Chapter 38

Wave Optics (II)

Huygens' Principle

Huygens in 1678 came up with the idea that light waves (or in general waves) propagate in the form of wave fronts.  The waves generated from a source put the immediate points around that source into oscillation.  Such secondary points form a wavefront.  Each point of this wavefront will then act as an independent source that will send wavelets or waves in all directions.  The next series of points around these secondary points will receive wavelets from many of such secondary points that will interfere and will have a resultant that is maximum in the propagation direction of the original wave.  This next series of points (the tertiary points) form a new wavefront each point of which will again act as an independent source.  The process will continue in this manner (wavefront by wavefront) as the waves propagate in a medium or even vacuum.

Note that only electromagnetic waves propagate in vacuum.  The points in vacuum are not occupied by matter (molecules or atoms).  We look at points in vacuum as geometric points.  The following figure shows a series of secondary points that form a wave front (wavefront AB).  Each point on wavefront AB acts as an independent source sending wavelets in all directions; however, the next series of points (the tertiary points) form the new wavefront CD.

Figure 1

Snell's Law of Refraction:

In Chapter 37, the Snell's law was introduced without proof.  In this chapter, the wavefront concept of Huygens principle will be used to derive the Snell' law.

In Fig. 2, wavefront AB carries the incident rays and wavefront A'B' carries the refracted rays.  Note that each wavefront is perpendicular to the rays it carries.  The A end of wavefront AB arrives at the interface first.  Its B end arrives at the interface Δt seconds later because of distance BB'.  Of course, BB' = v1Δt  where v1 is the wave speed in medium 1.  Meanwhile, the ray that arrives at A first travels a distance AA' in medium 2 that is shorter than BB' of  medium 1.  Of course, AA' = v2Δt  where v2 is the wave speed in medium 2.  The assumption is that the speed of light in medium 2 is less than that of medium 1.  For this reason, light bends and gets closer to the normal line.  Triangles ABB' and AA'B' share the same hypotenuse AB'.  The angle opposite to BB' is equal to θ1, the angle of incidence.  The angle opposite to AA' is equal to θ2, the angle of refraction. The following trigonometric ratios may be written:

Figure 2

Diffraction:

Diffraction is the bending of light upon passing through small openings or by sharp edges.  Diffraction occurs in ocean waves as well.  Actually, it is beneficial to first examine the diffraction concept with ocean (mechanical) waves arriving at a jetty.  One role of a jetty where boats are parked is to separate the body of the water in the jetty from the heavy waves coming in from the ocean side.  That way the parked boats receive much weaker (in amplitude) waves and do not bump into each other.  Each jetty has a small opening to the ocean compared to its body of water.  See Fig. 3.  The opening is made large enough to allow all size boats to comfortably pass through ; however, its size is much smaller than the jetty itself.

Figure 3

As shown in Fig. 3, the wavefronts arriving from the ocean hit the jetty like almost straight lines one after each other.  See Remark 1 in Fig. 3.  The huge rocks bordering the jetty break the arriving fronts except at the opening.  The straight fronts make the lump of water at the jetty's opening oscillate up and down accordingly.  The good size lump of water at the opening becomes an independent source that sends waves in all directions especially into the jetty.  That lump of water being the originator of new waves is at the center of this wave generator.  The new waves center is the jetty's opening itself.  The new wavefronts are not straight fronts anymore.  The are very curved.  See Remark 2 in Fig. 3.   If they are given a huge area to propagate in, they become more straight similar to the ones that keep coming from the ocean.  The new waves into the jetty propagate in different directions compared to the waves arriving from the ocean side.  For an observer looking down from a helicopter into the jetty, it appears that the ocean waves bend when they pass through the opening.  The observer will call this phenomenon the "diffraction" of the ocean waves The same concept applies to light waves as they pass through small openings or by sharp edges.

 When light waves arrive at an opening or an aperture, each point of the opening becomes an independent source sending wavelets in all directions. The wavelets emerging from such independent sources interfere at different points of the region past the aperture causing interference patterns (due to diffraction) on a wall or a screen (Fig. 4).  If the aperture or the hole that a wavefront arrives at is big compared to the wavelength of the light, the diffraction pattern will not be very clear and approximates the shape of the aperture (hole) itself; however, if the opening's diameter is relatively small (as much as a few wavelengths), sharper diffraction patterns are formed. Figure 4 The diffraction pattern (on the wall) of light passed through a small aperture.

Fresnel Diffraction:

 When either the source or the screen is near an aperture or an obstruction (a sharp edge),  the wavefronts are spherical (not straight) and the diffraction pattern they form is quite complex.  This is called "Fresnel Diffraction."  One case is shown in Fig. 5.    In this case some light enters the region of geometrical shadow.      The curve shows how the intensity and width of the bright fringes change. Figure 5

Fraunhofer Diffraction:

Fraunhofer diffraction is one in which source is far from the slit or aperture. The wavefronts arriving at the slit or aperture are almost like straight lines.  In single-slit diffraction, the diffraction pattern on the screen depends on the aperture diameter or width, α.  Note that a single slit has a width only and not a diameter.    If the aperture diameter or the single slit width is large (several times the wavelength λ), the lit area over the screen fairly defines the shape of the aperture or the slit.  If the aperture diameter or the slit width is small and comparable to the wavelength of the light used, the diffraction patterns will be more pronounced or have better contrast as shown in Fig. 6.

Figure 6

The following formula gives the

The following formula gives the positions of the minima with respect to the center of the pattern on the screen.

αsinθn = nλ where n = ±1, ±2, ±3, ... .

In this formula, α  is the width of the single slit, and θn is the angle that the n-th minimum makes with the axis of symmetry that extends from the center of the slit to the screen (perpendicular to the screen) similar to the one in Young's double-slit experiment.  For small angles sinθn ≈ θn, and the formula becomes αθn = nλ from which θn = nλ/α.   This is consistent with the positions of the minima shown in Fig. 6 as λ/α, 2λ/α, and 3λ/α on both sides of the center in the pattern.

X-Ray Diffraction:

 X-ray was discovered during Cathode-Ray experiments done by Roentgen in 1895.  It was also determined that the wavelength of X-rays are much smaller than those of visible light.  This is because of the fact that when X-rays were passed through very thin layers of NaCl and ZnS crystals, symmetric patterns were observed on photographic films.  Not only this suggested that X-rays were smaller than inter-atomic distances, it also verified the wave nature of them because of the observed diffraction patterns on photographic films. When X-rays pass through a crystal, they produce a diffraction pattern that is a characteristic of that crystal. Figure 7

In 1913, W. H. Bragg and his son W. L. Bragg suggested the following analysis:  "If the atoms in a crystal are orderly arranged, and X-rays can penetrate into lower layers of atoms, after reflection from lower layers, they must interfere constructively or destructively with other reflected rays resulting in interference patterns from which inter-atomic distances can be measured."  The following figure shows an X-ray wavefront incident on a flat surface of a certain material.

Bragg's Formula:

In Fig. 8, one wavelet hits an atom M of the top layer of the material and another wavelet penetrates in and hits a lower layer atom B.  The two incoming rays move together up to line segment MA.  Note that MA must be drawn perpendicular to the direction of the arriving rays (front).  The reflected rays will also move together starting from MC up.  MC must be drawn perpendicular to the reflected rays.

The distance AB + BC is the path difference traveled between the arriving wavelets and the reflected wavelets.  If they interfere constructively, they result in pattern formation on the photographic film. The condition for forming a pattern on the  photographic film is

 2d sin θ = n λ

where n is an integer.  Angle θ must be changed until a pattern is obtained.

If the path difference = nλ ,

constructive interference occurs.

Equating  AB + BC = nλ , we get:

2d sin θ = n λ

Polarization:

As we already know, electromagnetic waves or light waves are generated as a result of electric charge oscillations.  If the frequencies of such waves are in the visible range, we can see them.  We have also learned that electromagnetic waves are of transverse type and there are two oscillating vectors that travel with each pulse perpendicular to the propagation direction. One is the electric field vector E and the other is a magnetic field vector B as shown below:

Figure 9

If a charge keeps oscillating up and down as shown in this figure, the generated pulses of E&M waves travel along the x-axis as shown.  Note that only the pulses traveling to the right are shown.  The pulses traveling to the left and any other horizontal direction are not shown here.  We know that as the electrons of atoms spin around their nuclei and make transitions to their higher orbits, they keep changing their planes of oscillation, some several hundred billion times or more per second.  The up and down vectors shown for charge q at the origin, in the above figure, is only one direction of oscillation out of some trillions possible directions of oscillation.  Accordingly, the electric field vector E does change its oscillation direction the same number of times.  It is the electric field vector that we emphasize on, in this topic.  For our current discussion, we will set aside the magnetic effect.  Fig. 10 show 3 possible directions of oscillation that appear to be like 6 possible directions of oscillation; in fact there are only 3.

Figure 10

When a light source emits light, several hundred trillions or more of such oscillation directions (pulses) are generated per second and come to you.  If you divide a circle into 360 oscillation directions, how many of these hundred trillions oscillation directions per second will you receive within each one-degree sector of the circle?  Still some trillions, you would say.  We are going to approximate all oscillation directions that fall within one degree as one direction only.  Now if there is a way to receive light in approximately one direction out of say 360 directions, we say that such light is polarized.  It means that the electric field oscillations occur in one direction only as shown below:

 Note that we are approximating all directions that fall within say, a 1o sector , for example, as one direction.     In a vertical 1o sector , for example, such as E1 on the right, there are still billions of directions that can all be treated as vertical and one direction only! Figure 11

Polarization Methods:

A few methods of generating polarized light is explained below:

1) Polarization by Reflection:

When light is incident on a transparent surface, it partially reflects and partially refracts.  If we keep changing the angle of incidence θi , the reflection angle, θrfl keeps changing but remains equal to θi .  The refraction angle, θrfr, keeps changing as well, but according to Snell's formula.  There comes a critical angle of incidence θi = θp , at which the reflected ray is perpendicular to the refracted ray.  When this happens, the reflected ray is polarized and the electric fields of its pulses oscillate in one direction only.  Fig. 12, shows the critical angle of incidence θp at which polarization occurs.

θp, the polarization angle, can be expressed in terms of the refraction indices of the involving media.  Such formula is attributed to Brewster and is called the "Brewster's law."

When the reflected and refracted rays are at 90° due to that special angle of incidence θp , the reflected ray is polarized, and we may write:

θrfl +90° + θrfr = 180° or,  θrfr = 90°- θrfl .

Since   n1sin θi = n2sin θrfr    or,

n1sin θp  =  n2 sin (90° - θrfl )   or,

n1sin θp  =  n2 cos θrfl   or,

n1sin θp  =  n2 cos θp   or,

 tan θp = n2/n1. (Brewster's Law)

Figure 12

2) Polarization by Selective Absorption:

3) Polarization by Scattering:

Chapter 38 Test Yourself 1:

1)  According to Huygens' principle, waves travel in the form of (a) particles   (b)  wavefronts   (c) a hump-like disturbances.   click here.

2)  A wavefront is formed as a result of (a) wavelets oscillations on a previous wavefront   (b) a change in frequency   (c) a change in wavelength.

3)  Each point on a wavefront acts as (a) a wavefront    (b) a wavelet   (c) an independent source sending out waves in all directions    (d) both b and c.

4) Wavefronts (a) do not form in vacuum   (b) do form in matter only   (c) form both in matter and vacuum.

5) When a wavefront of light is incident on the interface of two transparent media at a nonzero angle with the normal line, wavelets arrive at the interface (a) simultaneously   (b) at different instances    (c) at different points  (d) both b and c.    click here.

6) When monochromatic light is refracted at a flat interface, all refracted wavelets travel (a) at the same speed   (b) at the same angle   (c)  both a and b   (d) on the same wavefront.

7) When light is refracted into a new medium, (a) its frequency changes only   (b) its wavelength changes only   (c) both of its frequency and wavelength change in order to keep the same speed.   click here.

8) Diffraction is (a) the separation of light into its constituent colors   (b) the bending of light upon changing medium   (c) is the bending of light upon passing over sharp edges or through small holes or apertures.

9) The bigger the diameter of a hole that light wavefronts arrive at (a) the more pronounced the diffraction occurrence on the screen ( b) the more the shape of the hole is defined on the screen instead of a clear diffraction pattern    (c) the more pronounced the refraction phenomenon.

10) A clear diffraction pattern on a screen is considered to be (a) the unevenly spaced concentric dark and bright circles   (b) the fairly evenly spaced dark and bright fringes similar to a Young's double-slit experiment   (c) neither a nor b.

11) Fresnel diffraction occurs when (a) the source is far from the aperture or obstruction    (b) the screen is far from the aperture or obstruction   (c) either the source or screen is near the aperture or obstruction (d) both a and b.   click here.

12) In Fresnel diffraction, the spacings and widths of the interference patterns (a) remain constant for different angles of diffraction   (b) become smaller for greater angles of diffraction   (c) become greater for greater angles of diffraction.   click here.

13) Fraunhofer diffraction is the same thing as (a) single-slit diffraction with source far from screen   (b) double-slit diffraction with source far from screen   (c) neither a nor b.

14) In Fraunhofer diffraction, the smaller the slit or the aperture, (a) the more spaced the patterns   (b) the less spaced the patterns   (c) the closer the shape of the pattern to the shape of the aperture.   click here.

15) The reason X-rays were accepted to be waves was that (a) they passed through thin films of NaCl or ZnS  (b) they formed diffraction patterns on a photographic film after passing through thin films of NaCl or ZnS   (c) they were called cathode rays.

16) X-rays are suitable for inter-atomic measurements because (a) they are cathode rays   (b) they form diffraction patterns after reflection from atomic layers of material surfaces   (c) their wavelengths are short enough to penetrate matter   (d) both b and c.  click here.

17) In Bragg's formula, for constructive interference and formation of diffraction patterns, the total distance difference must be (a) an odd multiple of λ   (b) an even multiple of λ   (c) an integer multiple of λ.

18) In the derivation of Bragg's formula, dsinθ is equal to (a) 1/2 distance difference   (b) distance difference   (c) twice the distance difference.   click here.

19) Polarized light (a) has a magnetic field that oscillates normal to its electric field (b) lacks an oscillatory magnetic field because of polarization (c) neither a nor b.

20) Polarization by reflection occurs (a) at all angles of incidence   (b) only at an incidence angle of 45 degrees that makes the reflected light normal to the incident light   (c) at a special angle of incidence that depends on the refraction indices of the media involved.

Problems:

1) Light of wavelength 589nm is incident normally on a slit of width 0.054mm.  The diffraction pattern is observed on a screen 2.00m away.  Determine (a) the width of the central peak, and (b) the distance between the 1st order and second order minima.  Note:  Refer to the figure under Fraunhofer Diffraction.

2) When light of wavelength 436nm passes through a single slit, the width of the central diffraction peak on a screen is 2.22cm.  What would the width be if light of wavelength 589nm is used?  Note:  Refer to the figure under Fraunhofer Diffraction.   click here.

3) In a single-slit diffraction, the distance between the 1st and 2nd minima is 4.0cm on a screen that is 3.73m from the slit.  If the slit width is 4.50x10-5m, find the wavelength of the light used.

4) A diffraction grating with 200 lines/mm is used to analyze the light from a hydrogen discharge tube that emits wavelengths of  410.1nm and 656.3nm.  What is the angular separation between the fringes (a) of the first order, and (b) of those of the 2nd order?   click here.

5) X-rays of wavelength 0.18nm are incident on atomic planes of a crystal that are 0.34nm apart.  Calculate the first angle in the Bragg's formula at which constructive interference occurs and a diffraction pattern will be formed.

6) Monochromatic X-rays are incident on certain atomic planes of a crystal that are 0.30nm apart.  The 3rd order Bragg diffraction maximum forms at 27.0°.  Calculate the wavelength of the X-ray used.

7) Show that at the interface between two transparent media, the relation between θc, the critical angle for total  reflection, and θp , the polarization angle, is given by (sinθc)(tanθp) = 1.

8) The critical angle (the angle at which total reflection occurs) between two transparent media is 42°.  Calculate the polarization angle, θp, for such two media.

9) A beam of light is incident on a glass surface with a refraction index of 1.46 such that the reflected light is polarized.  At what refraction angle does it enter glass?

10) Show that the angle of separation between the mth order principal maxima of a grating for wavelengths λ and λ+Δλ is given by Δθ = mΔλ /(dcosθ).