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VII–3 Introduction into Wave Optics

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Main Topics

• Huygens’ Principle and Coherence.• Interference

• Double Slit• Thin Film

• Diffraction• Single Slit• Gratings• X-Rays, Bragg Equation.• Wave Limits of Geometrical Optics.

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Huygens’ Principle I

• Up to now, we have have treated situations, where many of wave properties could be neglected. In our rays model, we actually needed only their straight propagation.

• Now, we shall concentrate to typically wave properties of light, which are generally valid for all electromagnetic (and other) waves.

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Huygens’ Principle II

• The basis for studying wave effects is Huygens’ (Christian 1629-1695 Dutch) principle of wave propagation. It states:

• Every point reached by a wave can be considered as a new source of tiny wavelets that spread out in all directions at the speed of the wave itself.

• The new wave is superposition of all the wavelets, which usually cancel in other direction then the wave front, which is their envelope, propagate.

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Huygens’ Principle III

• If light is traveling through homogeneous isotropic media without obstacles Huygens’ principle gives us the same results as ray (geometrical) optics including effects as reflection and refraction.

• However, when there is e.g. an obstacle then wave fronts will be not only distorted but new effects of interference and diffraction will appear. There will be for instance bright or light or colored regions even where shadow should be.

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Coherence I

• Typical wave properties are based on the principle of superposition. If several waves meet in one spot their common effect is the sum of all of them. But since waves are periodic, extremes may happen e.g. they are in phase and they will constructively interfere or they may be out of phase and they will interfere destructively.

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Coherence II

• Since the frequency of light is very high this adding of waves may have some stable result only if the interfering waves are coherent i.e. have constant phase difference. In the case of e.g. radio waves we can, in principle build two same oscillators and synchronize them. But electronic oscillators for visible light don’t exist. Light can be generated only by transitions in atoms.

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Coherence III

• It’s problem of accuracy and the fact that light is not continuous, in short time scale, but comes in ‘trains’.

• So ideally coherent light waves must stem from the same transition of the same atom.

• But also a partial coherence exists under much less strict conditions. Diffraction can be for instance obtained from a Sun light when it passes through a very small aperture.

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Double-Slit Interference I

• This experiment was the first convincing evidence of wave properties of light done in 1801 by Englishman Thomas Young (1773-1829).

• If a plane monochromatic light wave passes through two thin, closely spaced slits. The picture on a screen behind are not two bright lines but rather a series of them.

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Double-Slit Interference II

• According to the Huygens’ principle the slits are sources of new wavelets but now in every point of screen only two of these wavelets add, instead of infinity, what would be the case without slits.

• Suppose that the distance of the slits d is negligible to that of the screen so two rays entering a far point of it are almost parallel.

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Double-Slit Interference III

• If two waves leave the slits under some angle their path difference is:

d = d sin• Clearly, if d is an integral multiple of the

wavelength the waves constructively interfere. This condition for maxima is:

d sin = m• m = 0, 1 … order of the interference fringe.

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Double-Slit Interference IV

• If, however, d is odd multiple of /2 the waves will be completely out of phase and they will interfere destructively. The exact condition for minima is:

d sin = (2m+1)/2 = (m+1/2)• Again: m = 0, 1

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Double-Slit Interference V

• We have found the positions of the maxima and minima but there are also apparent changes in intensities which we have to explain.

• The treatment of these more subtle details is similar to that we used in AC circuits. We can employ the mathematics of phasors.

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Double-Slit Interference VI

• The intensity of light is proportional to the square of their electric field: I ~ E2 so we shall find the total electric field produced by both waves at some angle , which is a sum of fields:

E = E1 + E2

• The fields have the same but can be phase shifted:

E1 = E10sint and E2 = E20sin(t+)

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Double-Slit Interference VII

• The phase shift can be easily related to the path difference. From:

/2 = d sin/ = 2/ d sin

• If we expect that our point on the screen is equally illuminated by both slits we find:

E = 2E0 cos(/2)sin(t + /2)

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Double-Slit Interference VIII

• The same result can be found from the phasor diagram. The phase shift of E is clearly /2.

• We omit the fast changing term and relate the intensity to the one in the middle, where both waves are in phase:

I/I0 = E20 /(2E0)2 = cos2(/2)

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Double-Slit Interference IX

• Finally we substitute for the phase difference :

I = I0cos2(/2) = I0cos2(dsin/)

• And for the angle as a function of the screen distance L and distance of the point of interest from the center of the screen y:

20 )][cos(sin y

L

dII

L

yy

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Double-Slit Interference X

• y is an easy measurable variable.

• Again the same conditions for the positions of the maxima and minima are present in this formula but we have also obtained the information on intensities for any point on the screen described by or y.

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Interference in Thin Films I

• The principle is the same as for any interference generally:

• Now two waves reflected on the upper and lower surfaces of a thin film can interfere either constructively if the path difference in the film is equal to integer number of wavelengths or destructively if it is an odd number of half wavelengths.

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Interference in Thin Films II

We have to consider two new effects:

• Wavelength in the particular material of the film changes – we didn’t have to care about this when dealing with dispersion!

• Under a certain conditions there can be phase changes when the wave reflects.

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Interference in Thin Films III

• Experiments show that the waves have the same frequency in all materials and since the speed changes so must the wavelength:

• If we use white light the conditions for maximum at a certain angle will be valid always for some color – color interference.

nn

v

cvcf n

nn

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Interference in Thin Films IV

• Experiments show important property of reflection: • If a wave reflects on a surface with optically

denser media it changes its phase by .

• If the second medium is less optically dense there is no phase change.

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Interference in Thin Films V

• An important application of thin-film interference is a non-reflective coating of optical elements.• In the case of destructive interference more

light gets through ~ 99%.• A single layer works well for one wavelength,

usually 550 nm.

• Actually whole field of layer reflective optics exists.

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Interference in Thin Films VI

• What thickness t must have a coating of MgF2 with n = 1.38 on glass with ng = 1.5 to give destructive interference for green light = 550 nm?

• There will be a phase shift of /2 on the air-coating boundary and the path difference is 2t. For minima: 2t - n/2 = (2m+1) n/2 t = n/4 = /4n = 99.6 nm

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Diffraction I

• Wave theory predicts that waves can be diffracted around edges of obstacles and interfere in the shadow behind them.

• Only after diffraction was observed the wave nature of light was fully accepted.

• The main ideas are again based on the Huygen’s principle.

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Diffraction II

• Let’s consider a diffraction pattern produced by a single narrow slit of the width a.

• Every point in the slit is a source of wavelets which add on some screen behind.

• Let’s find conditions for constructive and destructive interferences:

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Diffraction III

• The condition for the first minimum is:sin = /a

• A wave from a point in the middle of the slit has a path difference of /2 from the point on the lower edge. These waves are out of phase and thereby cancel themselves. Similarly, if we proceed up, all waves will cancel in-pairs so we get a minimum.

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Diffraction IV

• The condition for the first maximum is:

sin = 3/2 /a

• If we consider points in the two adjacent thirds of the slit, their waves will also cancel in-pairs but the waves from points in the last third will not, so we have a maximum intensity.

• Conditions for higher orders are found similarly.

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Diffraction V

• The conditions are opposite from those for two-slit interference.

• Calculation of intensities can be again performed using phasors.

• We can divide the slit to equivalent strips y and find the phase difference of the waves from adjacent strips:

= 2/ ysin .

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Diffraction VI

• We are interested what will be the resulting phasor of total field built from these small phasors.

• In the case of minima the phasors complete a whole circle so the total is zero.

• In other words for every phasor an opposite phasor exists, which cancels it.

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Diffraction Grating I

• It’s in principle multiple parallel slits. At present gratings can be made very precisely with very high densities of slits of the order of 104 lines per centimeter. Gratings can be made both for transmission and reflection.

• The condition for principal maxima is the same as with the double slits:

• sin = m/d here d is the adjacent spacing

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Diffraction Grating II

• The main difference between the double-slit and multi-slit pattern is that the maxima in the latter case are much sharper and narrower.

• For a high maximum we need that waves even from far slits are exactly in-phase. If there is even a slight difference waves will cancel in pairs with some those from distant slits.

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Diffraction Grating III

• Gratings can be used to decompose light. Spectrometers using gratings are actually better than those using dispersive elements. The resolution of gratings is higher and the response is linear.

• This has a great impact in spectroscopy.

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X-Ray Diffraction I

• If we use X-rays which are EMW with the wavelengths of the order of 10-10 m then actually crystal planes can serve as gratings with single atoms as single slits.

• The refractive index for these wavelengths is almost 1 so the condition for maxima can be described by a simple Bragg’s equation

2dsin = m

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X-Ray Diffraction II

• Beside positions of maxima, there is also much information in intensities which are interpreted by more complicated dynamical theory.

• Methods based on X-ray diffraction are important for characterization of structure of materials. Different methods exist for monocrystals, powders or even solutions.

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Polarization I

• In unpolarized light the electric vectors have random position in the plane perpendicular to the propagation of the wave.

• It is however possible to polarize light i.e. to select electric vectors only in a certain plane. This is done by polaroids but polarization is produced even by reflection.

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Polarization II

• If we cross two polaroids, ideally, no light gets through.

• Some materials with assymetric molecules are capable to turn the polarization plane.

• In polarographs we can measure e.g. concentration of these.

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Scattering I

• Actually any atom interacts with light. Its electrons become roughly sources of new wavelets. We can compare it to Huygens’ principle which is valid even in vacuum.

• Since atoms are different there will be some interaction due to superposition of waves even for less ordered structures. But it will be seen only in the vicinity of the 0-order maximum.

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Scattering II

• Scattering again contains important structural information.

• In atmosphere scattering behaves as 1/4.• The sky is blue since blue light scatters most

• The sunset is red since blue waves are scattered and just the red ones remain.

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Wave Limits to Geometrical Optics I

• An image done by e.g. a lens, is in fact a superposition of diffraction patterns. We would find this if we display just little point. Its image is not a point but rather diffraction circles.

• This matters only when the magnification is high.

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Wave Limits to Geometrical Optics II

• A resolution of some optical device is roughly a distance of two points which we are still able to distinguish.

• If we think about the diffraction patterns the point can be distinguished when the main maximum of one falls into the first minimum of the second.

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Homework

• No more homework from physics!

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Things to read and learn

• Chapters 35, 36

• Try to understand all the details of the scalar and vector product of two vectors!

• Try to understand the physical background and ideas. Physics is not just inserting numbers into formulas!

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