the phase relationship is not constant, the points where superposing light waves arrive in the phase at one instant might get light waves that are completely out of phase at the other instance. This outcome in uniform illumination of the screen and no interference fringe pattern can be noticed.

Temporal Coherence:

Most of the quasi-monochromatic sources of light encompass spectral intensity profile which can be approximated through a Gaussian curve. Temporal coherence is determined through the coherence length Lc that mainly depends on the line width Δλ (that is, wavelength uncertainty in FWHM of the relative irradiance) of central wavelength λo. It can as well be deduced in terms of frequency bandwidth (that is, uncertainty) Δf of the central optical frequency fo

(It is noted that if Δf << fo and Δλ << λo, then Δf/fo = Δλ/λo).

In free space, the coherence length Lc = c/Δf = λo2/Δλ.

The Coherence length Lc is the distance in the direction of wavefront propagation in which the amplitude and phase of the wave can be considered well stated, predictable and thus subject to possible wave interference. An ideal light source would encompass an infinite coherence length. On the other hand, thermal sources of light (like the sun or a light bulb) cover a relatively wide range of wavelengths Δλ and thus encompass extremely small coherence length. Due to that reason, under normal lighting conditions, we can observe interference merely in very thin regions, like the thickness of the soap bubble.

It is noted that in a dielectric medium having refractive index 'n', the speed of light, wavelength, and therefore as well the coherence length, is reduced by the factor 'n'

Therefore, Lc= c/(nΔf) = λ2/(nΔλ).

The Michelson interferometer is an instrument in which a collimated beam of light is divided into two beams travelling separate paths (1) and (2), and then reassembled as (3). If the path difference ΔL = 2L = path (2) - path (1) is smaller than the coherence length Lc, then interference can be observed. If ΔL > Lc then the interference pattern disappears (that is, the phase relation among the reassembled beams no longer exists). We can therefore find out the coherence length Lc = λo2/Δλ by increasing the distance (2) and finding the path difference ΔL = 2L at which the interference pattern disappears. From this measurement we can find out the line-width Δλ and the quality of the quasi-monochromatic light source λo.

Spatial Coherence:

The Spatial coherence is found out by the coherence width:

Wc = k (λ/D) R

Here, k is the constant dependent on the shape of the source (that is, for a circular source k = 1.22), 'D' is the approximate diameter of the source and 'R' is the distance from the source. Coherence width is the distance all along the wavefront (that is, perpendicular to the direction of propagation), in which the amplitude and phase of the wave can be considered well defined and thus predictable. The ideal light source would be a point (D = 0) producing ideal spherical wave-fronts. An ideal point source would encompass infinite coherence width.

The degree of spatial coherence can be predictable by means of inspection of a shadow cast through an illuminated object. The sharper the shadow, the better spatial coherence of the source.

Note: The Coherence of a source can be enhanced through different physical arrangements and optical components (that is, raising the distance from the source, focusing and passing light via a small pinhole aperture and so on). Improved coherence, though, outcomes in the drastic reduction of light intensity.

Spectral coherence:

The waves of various frequencies (that is, in light these are different colors) can interfere to form or make a pulse if they encompass a fixed relative phase-relationship. Conversely, whenever waves of different frequencies are not coherent, then, when combined, they make a wave which is continuous in time (example: white light or white noise). The temporal duration of the pulse Δt is limited through the spectral bandwidth of the light Δf according to:

Δf Δt ≥ 1,

that obeys from the properties of the Fourier transform and outcomes in Kupfmuller's uncertainty principle (that is, for quantum particles it as well outcomes in the Heisenberg uncertainty principle).

If the phase depends linearly on the frequency (that is, θ(f) ∝ f) then the pulse will encompass the minimum time duration for its bandwidth (that is, a transform-limited pulse), or else it is chirped.

Measurement of spectral coherence:

The measurement of the spectral coherence of light needs a nonlinear optical interferometer, like an intensity optical correlator, frequency-resolved optical gating (or FROG), or spectral phase Interferometry for direct electric-field reconstruction (or SPIDER).

Polarization coherence:

Light as well consists of a polarization, that is the direction in which the electric field oscillates. Unpolarized light is comprised of the incoherent light waves having random polarization angles. The electric field of the unpolarized light wanders in each and every direction and changes in the phase over the coherence time of the two light waves. The absorbing polarizer rotated to any angle will for all time transmit half the incident intensity if averaged over time.

Whenever the electric field wanders through a smaller amount the light will be partly polarized so that at certain angle, the polarizer will transmit more than half the intensity. Whenever a wave is joined through an orthogonally polarized copy of itself delayed by less than the coherence time, partly polarized light is made.

The polarization of a light beam is symbolized through a vector in the Poincare sphere. For polarized light, the end of the vector lies on the surface of the sphere, while the vector consists of zero length for unpolarized light. The vector for partly polarized light lies in the sphere.

Applications of coherence:

1) Holography:

The coherent superposition of optical wave fields comprises holography. The holographic objects are employed often in everyday life in bank notes and credit cards.

2) Non-optical wave fields:

Additional applications concern the coherent superposition of the non-optical wave fields. In quantum mechanics for illustration one considers a probability field that is associated to the wave function Ψ(r) (interpretation: density of the probability amplitude). In this, the applications concern, among others, the future technologies of the quantum computing and the already available technology of quantum cryptography.

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