Fundamental Physics II - Optics

From USTH Learning Support

April 2025

Contents
I. Geometrical Optics 3

1 Reflection and Refraction 3

1.1 Laws of reflection and refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Chromatic Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Total internal reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Reflection and Refraction at a plane surface 5

2.1 Image formation by a plane mirror . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Image by plane reflecting and refracting surface . . . . . . . . . . . . . . . . . . . 5
2.1.2 Find location of the virtual image P’ . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Sign rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Magnification (m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Reflection and Refraction in Spherical surface 8

3.1 Sign rule for radius of curvature of a spherical surface (R) . . . . . . . . . . . . . . . . . 8
3.2 Reflection in spherical surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.3 Focal point and focal length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.4 Refraction in spherical surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.4.1 Graphical methods for mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.5 Graphical methods for mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 Optical Devices 11
4.1 Thin lenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.1.1 Graphical methods for lenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.2 Camera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.2.1 Focal length of the camera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.2.2 f-Number of a lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2.3 Zoom Lenses and Projectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.3 Eyes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.3.1 Myopic (near-sighted) eye . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.3.2 Hyperopic (far-sighted) eye . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.3.3 Astigmatism eye . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.3.4 Diopter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.3.5 Magnifying glass (simple magnifier) . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.4 Microscope and telescope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.4.1 Microscopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.4.2 Telescopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

II. Wave Optics 20

5 Electromagnetic waves 20

6 Electromagnetic spectrum 20

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7 Dispersion (also in 1.2) 21

8 Polarization 22

9 Interference 23

10 Diffraction 25

11 Diffraction grating 27

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I. Geometrical Optics
Geometrical optics is the study of properties of light under the approximation that light travels in
a straight line.

1 Reflection and Refraction

1.1 Laws of reflection and refraction
When light hits the boundary between two materials, part of it is reflected and the other part is
refracted. Reflection is when light bounces back off a surface. Refraction is when light changes
direction as it enters another material, because of a change in the refractive index.

Figure 1: A beam of light is reflected and refracted by a horizonal water surface.

Reflection and refraction are governed by two laws:

• Law of reflection: The angle of incidence and the angle of reflection are equal (both angles are
relative to the normal).
θi = θr (1)

• Law of refraction (also called Snell’s Law): Relation of the incidence angle and the refracted
angle are given by:
n2 sin θ2 = n1 sin θ1 (2)
where n1 , n2 are refractive indices.

Refractive index are defined as the ratio of the speed of light in a vacuum to its speed in a specific
medium.
c
n= (3)
v
where c is the speed of light and v is the speed of light in a material. Some refractive indices
4
that you should memorize: nair = 1, nwater = = 1.33
3
REMARK: Light traveling from medium a to medium b:

3
 • If na > nb , then the light ray is bent away from the normal.
• If na < nb , then the light ray is bent toward the normal.

Figure 2: Refraction of light

1.2 Chromatic Dispersion

• The refractive index encountered by light in a medium (except vacuum) depends on the wave-
length of light. That means different wavelengths will have different refractive indices in a
medium.
• If a beam of light is made up of many wavelengths, each wavelength will be refracted at different
angle when it enters the new medium, leading to a spread-out of light, called chromatic dispersion.
• Generally, the refractive index is higher for a shorter wavelength.

We have the velocity of light given by the equation: v = f λ, where f is constant.

v c
⇒f = = (λ0 is the wavelength of light in vacuum)
λ λ0
c
We also have n = , so:
v
λ0
⇒λ= (4)
n

For example, blue light (4̃50 nm) has shorter wavelength than red light (6̃20 nm), so the blue ray will
be more bent than the red ray.

1.3 Total internal reflection

Total internal reflection is when all the light is reflected and none is refracted. It happens with two
conditions:
• The light travels from a medium with higher refractive index (na < nb )
• The incidence angle is larger than the critical angle, which is defined by:
nb
sin θcrit = (5)
na
This equation is derived from Snell’s Law with refracted angle is:

θb = 90◦

⇒ na sin θcrit = nb sin θb = nb sin 90◦ = nb

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 Figure 3: Total internal reflection

2 Reflection and Refraction at a plane surface

Concept of object in optics: anything from which light rays radiate.

• Light emitted by the object itself (Self-luminous objects).

• Light from another source, reflected by the object.

2.1 Image formation by a plane mirror

2.1.1 Image by plane reflecting and refracting surface

Light rays from the object strike a smooth,

plane reflecting surface. The directions of the
reflected rays appear to originate from point
P ′ . We call P the object point and P ′ the
image point. If the surface is not smooth,
the reflection will not be specular but diffuse,
and there is no definite image point P ′ .

Figure 4: Image by reflecting plane surface

Light rays from the object are refracted at

the boundary of two materials. When the an-
gles of incidence are small, the final directions
of the rays after refraction are the same as
though they had come from an image point
P

Figure 5: Image by refracting plane surface.

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In both figures, the rays do not actually pass through the image point P ′ . We have the concepts of
real and virtual images:
• Real image: the outgoing rays really do pass through an image point.
• Virtual image: the outgoing rays do not pass through the image point.

2.1.2 Find location of the virtual image P’

Figure 6: Construction for determination of the location of the image formed by a plane mirror

The diagram shows two rays diverging from an object point P at a distance s to the left of a plane
mirror. We call s the object distance. When we extend the two reflected rays backward, they intersect
at point P ′ , at a distance s′ behind the mirror. We call s′ the image distance.
It can be easily shown that the image distance is equal to the object distance:

|s| = |s′ |

The sign of s and s′ , however, will have to follow sign rule. (Spoiler alert: s = −s′ )

2.2 Sign rule

1. Object distance s:
• s > 0 when the object is on the same side with the incoming light.
• s < 0 when the object is not on the same side with the incoming light.
2. Image distance s′ :
• s′ > 0 when the image is on the same side with the outgoing light.
• s′ < 0 when the image is not on the same side with the outgoing light.

For plane mirror: s = −s′

2.3 Magnification (m)

Lateral maginification: ratio of the height of image to the height of object.

y′
m= (6)
y
• m > 0 ⇒ The image is erect.

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 • m < 0 ⇒ The image is inverted.
For plane mirror: Lateral magnification: m = 1 (unity).

Figure 7: Height of image formed by reflection at at plane reflecting surface

CAUTION: For a three-dimensional object, only the front–back dimension is reversed (PR and P’R’).

Figure 8: Only PR and P’R’ are in opposite directions.

An important property of all images formed by reflecting or refracting surfaces is that an

image formed by one surface can serve as the object for a second surface.
In this case (Figure 9), the image P1′ formed by mirror 1 can act as the object for mirror 2 to produce
the image P3′ . Similarly, the image P2′ formed by mirror 2 can serve as the object for mirror 1.

Figure 9: Image P3′ is formed by a double reflection of each ray.

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3 Reflection and Refraction in Spherical surface
3.1 Sign rule for radius of curvature of a spherical surface (R)
• R > 0 when C is on the same side as the outgoing light.

• R < 0 when C is not on the same side as the outgoing light.

Figure 10: The sign rule for the radius of a spheical mirror.

3.2 Reflection in spherical surface

We apply Law of reflection to every incidenct ray from point P to mirror. Then all the reflected rays
will intersect at point P’, which is the real image of point P.

Figure 11: Construction for finding the position P of an image formed by a concave spherical mirror.

The relation of object and image (relating s, s′ and R):

1 1 2
+ = (7)
s s′ R
When R = ∞, the mirror becomes plane mirror.

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3.3 Focal point and focal length
When s = ∞, the incoming rays are so far away that they can be considered parallel to the optical axis.

Figure 12: All parallel rays incident on a spherical mirror reflect through the focal point.

In this case, the reflected rays will converge at a point called the focal point (F), and the distance
from the mirror’s vertex to point F is called the focal length (f ).
R
f= (8)
2

Substituting s = ∞ to Equation (7):

R R
s′ = =⇒ s′ = f = (9)
2 2

R
Reversely, when s′ = f = , we will have s′ = ∞.
2
We also have the important relation between s, s′ and f :
1 1 1
+ ′ = (10)
s s f

3.4 Refraction in spherical surface

3.4.1 Graphical methods for mirrors
As we already know, a spherical surface divides space into two media: one with refractive index na ,
which contains the incoming light, and the other with a higher refractive index nb . After refraction at
the spherical surface, the rays converge to form a real image located at a distance s′ from the surface.
From that we can derive a formula that relates s′ , s, the radius of curvature R and the refractive
indices na and nb :
na nb nb − na
+ ′ = (11)
s s R
Note: Sign of s′ , s, and R follow the sign rule. Example: Figure 13

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Figure 13: Construction for finding the position of the image point P of a point object P formed by
refraction at a spherical surface.

Magnification of spherical surface:

y′ na s′
m= =− (12)
y nb s

Special case: R = ∞, spherical surface becomes a plane surface, then we have the plane refracting
surface equation:
na nb
+ ′ =0 (13)
s s
Magnification in this case is m = 1 (the image formed by the refracting plane is always erect and of
the same orientation as the object).

3.5 Graphical methods for mirrors

1. A ray parallel to the axis, after reflection, passes through the focal point F of a concave mirror
or appears to come from the (virtual) focal point of a convex mirror.
2. A ray through (or proceeding toward) the focal point F is reflected parallel to the axis.

3. A ray along the radius through or away from the center of curvature C intersects the surface
normally and is reflected back along its original path.
4. A ray to the vertex V is reflected, forming equal angles with the optic axis.

Figure 14: The graphical method of locating an image formed by a spherical mirror.

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4 Optical Devices
4.1 Thin lenses
Definition: A thin lens is a lens with a small thickness compared to its radius of curvature. 2 types:
• Convex lens (Converging lens): A lens that causes light rays initially parallel to the central axis
to converge
• Concave lens (Diverging lens): A lens that causes parallel rays of light to diverge.
Equations:
• Object-image relationship, thin lens:

1 1 1
+ ′ = , (14)
s s f
where s: object distance, s′ : image distance, f : focal length of lens.
• Lensmaker’s equation for a thin lens:
1 1 1
= (n − 1)( − ) (15)
f R1 R2
where f : focal length, n: index of refraction of lens material, R1 : the radius of curvature of the
lens surface nearer the object and R2 : the radius of curvature of the second surface.
Some variations of converging lenses and diverging lenses:

Figure 15: Various types of lenses.

For Converging lenses:

• The image is real (s′ > 0) if s > f
• The image is imaginary (s′ < 0) if s < f
For Diverging lenses: The image is always imaginary, erect and smaller than the object.

4.1.1 Graphical methods for lenses

1. A ray parallel to the axis emerges from the lens in a direction that passes through the second
focal point F2 of a converging lens, or appears to come from the second focal point of a diverging
lens.
2. A ray through the center of the lens is not appreciably deviated; at the center of the lens the
two surfaces are parallel, so this ray emerges at essentially the same angle at which it enters and
along essentially the same line.
3. A ray through (or proceeding toward) the first focal point F1 emerges parallel to the axis

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 Figure 16: Graphical method of locating an image formed by thin lens.

4.2 Camera
A camera inclues 5 main parts:
• Light-tight box or camera body
• Converging lens
• Shutter
• Light-sensitive recording medium (electronic sensor or photographic film): can be understood as
a surface for capturing light and converting it into an image.
• Aperture: the opening of the lens that allow the amount of light passes through.
Since being created by a converging lens, the image produced by the camera is always a real image,
inverted, and usually smaller than the object.

4.2.1 Focal length of the camera

As we know, the focal length is the distance from the lens to the image when the object is at an
infinite distance. Thus, a camera with optical lenses that have a longer focal length will have a greater
distance to the image. Combined with magnification m (lateral magnification), a longer focal length
will result in a larger image.

Figure 17: Three photographs taken with the same camera from the same position, using lenses with
different focal lengths.

• Telephoto Lens:

– Lenses with a long focal length are called telephoto lenses.

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 – They provide a smaller field of view but a larger image of a distant object.

• Wide-angle lens:

– Microscopes with a shorter focal length are called wide-angle lenses.

– They offer a wider field of view but produce a smaller image.

4.2.2 f-Number of a lens

f
f -number of a lens = (16)
D
where f is focal length of lens and D is aperture diameter
Examples:
1. A lens with a focal length f = 50 mm and an aperture diameter D = 25 mm

50 f
f-number = =2 or we can say it has an aperture of
25 2
2. A lens with focal length f from 100 mm to 200 mm, an aperture diameter D1 = 25 mm at
f1 = 100 mm, and D2 = 35.7 mm at f2 = 200 mm.

f1 -number = 4, f2 -number = 5.6

f f
⇒ the aperture of the lens is −
4 5.6
Note: The values 2, 4, 5.6 are called f-stops. The smaller the f-stop number, the larger the aperture
f f
of the lens. For example: >
4 5.6
Characteristic of f-Number: The light intensity reaching the film is inversely proportional to
the square of the f-number.
1
I= (17)
(f -number)2
√ √
Example: if D increases by 2 times then f-Number decreases by 2 times and the intensity of light
increases by 2 times.

Figure 18: Changing D and f-Number

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4.2.3 Zoom Lenses and Projectors
Zoom Lens A zoom lens is a complex collection of several lens elements that gives a continuously
variable focal length, often over a range as great as 15 to 1. Its operation can be summarized as:

• When the two key lenses are close together, they function like a single lens with a longer focal
length.
• When the two lenses move farther apart, they behave like a single lens with a shorter focal length.

Figure 19: A simple zoom lens uses a converging lens and a diverging lens in tandem

Projector A projector—used to display slides, digital images, or moving pictures—works much like a
camera in reverse: it shines light through an optical system to project an enlarged image (real, m > 1,
inverted) onto a screen.

4.3 Eyes
• Sphere-like shape with a diameter of about 2.5 cm.
• To be able to observe objects from different distances, the eyes will adjust their focal length. So,
s can vary while s′ remains constant (the image must form on the retina, and the distance
from the lens to the retina does not change).

• An object at infinity (very far away) is brought into focus when the ciliary muscles are relaxed,
allowing the lens to assume its minimum power (longest focal length).
• As an object moves closer, the ciliary muscles contract, reducing the tension on the zonular fibers
around the lens. This allows the lens to become more convex, thereby shortening its focal length.

Figure 20: Diagram of the eye

The eye forms a real image: refraction at the cornea and the surfaces of the lens.
For normal eye:

• The far point is at infinity.

• The near point is 25 cm.

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Figure 21: (a) An uncorrected myopic (nearsighted) eye. (b) A negative (diverging) lens spreads the
rays farther apart to compensate for the excessive convergence of the myopic eye.

4.3.1 Myopic (near-sighted) eye

• In myopia (nearsightedness), the eyeball is too long relative to the curvature of the cornea (or
the cornea is too curved).
• As a result, incoming parallel light rays converge to a focal point in front of the retina rather
than on it.
• The far point of a myopic eye is therefore at a finite distance (objects beyond that point appear
blurred), and the near point is closer than normal.

• Solution: wear a divergent (concave) lens with focal length f = −dfar , where dfar is the
distance to the far point of the myopic eye. This lens shifts the focal point back onto the retina
so that distant objects can be seen clearly without accommodation.

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 Figure 22: One type of astigmatism and how it is corrected

4.3.2 Hyperopic (far-sighted) eye

• In hyperopia (farsightedness), the eyeball is too short relative to the curvature of the cornea (or
the cornea is insufficiently curved).
• Consequently, incoming parallel light rays are focused at a point behind the retina rather than
on it.

• The near-point of a hyperopic eye is therefore located farther away than normal (objects closer
than this point appear blurred).
• Solution: wear a converging lens (convex), which brings the focal point forward so that
images of near objects form on or just inside the near point of the eye.

4.3.3 Astigmatism eye

• In astigmatism, the cornea surface is not perfectly spherical but has different curvatures in
different planes, causing incoming light rays to focus over a line or area rather than at a single
point on the retina.
• As a result, images appear blurred or distorted because rays in one plane (e.g., vertical) come to
focus at a different distance than rays in the perpendicular plane (e.g., horizontal).

• A person with astigmatism may be unable to focus both the vertical and horizontal edges of a
window frame sharply at the same time.
• Solution: a cylindrical lens, which has focusing power in only one meridian.
• Explanation: By orienting the cylinder axis appropriately, the lens leaves rays in one plane
unchanged while adding or subtracting power in the perpendicular plane, so that both meridians
are brought into sharp focus on the retina.

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4.3.4 Diopter
Lenses used for vision correction are characterized by their optical power, called diopters, defined as
the reciprocal of the focal length (in meters):
1
P = (18)
f

Examples:
• A lens with f = +0.50 m has P = +2.0 D.
• A lens with f = −0.25 m has P = −4.0 D.

4.3.5 Magnifying glass (simple magnifier)

• Definition: A magnifier is a single converging lens used to produce a virtual image that is both
larger than the object and appears farther from the eye than the object itself.
• Angular size: The apparent size of an object is determined by the angle θ it subtends at the
eye. When an object is held close to the eye (within the near point), its angular size increases,
making it appear larger.

As we learned, a virtual image is most comfortable to observe when it is at infinity, at which point
the ciliary muscle of the eye is relaxed. Therefore, in the figure above, we place the object at the focal
point f of the magnifying glass, so that the image formed is at infinity. This virtual image can then
be treated as a real object subtending an angle θ′ at the eye. We have:
y y
θ′ = = (19)
s f

Angular magnification M for a simple magnifier:

θ′ y/f 25cm
M= = = (20)
θ y/25cm f
where θ′ is the angular size of object seen with magnifier, θ is angular size of object seen without
magnifier, y is the object height, and 25 cm is the Near point.

4.4 Microscope and telescope

As mentioned above, we discussed optical instruments that use only a single lens to form an image.
However, there are two additional optical instruments — the microscope and the telescope — that
employ two lenses to produce images. In these instruments:
• The objective lens is used to form a real image.
• The eyepiece acts as a magnifier, creating an enlarged virtual image.

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4.4.1 Microscopes

Figure 23: The microscope

• To understand how an image is formed by telescopes, we apply the image formation rule of a
single lens that the image formed by the first lens acts as the object for the second lens.
• Basically, the object O is placed just before the focal point F1 (close enough that we can approx-
imate f1 ≈ s1 ) of the objective lens, producing a real, inverted, and enlarged image I (located
at the focal point F2 of the eyepiece). The eyepiece then acts as a magnifying glass to produce
a virtual image l′ .
• Just like with a magnifying glass, we are interested in the angular magnification M . The
angular magnification of a simple microscope can be calculated as the product of the lateral
magnification m of the objective lens and the angular magnification M of the eyepiece.
• We have the magnification:

s′1 s′1
m= , with f1 ≈ s1 ⇒ m =
s1 f1

Eyepiece magnification:
25
M=
f2
⇒ Angular magnification for a microscope:

25s′1
M′ = M × m = (21)
f2 f1

4.4.2 Telescopes
• Basically, a telescope operates similarly to a microscope.
• The main difference between these two optical instruments is that telescopes are used to observe
large objects at great distances, while microscopes are used to observe small objects close
at hand.

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 • Additionally, some telescopes use curved mirrors (concave mirrors) as their objective lens.

Figure 24: Optical system of an astronomical telescope

• In this case, the telescope uses a converging lens as its objective lens, so it is also called a
refracting telescope or refractor.
• The image formed by the objective lens is real and smaller than the object. Since the object is
assumed to be at infinity, we consider l ≈ F1′ .

• The eyepiece acts as a magnifier that creates a larger virtual image from the real image I ′ , and
since I ′ is very close to the eyepiece, we can assume I ′ ≈ F2 .
• The total length of the telescope is f1 + f2 .

Angular magnification M of a telescope is defined as the ratio between the final angular size of the
image θ′ and the angular size of the object θ:

−y ′ y′
θ= , θ′ =
f1 f2
θ′ y ′ /f2 f1
M= =− ′ =− (22)
θ −y /f1 f2

Note: For telescopes using a concave mirror as the objective lens, they are called reflecting
telescopes.

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II. Wave Optics

5 Electromagnetic waves

A time-varying magnetic field induces an elec-

tric field, and conversely, a time-varying elec-
tric field induces a magnetic field.
This continuous mutual induction repeats,
forming the mechanism by which electromag-
netic waves are generated.

Figure 25: EM waves formation

Imagine if we throw a stone into a pond, it will create ripples that spread out through the surrounding
medium. Similarly, if we take a magnet and “stir” it around, it will create ripples in the electromagnetic
field.

6 Electromagnetic spectrum
The fundamental nature of light is that our eyes can detect electromagnetic waves with wavelengths
in the range 0.4–0.7 µm. Waves with wavelengths longer or shorter than this range are not visible to
us.

Figure 26: Visible wavelengths

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 Within the visible region, each wavelength
corresponds to a distinct perceived color (as
in the table of spectral colors). What will we
see if we mix all the colors together?

Figure 27: Visible colors by wavelength

Answer: We have 3 primary colors: red, green
and blue. By mixing these 3 colors we can see
secondary colors:

• Yellow = red + green

• Magenta = red + blue
• Cyan = green + blue
White is produced by mixing all three primary
colors. Black corresponds to the absence of
light (try closing your eyes to ”see” black).
Figure 28: Color mixing

7 Dispersion (also in 1.2)

Electromagnetic waves of different wavelengths propagate through any medium other than vacuum
(e.g., glass, water) with different refractive indices, causing each wavelength to refract by a different
angle.
• As a result, a polychromatic beam is split into its monochromatic components.
• This principle is used in spectrometers, instruments that separate and identify the spectral
components of a light beam.

• The same effect produces rainbows: instead of a prism, raindrops act to disperse sunlight into
its constituent colors.

Figure 29: Raindrop as prism that disperses light.

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8 Polarization
In this section, we focus on the Electric-field component of an electromagnetic wave.

• Polarized light: the electric field oscillates entirely within a single plane.

Figure 30: Polarized waves

The electric field is a vector quantity and can be added together. So if two polarized waves that
are perpendicular to each other are superimposed, what kind of wave do we have?
• Unpolarized light: the electric field oscillates in all directions perpendicular to the direction
of propagation; it can be viewed as a superposition of two orthogonal polarized waves.

• Polarizing filter: an optical element that blocks electric-field oscillations in one direction and
transmits those in the perpendicular direction.

– When unpolarized light passes through, the transmitted light is polarized and its intensity
is reduced by half.
– When polarized light with initial intensity I0 and electric-field oscillation at angle Θ to the
filter’s axis passes through, the transmitted intensity I follows Malus’s law:

I = I0 cos2 Θ. (23)

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 Figure 31: The polarizing filter

• Polarization by reflection
Reflection and refraction at interfaces can also alter the polarization state of light.

Figure 32: Polarization by Reflection

At a specific angle of incidence, called the polarizing angle θp , the reflected light is completely
polarized perpendicular to the plane of incidence.
According to Brewster’s discovery, when the reflected and refracted rays are perpendicular to
each other, i.e.,
θb = 90◦ − θp ,
then by Snell’s law:
na sin θp = nb sin θb = nb cos θp .

Dividing both sides by cos θp , we get the Brewster’s law:

nb
tan θp = (24)
na
which gives the angle of incidence θp at which reflected light is 100% polarized.

9 Interference
• Definition: The superposition of two or more coherent waves, resulting in a new wave pattern.
• Coherent waves: Waves having the same frequency ν, wavelength λ, and a constant phase
difference.

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• Constructive interference:

– Occurs when two waves arrive in phase at a point.

– Path difference: ∆r = r2 − r1 = m λ m ∈ Z , where r1 and r2 are distances from source
to the surface.

– Resultant amplitude: Ares = A1 + A2 .

• Destructive interference:

– Occurs when two waves arrive out of phase by π.

– Path difference: ∆r = r2 − r1 = m + 21 λ m ∈ Z .

– Resultant amplitude: Ares = A1 − A2 (zero if A1 = A2 ).

Figure 33: Constructive and Destructive waves

• Constructive and destructive two-slit interference:

Figure 34: Example of a two-slit interference experiment.

– Condition for bright fringes (constructive interference):

∆r = d sin θ = m λ, m ∈ Z. (25)

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 – Conditions for dark fringes (destructive interferences):

∆r = d sin θ = m + 21 λ, m ∈ Z.


(26)

Note: In the test, it might ask about the number of bright/dark fringes, which is m.

– Distance from the center to mth bright fringe:

ym = R tan θm

ym ≪ R ⇒ tan θm ≈ sin θm
⇒ ym = R sin θm
We finally have:
mλ
ym = R (27)
d
R is the very small distance between the screen and the light source, d is the distance
between 2 slits, λ is the wavelength.
– Amplitude in 2-source interference:

ϕ
Ep = 2E cos (28)
2

where E is the amplitude of the wave from one source, ϕ is the phase difference between
the waves, and EP is the resulting electric-field amplitude.

10 Diffraction
• Huygens–Fresnel principle:
Every point on a wavefront acts as a new source of waves (called secondary wavelets). The sum
of these wavelets determines the subsequent propagation and explains diffraction phenomenon.

Figure 35: Illustration of Huygen’s principle.

Note: In interference problems, we usually deal with very narrow slits. In reality, slits often have
significant width, and what we need to do in this section is to construct a more accurate model.

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• Single-slit diffraction: A slit of finite width a can be modeled as a collection of infinite number
of point sources, oscillating with a very, very small amplitude.

Figure 36: Single-slit from a phasor diagrams.

The intensity distribution on a distant screen is

 2
sin α 1 π a sin θ
I(θ) = Im , α= ϕ= , (29)
α 2 λ
where λ is the wavelength and θ the diffraction angle.

• Double-slit diffraction (narrow slits):

We have the intensity of:

– Two very narrow slits interfering with each other:

– A single slit with significant width causing diffraction:

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 Then 2 slits with significant width will be the combination of the two:

We have the formula for double-slit diffraction:

 2
2 sinα π d sin θ π a sin θ
I(θ) = Im cos β , β= and α = (30)
α λ λ

• Circular aperture (Airy pattern):

When light passes through a circular opening, it spreads out or diffracts. This diffraction
pattern forms a series of bright and dark rings, known as the Airy pattern.

Figure 37: Circular aperture

λ λ λ
The angular radii of bright rings: sin θ1 = 1.22 , sin θ2 = 2.23 , sin θ3 = 3.24 ,
2r 2r 2r
λ λ λ
The angular radii of dark rings: sin θ1 = 1.63 , sin θ2 = 2.68 , sin θ3 = 3.70 .
2r 2r 2r

11 Diffraction grating
• Diffraction grating:

– Consider a system of many very narrow slits illuminated by a white-light beam.

– Each slit acts as a secondary source, and the diffracted waves interfere.
– At an observation angle Θ, constructive interference for wavelength λ occurs when

d sin Θ = m λ, m ∈ Z,

where d is the slit spacing.

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 Figure 38

Figure 39: Diffraction grating

• X-ray diffraction (Bragg diffraction):

– Similar to diffraction grating but instead of letting light goint through the slits, we let
X-ray reflect from the atomic planes to obtain images.
– For a glancing angle Θ, maximum reflected intensity (Bragg peak) satisfies

2 d sin Θ = m λ, m∈Z (31)

where d is the interplanar spacing.

Figure 40: X-ray diffraction.

– This method is used to determine crystal structures.

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