Geometric Optics Lecture Notes - Biophysics 1st Year

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Biophysics
Table of contents
Introduction 3
I - Geometric Optics Principles 4
1. Rectilinear Propagation of Light Principle 5
1.1. Light as an Electromagnetic Wave (Key Charactristics) 5
1.2. Refraction Index of a Medium 6
1.3. Cauchy's Law 7
2. Snell-Descartes' Laws 7
2.1. Snell's Laws 8
2.2. Deviation of Light 9
2.3. Total Internal Reflection - Critical Angle 9
2.4. Dispersion of Light 11
3. Notions of Objects/Images and Stigmatism 11
3.1. The Optical System 11
3.2. Notion of Objet / Image 12
3.3. Stigmatism and Approximate Stigmatism 13
II - Geometric Optics Elements 15
1. Planar Systems 15
1.1. Refractive Systems: Planar Diopters 15
1.2. Reflective Systems: Plane Mirrors 17
2. Spherical Systems: Dioptres 17
2.1. Geometrical Construction Of an Image 18
2.2. The Conjugation Relation 19
2.3. Magnififaction 20
3. Lenses 20
3.1. The Optical Center 21
3.2. Thin Lenses 21
3.3. Types of Lenses 22
3.4. The Conjugation Relation 23
3.5. Focal Length 24

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3.6. Magnification 24
3.7. Geometric Construction of an Image 25
3.8. Adjacent Lenses 25
Glossary 26
Abbreviation 27
Introduction
Optics Overview:
Optics is the branch of physics that studies light, its behavior, and properties.
It includes models like geometric optics, wave optics, and quantum optics, which are
complementary.
Geometric Optics:
Focuses on light propagation in straight lines (rays) and its interaction with media via
reflection and refraction.
Useful for explaining image formation in refractive and reflective systems.
The geometric optics approximation applies when the wavelength of light is much
smaller than the dimensions of the objects it interacts with.
If we denote D as the characteristic dimension of the obstacles (medium light interacts
with), and λ as the wavelength, the geometric optics approximation holds if D >> λ.
It is taught as part of the Geometrical Optics lecture in the Physics-Biophysics course,
covered over three chapters in the first semester. These chapters are clearly outlined in
the following Mind Map (cf. New Map1-copy.pdf):
I Geometric Optics Principles
Introduction
What is Light?
The first serious theories regarding the nature of light were formulated in the 17th
century. Two seemingly contradictory theories emerged: one based on the corpuscular
aspect and the other on the wave mechanism. These theories sparked a controversy that
lasted until the early 20th century, as each explained some phenomena but left others
unexplained or even contradicted by them.

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Newton's corpuscular theory considered light as a collection of particles (without
specifying their nature). This theory, however, did not explain interference phenomena
(observed by Newton), where light superposition can produce darkness (overlapping
light can create darkness), nor diffraction phenomena, where light appears in geometric
shadowed areas.
The corpuscular theory was only
fully developed in the early 20th
century with Einstein (1905) and
Compton (1921), who described light
as energy quanta called photons that
follow conservation laws of
mechanics.
The wave theory was proposed in
1665 by Hooke. Young and Fresnel
further developed it by explaining
light wave interferences and
associating wave frequency with color.
Despite its strengths, the wave theory couldn't explain certain phenomena, such as the
photoelectric effect, where electrons are expelled from a metal plate exposed to light.
In 1864, Maxwell introduced electromagnetic waves, giving the wave theory its definitive
form:
light is an electromagnetic wave. According to this theory, light is an electromagnetic
wave made up of electric and magnetic fields that vibrate at a frequency ν and move
together at the speed of light, c. In vacuum, this speed is c = 3 × 108 m/s.
Maxwell's electromagnetic theory does not limit the frequency of electromagnetic waves.
The electromagnetic spectrum ranges from "radio waves" to "gamma rays," with visible
light (0.4 μm < λ < 0.8 μm) occupying only a small part of this spectrum.
(1 μm = 10−6m).
1. Rectilinear Propagation of Light Principle
At the heart of geometric optics is the principle of rectilinear propagation, which states
that:
In a transparent, homogeneous, p.26 and
isotropic medium (like air or vacuum p.26),
light propagates in the form of straight-line
rays.
This behavior can be easily observed when a
flashlight is shone in a dark room, producing
a straight beam of light
A medium is said to be:
Transparent: if it allows light to pass through without attenuation; otherwise, it is called
opaque.
Homogeneous: when the refractive index is the same at all points; otherwise, it is called
inhomogeneous.
Isotropic: when the refractive index is the same in all directions; otherwise, it is called
anisotropic.

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Note :
Other old principals that are worth mentioning which are based on simple common
observations such as the formation of shadows or solar eclipses include:
Principle of the Reversibility of Light: If light follows any path from point A to point B
(including through an optical system), then light can follow exactly the same path in
reverse from B to A. It is said that the path followed by the light is independent of the
direction of its propagation.
Principle of Independence of Light Rays: When two light rays meet, they do not interact:
one light ray cannot be deflected by another light ray.
1.1. Light as an Electromagnetic Wave (Key Charactristics)
Transverse Nature: Light is an electromagnetic (EM) wave, consisting of oscillating
electric and magnetic fields perpendicular to each other and to the direction of
propagation.
Speed (c): In vacuum, all EM waves travel at the speed of light, c = 3 × 108 m/s.
Wavelength (λ): Distance between two successive peaks of the wave; typically measured
in meters (m), nanometers (nm) for visible light.
Frequency (ν): Number of oscillations per second; measured in hertz (Hz). Related to
wavelength by c = λν.
Period (T): Time for one complete oscillation; T = 1/ν.
Energy (E): Photons carry energy proportional to frequency, E = h ⋅ ν, where h is
Planck's constant (6.626 × 10−34 J⋅s).
Note :
A detailed discussion of these properties can be found in Chapter 1: Fundamentals of
Radiation Physics, in the Biophysics of Radiation course.
1.2. Refraction Index of a Medium
While the rectilinear propagation of light principle generally holds, an intriguing
question arises:
Does light always stick to a straight path?
Answer: Yes and No!
Light does follow a straight path as long as the medium’s properties remain unchanged.
Let's explore the situations where the straight-line path of light changes.
Light exhibits fascinating behaviors when it encounters different surfaces or media.
In fact, the electromagnetic wave, and thus light, propagates in a vacuum at the speed
c = 3×108 m/s. Its propagation in a medium, however, gets affected by the refractive
index n of that medium, due to which, the velocity of the EM wave slows down to a value
v < c. As a consequence, light changes its direction: it either reflects or refracts.
The refractive index of a medium is defined as the ratio between the speed of light in a
vacuum C and its speed in that medium v:
The refractive index is a dimensionless physical parameter that is greater than or equal
to 1 (n ≥ 1).

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Example :
1.3. Cauchy's Law
While the behavior of a light ray is governed by the medium it passes through, another
intriguing question arises:
Do all wavelengths of light behave uniformly in a given medium? And does the refractive
index interact uniformly with all wavelengths of light?
The answer is revealed through Cauchy's Law, which states that the refractive index of a
medium (n) varies with wavelength.
Cauchy Law is an empirical relation that gives the refractive index n as a function of the
wavelength of light for a transparent medium
Where:
A and B are experimental constatnts that depend on the medium.
λ0 is the wavelength in the vacuum.
Specifically, it indicates that the refractive index decreases as the wavelength increases,
leading to different degrees of bending for various colors of light. This results in a
phenomenon, known as dispersion, in the case of polychromatic radiation (see below).
2. Snell-Descartes' Laws
when encountering a surface separating two media (with two different refractive
indices), Snell's Law, or Snell-Descartes Law, explains how light rays bend as they
transition from one medium to another.
Reflection occurs when light bounces off a surface, following the law of reflection. This
phenomenon is easily observed when looking into a mirror.
Refraction occurs when light moves from one medium to another, like from air to water
or glass. The change in speed as light enters a new medium causes it to bend, following
the law of refraction, with the degree of bending determined by the medium's refractive
index. A common example of refraction is a straw looking bent or broken when it's partly
submerged in a glass of water.
A diopter is the surface separating two transparent media with different refractive
indices, through which, light can either be refracted or reflected. It is more often of a
form planar or spherical.
2.1. Snell's Laws
Note :
Light's behavior depends on the type of the surface (optical system) it encounters:
In a diopter, which is a surface separating two transparent media with different
refractive indices, light can undergo both reflection and refraction. Refraction occurs as
light passes through the diopter, bending according to the material's refractive index,
while some portion of light is also reflected off the surface. A diopter is more often of a
form planar or spherical.