Unit 1 - Geometrical optics | 1st Semester Bachelor of Optometry

 

Introduction to Geometrical Optics in Optometry

Geometrical optics, also known as ray optics, is a fundamental branch of optics that deals with the behavior of light as it travels through different media. It assumes that light moves in straight lines called rays and explains optical phenomena such as reflection, refraction, image formation, and the design of optical instruments using these rays.

In optometry, geometrical optics plays a vital role in understanding how light interacts with the human eye and corrective lenses. From diagnosing refractive errors like myopia and hypermetropia to designing spectacles, contact lenses, and advanced visual aids, the principles of geometrical optics form the scientific foundation of clinical optometric practice.

Key Applications in Optometry

• Understanding Vision: How the eye focuses light on the retina and the causes of blurred vision.
• Corrective Optics: Designing lenses to correct refractive errors using vergence and refraction principles.
• Optical Instruments: Functioning of devices like the retinoscope, phoropter, and slit lamp, which rely on reflection and refraction.
• Practical Skills: Applying mirror and lens formulae to determine image location, size, and magnification in clinical setups.

Unlike physical optics, which deals with wave phenomena like interference and diffraction, geometrical optics simplifies analysis by treating light as rays. This makes it an ideal model for optometric calculations and vision correction techniques.

In the following sections, we will explore key concepts like the nature of light, wavefronts, refractive index, and principles of reflection/refraction as part of Unit 1 of the syllabus. Each topic will be explained in detail with diagrams and clinical relevance to optometry.

1. Nature of Light – A Foundation for Geometrical Optics

Light is a form of electromagnetic radiation that is visible to the human eye and essential for vision. In geometrical optics, we study light by treating it as a stream of rays that travel in straight lines and obey laws of reflection and refraction. However, to fully understand its nature, it is important to explore its electromagnetic wave properties, including oscillations, amplitude, phase, and speed in different media.

1.1 Light as an Electromagnetic Oscillation

Light consists of oscillating electric and magnetic fields that are perpendicular to each other and to the direction in which the light is propagating. This transverse wave behavior is what categorizes light as an electromagnetic wave. These oscillations are sinusoidal in nature and can vary in frequency and wavelength, giving rise to different colors in the visible spectrum.

Although geometrical optics simplifies light as rays, the wave nature becomes important in understanding phenomena such as optical resolution, glare, and diffraction, especially in clinical situations like retinal imaging or corneal topography.

1.2 Sinusoidal Oscillations: Amplitude and Phase

A sinusoidal wave is defined by its amplitude (height of the wave), which determines the brightness or intensity of light, and phase (position of a point in the wave cycle), which plays a role in wave interference and coherence. In optometry, phase differences can impact binocular vision and stereoacuity assessments.

• Amplitude: Higher amplitude = brighter light
• Phase: Determines how waves interact (constructive or destructive interference)

1.3 Speed of Light in Vacuum and Media

In a vacuum, light travels at a constant speed of approximately 3 × 10⁸ meters/second. However, when light passes through materials such as air, water, glass, or cornea, it slows down due to interactions with particles in the medium. This change in speed causes light to bend or refract.

The speed of light v in a medium is given by:

v = c / n

Where:

• c = speed of light in vacuum
• n = refractive index of the medium

This slowing of light is the basis for refractive errors in the eye. For instance, if the axial length of the eye is too long or too short for its optical power, light does not focus precisely on the retina, causing myopia or hypermetropia.

1.4 Refractive Index (n)

The refractive index of a material is the ratio of the speed of light in vacuum to the speed of light in that material:

n = c / v

Each optical surface in the human eye — cornea, aqueous humor, lens, vitreous — has its own refractive index. Understanding these values is critical in clinical optometry for designing corrective lenses, intraocular lenses (IOLs), and in calculating intraocular power using biometry formulas.

Refractive Indices of Common Ocular Media:

• Cornea: ~1.376
• Aqueous Humor: ~1.336
• Lens (variable): ~1.39–1.41
• Vitreous Humor: ~1.336

These differences in refractive index cause the bending of light (refraction) and allow the eye to focus light sharply on the retina under normal circumstances.

Clinical Relevance in Optometry

• Understanding how light propagates and bends is the basis of prescribing correct lens powers.
• Knowledge of refractive indices is essential for interpreting keratometry, retinoscopy, and optical coherence tomography (OCT).
• Phase and amplitude knowledge helps in understanding optical coherence in advanced imaging techniques.

In summary, while geometrical optics simplifies light as rays for practical analysis, its wave nature and interaction with various media provide the foundation for understanding many clinical and diagnostic principles in optometry.

2. Wavefronts and Vergence: Understanding Light Propagation

In geometrical optics, we often talk about rays of light, but there is another important concept that helps explain how light spreads out — wavefronts. A wavefront is an imaginary surface that connects all the points of a light wave that are in the same phase — that is, vibrating together.

Think of it this way: if you drop a stone into a calm pond, ripples spread out in circles. Each circle is like a wavefront — a surface over which the wave has moved the same distance. In the case of light, wavefronts tell us how the light is moving — whether it's spreading out, coming together, or traveling straight.

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2.1 Types of Wavefronts

There are three main types of wavefronts in optics:

1. Spherical Wavefront
2. Elliptical Wavefront
3. Plane Wavefront

1. Spherical Wavefront

A spherical wavefront is produced by a point source of light. The light spreads out equally in all directions, forming concentric spherical shells around the source.

Example: Light coming from a torch bulb or the retina’s photoreceptors (point sources).

2. Elliptical Wavefront

These wavefronts occur when light is emitted from an elliptical source or focused through specific optical systems like elliptical mirrors. They are less common in basic geometrical optics and more in advanced systems.

Example: Special lenses that correct aberrations or high-end optical equipment.

3. Plane Wavefront

If a light source is very far away (like the Sun), the spherical wavefronts become almost flat — this is a plane wavefront. All the rays in a plane wavefront are parallel to each other.

Example: Sunlight reaching the Earth or laser light in clinical optometry.

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2.2 Rays and Wavefronts

Rays are perpendicular lines drawn from a wavefront. If the wavefront is plane, the rays are parallel. If the wavefront is spherical, the rays spread out (diverge) or come together (converge).

Quick Tip: Rays are used to trace the direction of light. Wavefronts describe the shape of the light at any instant.

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2.3 Curvature and Vergence of Wavefronts

The curvature of a wavefront tells us how much it is bent. A very curved wavefront is from a near point source. A flatter wavefront means the source is far away.

Vergence is a term used to describe the degree to which light is converging (coming together) or diverging (spreading out). It's a very important idea in optometry!

Vergence (V) Formula:

V = ±100 / d

Where:

• V = Vergence in Diopters (D)
• d = Distance in centimeters from the point of focus to the wavefront

Positive vergence (+): Converging light (as in plus lenses)
Negative vergence (−): Diverging light (as in minus lenses)

Zero vergence: Parallel rays, i.e., a plane wavefront

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2.5 Convergence and Divergence in Terms of Rays and Vergence

Convergence and divergence describe the way light rays travel relative to one another, and they are directly related to the concept of vergence. These terms are essential for understanding how lenses correct vision problems.

Converging Rays

Converging rays are rays of light that are moving towards each other and eventually meet at a focal point. This usually happens after light passes through a convex (plus) lens. The wavefronts in this case are spherical and get flatter as they approach the focal point.

• Vergence (V): Positive (+), because the rays are focusing inward
• Clinical example: A hyperopic (farsighted) eye needs converging rays to focus light on the retina

Diverging Rays

Diverging rays are rays that spread apart as they move away from a point source or after passing through a concave (minus) lens. The wavefronts in this case become more curved as they move outward.

• Vergence (V): Negative (−), because the rays are spreading outward
• Clinical example: A myopic (nearsighted) eye needs diverging rays to reduce the focusing power of the eye

Parallel Rays

When light rays are neither converging nor diverging, they are called parallel rays. This occurs when the light source is at an infinite distance or has been corrected to focus perfectly on the retina.

• Vergence (V): Zero (0), because the rays are straight and do not change spacing
• Clinical example: In ideal emmetropic vision (normal eye), the eye focuses parallel rays directly on the retina

Quick Summary Table:

Type of Rays	Vergence	Wavefront Shape	Lens Type	Clinical Example
Converging	Positive (+)	Concave wavefront (flattening)	Convex lens	Hyperopia correction
Diverging	Negative (−)	Convex wavefront (steepening)	Concave lens	Myopia correction
Parallel	Zero	Plane wavefront	No lens or emmetropic eye	Normal vision

2.4 Clinical Relevance in Optometry

• Vergence is central to lens prescriptions: The amount of convergence or divergence needed to focus light on the retina determines the power of glasses/contact lenses.
• Plane wavefronts: Used in retinoscopy to simulate distant vision.
• Converging/diverging wavefronts: Essential in understanding myopia, hyperopia, and presbyopia.
• Wavefront analysis: Advanced machines like aberrometers measure the shape of wavefronts to detect optical errors like coma and astigmatism.

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Conclusion

Wavefronts give a more complete picture of how light behaves, beyond simple rays. Understanding their types, curvature, and vergence is essential in optometry to prescribe lenses and interpret optical findings. Whether it’s a patient with blurry vision or fitting a custom lens, knowing how light bends and spreads helps provide clear sight and effective treatment.

3. Refractive Index and Its Dependence on Wavelength

Refractive index is one of the most important concepts in geometrical optics, especially in optometry. It explains why light bends when it moves from one material to another — like from air into the cornea, or from the lens into the vitreous humor. This bending is known as refraction.

Understanding the refractive index helps us design corrective lenses, contact lenses, and even artificial intraocular lenses used in cataract surgery.

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3.1 What is Refractive Index?

The refractive index (n) of a material is a measure of how much light slows down when it enters that material from a vacuum. Light travels fastest in a vacuum (about 3 × 10⁸ m/s). When light enters a denser material like glass or water, it slows down, and this slowing causes it to bend or change direction.

Formula:

n = c / v

Where:

• n = Refractive Index
• c = Speed of light in vacuum
• v = Speed of light in the material

Example: If light slows down more, the refractive index is higher. If it barely slows, the refractive index is closer to 1.

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3.2 Refractive Indices of Common Materials

Different materials have different refractive indices:

• Air: ~1.0003 (almost like a vacuum)
• Water: ~1.33
• Glass: ~1.50
• Cornea: ~1.376
• Aqueous humor: ~1.336
• Lens: ~1.39–1.41 (varies by age and region)
• Vitreous humor: ~1.336

These values are very important when calculating the total optical power of the eye using formulas in optometry.

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3.3 Wavelength Dependence of Refractive Index (Dispersion)

The refractive index of a material is not constant — it actually depends on the wavelength or color of the light passing through it. This phenomenon is called dispersion.

Shorter wavelengths (like blue light) slow down more and bend more than longer wavelengths (like red light). That means the refractive index is slightly higher for blue light and lower for red light.

Why does this matter? Because different wavelengths bend differently, white light can spread out into its component colors when it passes through a prism — this is how we see a rainbow!

Clinical Relevance:

• Chromatic Aberration: Optical systems like the human eye or lenses may focus different colors at slightly different distances. This can cause visual distortions in high-contrast environments.
• Multifocal Lenses: Lens designers must account for dispersion to reduce color fringes in progressive and multifocal lenses.

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3.4 Snell’s Law and Refractive Index

Snell’s law describes how light bends when it crosses between two media with different refractive indices:

n₁ × sin(θ₁) = n₂ × sin(θ₂)

Where:

• n₁ and n₂ = Refractive indices of medium 1 and 2
• θ₁ = Angle of incidence (incoming ray)
• θ₂ = Angle of refraction (bent ray)

Example in the eye: When light enters from air into the cornea (n ≈ 1.376), it bends inward toward the normal due to a higher refractive index. This is the first and strongest refractive surface of the eye.

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3.5 Clinical Relevance in Optometry

• Corneal Refraction: Corneal curvature combined with refractive index determines corneal power in diopters.
• Lens Design: Refractive indices of lens materials (plastic, polycarbonate, glass) affect lens thickness and optical performance.
• Contact Lenses: Must match or complement corneal refractive index for proper fitting and vision correction.
• Optical Biometry: Accurate refractive index values of ocular media are used in IOL power calculations during cataract surgery.

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Conclusion

The refractive index is not just a number — it's a key to understanding how light behaves in the eye. Whether diagnosing refractive errors or selecting lens materials, optometrists rely on this concept every day. By understanding its dependence on wavelength, we can better predict visual outcomes and reduce unwanted effects like chromatic aberration in lenses.

4. Fermat’s Principle and Huygens’ Principle

To understand why light behaves the way it does — especially during reflection and refraction — we use two foundational theories in geometrical optics: Fermat’s Principle and Huygens’ Principle.

These principles help us derive the laws of reflection and refraction. They are more than just physics rules — they form the basis of how light interacts with mirrors, lenses, the cornea, and other optical structures used in optometry.

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4.1 Fermat’s Principle (Principle of Least Time)

Fermat’s Principle states that:

“The path taken by light between two points is the one which requires the least time.”

This means that when light travels from point A to point B, it always chooses the path that takes the shortest possible time — not necessarily the shortest distance. This explains why light bends when it enters a new medium.

Example:

Imagine light moving from air into water. Even though moving straight might be the shortest distance, bending allows the light to take a path that minimizes travel time (since light moves slower in water).

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4.2 Huygens’ Principle (Wavefront Construction)

Huygens’ Principle views light as a wave. It states:

“Every point on a wavefront acts as a source of new secondary wavelets. The new wavefront is the surface that touches all these wavelets.”

This helps explain the direction in which a wavefront moves and why bending occurs at boundaries (like cornea–aqueous humor).

Wavefront Construction:

• When light hits a surface, every point generates tiny circular waves.
• The sum (envelope) of all these wavelets forms the new position of the wavefront.
• This shows how light reflects or refracts at surfaces.

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4.3 Derivation of Laws of Reflection Using Fermat’s Principle

Law of Reflection: “The angle of incidence equals the angle of reflection.”

Using Fermat’s Principle:

1. Light reflects off a surface to minimize the total travel time.
2. By geometry, the shortest time is when the path forms equal angles with the reflecting surface.
3. So, angle of incidence (i) = angle of reflection (r).

Clinical Example:

Used in optometric devices like retinoscopes and keratometers, where reflection angles determine refractive properties of the eye.

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4.4 Derivation of Snell’s Law (Law of Refraction) Using Fermat’s Principle

Snell’s Law:

n₁ × sin(θ₁) = n₂ × sin(θ₂)

Derivation Idea:

1. Let light go from point A (in air) to point B (in water).
2. Light travels faster in air (n₁) and slower in water (n₂).
3. Fermat’s Principle says it will take a bent path that minimizes time.
4. Using calculus or geometric methods, we find the above equation — which is Snell’s Law.

Key Terms:

• n₁: Refractive index of the first medium (air, usually ~1.00)
• n₂: Refractive index of the second medium (e.g., cornea, ~1.376)
• θ₁: Angle of incidence (incoming ray)
• θ₂: Angle of refraction (bent ray)

This law governs how light enters the eye, or how it bends through a lens or cornea.

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4.5 Deriving Snell’s Law Using Huygens’ Principle

Huygens’ Principle can also derive Snell’s Law geometrically:

• As wavefronts move from air to glass, their speed decreases.
• The portion of the wavefront entering the slower medium lags behind.
• This causes the wavefront to rotate — bending the light ray.

This change in direction confirms Snell’s Law and gives a clear wave-based explanation of refraction.

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4.6 Clinical Relevance in Optometry

• Contact lens fitting: Depends on how light bends at the corneal interface.
• Visual acuity: Refraction determines focus and blur — key in glasses/contact prescriptions.
• Retinoscopy: Based on reflection laws to estimate refractive error.
• Lens design: Follows principles of refraction for proper image formation.

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Conclusion

Both Fermat’s and Huygens’ principles give us powerful ways to explain how light travels, reflects, and refracts. In clinical optometry, these theories help us design better lenses, understand eye optics, and use diagnostic tools more effectively. Together, they form the foundation of modern geometrical optics.

5. Plane Mirrors: Height and Rotation

A plane mirror is a flat, reflective surface that reflects light according to the law of reflection: the angle of incidence equals the angle of reflection. Even though plane mirrors are simple compared to lenses, they have important applications in vision therapy, optical instruments, and clinical testing environments.

In this section, we'll understand two important properties:

• Minimum height of a plane mirror needed to see full reflection
• Effect of mirror rotation on the reflected ray

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5.1 Image Formation in a Plane Mirror

Key properties of the image formed by a plane mirror:

• Virtual: It cannot be projected on a screen
• Upright: Same orientation as the object
• Same size: Image size = object size
• Laterally inverted: Left and right appear swapped
• Same distance: Image is formed as far behind the mirror as the object is in front

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5.2 Minimum Height of a Plane Mirror

How tall should a mirror be for a person to see their entire reflection?

Result: The mirror must be at least half the height of the person to see their full body, and it must be placed so its center is at eye level.

Why Half?

• The top of the person’s head reflects at the top half of the mirror
• The feet reflect from the lower half
• Because of symmetry and the law of reflection, each ray travels the same angle in and out

Example: A person 180 cm tall only needs a mirror 90 cm high to view their full reflection, if placed correctly.

Diagram Tip:

Draw a vertical mirror. Trace a ray from the person’s head to eye to mirror and back, and from the foot similarly. You'll see they reflect in the top and bottom halves.

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5.3 Rotation of a Plane Mirror

What happens to the reflected ray when a mirror is rotated?

Key Rule: If a mirror is rotated by an angle θ, the reflected ray rotates by 2θ.

This is because both the angle of incidence and the angle of reflection change by θ, adding up to 2θ total change in the outgoing direction.

Mathematical Form:

Change in reflected ray angle = 2 × angle of rotation of mirror

Clinical Example:

• In a mirror retinoscope, rotating the mirror changes the light path.
• In perimetry or vision therapy tools, mirror rotation alters field angles.

Quick Illustration:

If the mirror rotates 10°, the reflected ray changes by 20°.

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5.4 Clinical Relevance in Optometry

• Mirror positioning: Understanding minimum height is useful in setting up eye charts and mirrors in vision therapy rooms.
• Retinoscopy: Uses moving/tilting mirrors to estimate refraction through light reflection.
• Slit lamp and keratometer: Use internal mirrors to direct and reflect light precisely — mirror alignment is crucial.
• Vision Therapy: Rotating mirrors help simulate or train peripheral vision.

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Conclusion

Though simple, plane mirrors offer insights into reflection, symmetry, and light behavior. Understanding mirror height and rotation helps optometrists in clinical setups and diagnostics. It also strengthens core principles needed to interpret reflection-based instruments and therapy tools in modern optometry.

For more geometrical optics unit click below 👇 

👉 Unit 2

👉 Unit 3

👉 Unit 4

👉 Unit 5