**Unit 5: Light and Color**

**Problem Set B**

**Overview:**

Problem Set B targets your ability to analyze a thin film or air wedge and relate the wavelength of incident light to the thickness of the film or the number of visible bands to the wavelength of incident light and the thickness of the object used to create the wedge.

**Thin Film Interference**

Thin film interference is the phenomena which results when light
of a particular wavelength undergoes constructive interference
when it
reflects off the top and the bottom of a thin film. The interference
occurs between the ray reflected off the top of the film and the ray
which reflects off the bottom of the film and ultimately exits out of
the film across the original entry face. The diagram at the right
depicts the physical situation which results in constructive
interference with thin films. Ray 1 in the diagram represents the
wave which reflects off of the *top face* of the film (surface
A); ray 2 represents the wave which enters into the film and reflects
off the *bottom face* of the film (surface B). Ray 2 will
eventually transmit across surface A and exit the film. Upon exiting,
rays 1 and 2 will be capable of interfering constructively if they
meet a collection of conditions. One condition is that the two rays
must be relatively close together such that their crests and troughs
can meet up with each other. To meet this condition, the light must
be incident at angles close to zero with respect to the normal. (This
is not shown in the diagram above in order to space out the rays for
clarity sake.) A second condition which must be met is that the
*path difference* and the wavelength of light must have a whole
or half umber relationship between them such that wave 1 and wave 2
meet up with each other in phase - that is, such that crest is
meeting crest and trough is meeting trough. The path difference is
simply the difference in distance traveled between wave 1 and wave 2.
If these two conditions are met, then light of a particular
wavelength will interfere constructively and be reinforced upon
reflection from the top and the bottom of the film. In such cases,
the film might be illuminated with white light but a specific
wavelength in the mix of ROYGBIV (white) light will be reinforced in
such a manner that it stands out in brightness. This will cause the
film to take on an appearance of color.

Though thin film analysis is a rather complicated topic, a
systematic approach and a basic understanding of key principles will
facilitate the solution of any thin film problem. Having studied
two-point source interference, you might reason that the constructive
interference will occur when there is a relationship between the path
difference between waves 1 and 2 (refer to diagram above) and the
wavelength of light. Complicating this issue though is the fact that
the speed and the wavelength of a wave will change when it passes
from one medium to another. Thus the wavelength of light ()
is different in the film than it is out of the film. The extra
distance traveled by wave 2 is traveled in the film. Thus, it will be
important to relate the thickness of the film to the wavelength of
light in the film (not in air nor in any other medium). The equation
which relates the wavelength in the film (_{film})
to the wavelength in air (_{air})
is shown below:

In the above equation, n_{film} represents the index of
refraction of the material which the film consists of.

Relating the thickness of the film to the wavelength of light which is reinforced by the film demands an understanding of when a wave phase shifts upon reflection. Both wave 1 and wave 2 are reflecting of a boundary. When a wave reflects off a boundary, there are certain conditions under which it undergoes inversion or a 180-degree phase shift. If a wave is in a less dense medium and heading towards the boundary with a more dense medium, then it will undergo inversion (or a 180-degree phase shift) upon reflection. That is, its crest will become a trough and vice versa. Since constructive interference is about crests meeting crests and troughs meeting troughs, the inverting of a crest into a trough will have a tremendous impact upon the conditions for which wave 1 and wave 2 undergo constructive interference.

If wave 1 reflecting off the top of a film meets up with wave 2
reflecting off the bottom of the film, constructive interference
occurs when a crest of 1 meets with a crest of 2. If there were no
phase shift occurring for either of the two waves, then this
interference will take place when the difference in distance traveled
(PD) = m•_{film}
where m is a whole number. The difference in distance traveled for
the two waves is simply twice the thickness of the film (2t). Thus,
the following rule can be stated:

**Case 1**: If neither wave A nor B undergo a phase shift,
constructive interference occurs when:

or

**t _{film} = 0.5 • m •
_{film}where
m = 1, 2, 3, ...**

If one of the two waves undergoes a phase shift upon reflection, then a crest becomes a trough and the constructive interference criteria listed above change. Since a phase shift turns a crest into a trough, a path difference of a whole number of wavelengths will result in destructive interference. With a "180° phase shift", constructive interference will take place when the path difference (2•tfilm) is a half-number of wavelengths. Thus, the following rule can be stated:

**Case 2:** If either wave A or B undergo a phase shift,
constructive interference occurs when:

or

**t _{film} = 0.5 • m •
_{film}where
m = 0.5, 1.5, 2.5, ...**

If both of the waves undergo a phase shift upon reflection, then there is no overall phase shift of wave B relative to wave A. Since both waves become shifted by a half wavelength, a crest still meets a crest (and a trough meets a trough) when the path difference is a whole number of wavelengths. Thus, the criteria for constructive interference are the same as that stated in Case 1: The following rule can be stated.

**Case 3:** If both waves A and B undergo a phase shift,
constructive interference occurs when:

or

**t _{film} = 0.5 • m •
_{film}where
m = 1, 2, 3, ...**

The equations above for cases 1, 2 and 3 represent the relationships between the film thickness which would be required to reinforce light with particular wavelength. The equations form the basis for most of the Set B problems. The strategy for approaching the problems follows from the equations as follows

Step 1: Identify the material the film consists of and then determine the wavelength of light in the film.Step 2: Determine if Wave 1 and/or wave 2 undergo a phase shift upon reflection.

Step3: Based on step 2, determine if the physical situation corresponds to Case 1, 2, or 3; then utilize the corresponding equation to solve for the unknown quantity.

**Interference by Wedged-Shaped
Films**

The previous discussions of thin films have pertained to the interference by thin films of uniform thickness. Now we will develop the mathematics for cases in which the film has a varying thickness. The diagram below shows an air film formed when two flat plates are put together with a material (e.g., paper, cloth, saran wrap, etc.) wedged in between at one end. The result is a thin wedge of air positioned between the glass plates. When the glass plates are illuminated with green light (or any other color), a thin film interference pattern consisting of alternating bright (green) and dark (black) lines can be observed. The thin film interference pattern results from the interference of a wave reflecting off the bottom of the top glass plate (top of the air film) with a wave reflecting off the top of the bottom glass plate (bottom of the air film). The thickness of the air film between the glass plates is different for each bright line. The diagram expresses these thicknesses in terms of the wavelength of green light. The thickness of the material (paper, cloth, saran wrap) is related to the number of bright bands seen between the plates and to the wavelength of green light.

The derivation of the relationship between the number of bands seen along the glass plates and the wavelength of light and thickness of the object demands an understanding that destructive and constructive interference occur only for certain path difference values. Of the two waves reflecting back up to our eye, one of the waves would undergo an inversion (180-degree phase shift). This results in constructive interference (observable bright bands) for path difference values of half-number of wavelengths and destructive interference (dark bands) for path difference values of whole-number of wavelengths. The path difference is simply the difference in distance traveled for a wave reflecting off the top of the air wedge compared to a wave passing through the air wedge and bouncing off the bottom of the air wedge. This path difference distance is simply twice the thickness of the wedge at the location where a bright or a dark band is observed. This mathematical relationship can be expressed mathematically as

**m = 0.5, 1.5, 2.5, 3.5, ... (constructive
interference)**

**m = 0, 1, 2, 3, 4, ... (destructive
interference)**

The diagram below depicts the wedge of air with varying thickness along the length of the glass plates. A side view of the wedge is accompanied by a top view in which the dark and bright bands would be observed where the path difference criteria above are met. Observe that the path difference is twice the air or wedge thickness. And observe that destructive interference (D) or dark bands are observed for path differences equal to whole-number of wavelengths. Similarly, constructive interference (C) or bright bands are observed for path differences equal to half-number of wavelengths.

Since each consecutive bright band (and each consecutive dark band) is the result of the wedge becoming thicker by a uniform amount, it is possible to relate the number of bands in the interference pattern to the thickness of the film at the location of the last band. Since the thickness increases by one-half of a wavelength for each consecutive bright band (or dark band), then the thickness can be related to the number of bright (or dark bands) and one-half a wavelength. In the case of a film wedge trapped between two more dense materials (as shown above), one can conclude that the thickness-wavlength relationship would fit with the following equation

where n is the number of dark bands (not including the first dark band at location A in the diagram) or the number of bright bands.

To be successful with this collection of problems, you will need to:

- employ the suggested strategy for approaching thin film problems.
- understand the equations which relate to thin films and wedge-shaped films.
- be able to convert the units on distance or length quantities;
this includes an understanding of the following relationships:
10
^{9}nm = 1 m; 10^{6}micrometer = 1 m .

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