Unit 4- Geometrical optics | 1st Semester Bachelor of Optometry

Refraction by a Spherical Surface – Sign Convention, Spherical Aberration, Imaging, and Sag Formula

In geometrical optics, refraction by a spherical surface is a foundational concept for understanding how light is bent when it passes between different optical media. This principle governs the behavior of the human eye, spectacle lenses, and many diagnostic instruments in optometry.

This topic covers the physics of refraction through curved surfaces, the correct application of sign convention, the causes and effects of spherical aberration, the behavior of distant objects under such refraction, and the mathematical derivation and application of the sag formula.

1. Refraction at a Spherical Surface

A spherical surface is a part of a sphere that separates two media of different refractive indices. When light travels from one medium into another across this curved interface, it bends according to Snell’s law, but the curvature introduces additional optical effects compared to plane surfaces.

Refraction Formula for a Spherical Surface:

n₂ / v – n₁ / u = (n₂ – n₁) / R

Where:

n₁ = Refractive index of the medium from which the light is coming
n₂ = Refractive index of the medium into which light is entering
u = Object distance (from the pole)
v = Image distance (from the pole)
R = Radius of curvature of the spherical surface (positive if center of curvature is to the right)

This formula is used when one surface of a lens or an ocular structure (e.g., cornea) is curved. It allows us to determine the position of the image formed due to refraction at the surface.

2. Cartesian Sign Convention

The Cartesian (optical) sign convention standardizes the assignment of positive and negative values to distances and radii in optical calculations.

Rules:

All distances are measured from the **pole (P)** of the refracting surface.
Distances measured **in the direction of incident light** are negative.
Distances measured **in the direction of refracted light** are positive.
**Object distance (u):** Always negative (object is on the left of surface)
**Image distance (v):**
- Positive if image is real (on the right of the surface)
- Negative if image is virtual (on the left)
**Radius of curvature (R):**
- Positive if center of curvature is on the right side (convex surface)
- Negative if center is on the left side (concave surface)

This sign convention ensures consistent results while using the refraction formula.

3. Imaging by a Spherical Surface (Distant Object)

When an object is located very far from the refracting surface (like a star or the sun), we say the object is at “infinity.” In this case, u → ∞, so the term n₁/u becomes negligible (approaches zero).

The refraction formula becomes:

n₂ / v = (n₂ – n₁) / R

Solving for v:

v = n₂ × R / (n₂ – n₁)

This simplified formula allows us to compute the image position of a distant object such as parallel rays entering the cornea or hitting a curved lens surface in an instrument.

Clinical Example:

The cornea, with a curved anterior surface and a large difference in n between air (1.00) and cornea (1.376), forms the first refractive image in the eye.
When the eye is relaxed, it is focused on distant objects. The corneal surface acts as a single spherical refractive surface for parallel light rays from a distant object.

4. Spherical Aberration

Spherical aberration is an optical imperfection that occurs when light rays passing through the edges of a spherical surface focus at a different point than rays passing through the center.

Why it happens:

Ideal lenses would focus all rays at the same point, regardless of entry point.
Spherical surfaces bend edge rays more than central rays → leading to multiple focal points.

Types:

Positive spherical aberration: Peripheral rays focus in front of central rays (common in spherical lenses).
Negative spherical aberration: Peripheral rays focus behind central rays (less common).

Effects on Vision:

Blurred or distorted images
Glare and halos around lights (especially at night)

Clinical Relevance:

Intraocular lens (IOL) design considers spherical aberration to improve post-cataract vision.
Aspheric lenses are prescribed in spectacles to reduce aberration, especially in high prescriptions.

5. Sag Formula (Sagitta)

The sag (sagitta) of a curved surface is the height of the arc from the base chord to the top of the surface. It is important in lens design and fitting, especially in contact lenses and custom optics.

Sag formula:

s = h² / (2R)

Where:

s = sag (depth of the curved surface)
h = half-chord length (half the lens diameter)
R = radius of curvature

Used to determine:

Central thickness of a lens
Base curve selection for contact lenses
Corneal topography calculations

Clinical Example:

To fit a contact lens comfortably on the cornea, the sag of the cornea must match the sag of the lens. If mismatch occurs, the lens may:

Fit too tightly (causing discomfort or corneal edema)
Fit too loosely (leading to decentration or poor vision)

6. Clinical Relevance in Optometry

Refraction at corneal surface: First major refractive interface in the eye (air → cornea).
Aberration control: Aspheric lens designs reduce spherical aberration and improve contrast sensitivity.
Contact lens fitting: Requires accurate sag and curvature measurements.
Refractive surgery planning: In LASIK, understanding spherical refraction helps customize the ablation pattern.

Devices such as:

Keratometers: Measure corneal curvature, treating the cornea as a spherical refracting surface.
OCT (Optical Coherence Tomography): Maps curvature and sag values of cornea and lens.

Conclusion

Refraction at a spherical surface is a fundamental concept that explains how the eye, lenses, and instruments bend light to form clear images. By applying the correct sign convention, calculating image locations, managing spherical aberration, and using sag formulas, optometrists can design better lenses, fit contacts accurately, and diagnose vision problems with precision. This topic bridges basic geometrical optics with real-world clinical practice, making it essential for every optometry student and professional.

Paraxial Approximation and Derivation of Vergence Equation

In geometrical optics, the paraxial approximation is a simplification used when analyzing the path of light rays that travel close to the optical axis. These rays are called paraxial rays and they form the basis of most lens and optical system calculations, including vergence and image formation.

1. What is the Paraxial Approximation?

The paraxial approximation assumes that the light rays:

Make small angles (less than 10°) with the optical axis
Are close to the axis (near the center of the lens or cornea)

Under this approximation:

sin(θ) ≈ tan(θ) ≈ θ (in radians)

This greatly simplifies the use of Snell’s law and ray-tracing equations in lens and mirror systems.

Why it's useful: Most optical systems, especially in the human eye and spectacle lenses, work with central rays that meet the paraxial condition. It helps in deriving practical formulas used in vision science.

2. Definition of Vergence

Vergence (L) is a measure of how convergent or divergent a bundle of light rays is, with respect to a point source or point focus. It is defined as:

L = n / r

Where:

L = Vergence in diopters (D)
n = Refractive index of the medium (usually air = 1.00)
r = Distance from the wavefront to the point focus (in meters)

Signs:

Positive vergence: Rays converging toward a point (real image)
Negative vergence: Rays diverging from a point (virtual image)

Examples:

A light source 0.50 m away in air: L = 1/0.50 = +2.00 D (converging)
A virtual object 1.00 m behind a surface: L = 1/(–1.00) = –1.00 D (diverging)

3. Derivation of the Vergence Equation

The vergence equation links the vergence before and after refraction at a spherical surface:

L' = L + F

Where:

L = Vergence of the incident ray (object vergence)
L' = Vergence of the emergent ray (image vergence)
F = Power of the surface (in diopters)

This equation comes from the refraction formula at a spherical surface:

n₂ / v – n₁ / u = (n₂ – n₁) / R

Now multiply both sides by (n₁ × n₂):

n₁ × n₂ / v – n₁² / u = n₁ × (n₂ – n₁) / R

Then divide through by n₁ × n₂ and rearrange:

n₂ / v = n₁ / u + (n₂ – n₁) / R

Since:

L = n₁ / u
L' = n₂ / v
F = (n₂ – n₁) / R

We get:

L' = L + F

4. Clinical Application in Optometry

The vergence formula is fundamental in:

Lens design and calculation
Retinoscopy and objective refraction
Understanding image formation in the eye
Analyzing corrective lens powers for ametropia

Example:

If light of vergence –3.00 D strikes a lens of power +5.00 D:

L' = L + F = –3 + 5 = +2.00 D → image forms 0.5 m beyond the lens

Conclusion

The paraxial approximation allows us to simplify complex ray paths and derive useful, clinically applicable formulas like the vergence equation. It underpins how optometrists analyze and prescribe lenses to focus light accurately on the retina.

Imaging by a Positive Powered Surface and Negative Powered Surface

In geometrical optics, curved surfaces are not only characterized by their shape but also by their optical power, which directly influences how they bend light rays and form images. These surfaces can be either positively powered (converging) or negatively powered (diverging), and understanding how they affect image formation is essential in optometry, lens design, and visual correction.

This topic explains the behavior of light as it passes through surfaces with different optical powers, image characteristics formed by these surfaces, and their applications in clinical practice.

1. Optical Power of a Curved Surface

The optical power of a single spherical refracting surface is defined by:

F = (n₂ – n₁) / R

Where:

F = Power in diopters (D)
n₁ = Refractive index of the medium from which the light is incident
n₂ = Refractive index of the medium into which the light is refracted
R = Radius of curvature of the surface (in meters)

This formula describes how much the surface can converge or diverge light rays depending on the curvature (R) and the refractive indices on either side of the surface.

Sign convention reminder:

R > 0 for convex surfaces (center of curvature to the right)
R < 0 for concave surfaces (center to the left)

2. Positive Powered Surface (Converging Surface)

A positive powered surface has a convex shape relative to the incident light. These surfaces cause convergence of light rays.

Characteristics:

Convex surface (bulges toward incoming light)
F > 0 (positive power)
Converts parallel rays to converging rays
Forms real or virtual images depending on object position

Image formation rules:

Distant object (u → ∞): Image forms at focus, real and inverted
Object between ∞ and R: Real, inverted, smaller or magnified image
Object between P and F: Virtual, erect, magnified image on the same side

Clinical examples:

Anterior surface of the cornea (air to cornea transition)
Plano-convex or biconvex lenses in plus power spectacles
Magnifying lenses used in low vision aids

Vergence application:

L' = L + F

If L (object vergence) is small and F is large and positive, L' becomes strongly positive, causing convergence and a real image formation.

3. Negative Powered Surface (Diverging Surface)

A negative powered surface is typically concave with respect to the incoming light. These surfaces cause divergence of light rays.

Characteristics:

Concave surface (curves away from incoming rays)
F < 0 (negative power)
Converts parallel rays into diverging rays
Always forms virtual, erect, and reduced images for distant objects

Image formation rules:

Distant object: Virtual image formed between the surface and the focus
Object between F and surface: Virtual, erect, reduced image on the same side as the object

Clinical examples:

Posterior surface of the cornea (cornea to aqueous transition)
Plano-concave or biconcave lenses in minus power spectacles
Used in correcting myopia or diverging light in optical instruments

Vergence application:

L' = L + F

If L is already negative and F is also negative, the image vergence L' becomes more negative → more divergence.

4. Comparison Table

Feature	Positive Powered Surface	Negative Powered Surface
Shape	Convex	Concave
Optical Power (F)	Positive (+)	Negative (–)
Effect on rays	Converges light	Diverges light
Image type (distant object)	Real and inverted	Virtual and erect
Clinical use	Correcting hypermetropia, presbyopia	Correcting myopia

5. Imaging Examples and Diagrams

Positive Powered Surface:

Light rays from a distant object enter a convex surface
They bend toward the normal and converge to form a real image
Used in the cornea and biconvex lenses

Negative Powered Surface:

Parallel light rays strike a concave surface
They bend away from the normal and diverge
Virtual image appears behind the surface

Diagram Suggestions:

Ray tracing through convex (positive) surface showing real image
Ray tracing through concave (negative) surface showing virtual image

6. Clinical Relevance in Optometry

Understanding the image formation by positive and negative powered surfaces helps optometrists in:

Lens prescribing: Knowing when to use plus or minus lenses
Corneal modeling: Anterior and posterior corneal surfaces behave differently due to their power signs
Contact lens design: Custom curvature based on corneal power and direction of refraction
Diagnosing ametropia: Hypermetropes need more plus power; myopes need more minus power
Using diagnostic tools: Retinoscopy, keratometry, and auto-refractors rely on vergence and power principles

7. Summary

A positive powered surface converges light rays and forms real images (if object is at a distance).
A negative powered surface diverges rays and always forms virtual images.
Optical power (F) of a surface depends on curvature and refractive index difference.
Image vergence is determined using: L' = L + F

This fundamental understanding is essential for designing optical corrections, understanding refractive errors, and interpreting the behavior of the eye as an optical system.

Vergence at a Distance Formula and Effectivity of a Refracting Surface

In geometrical optics and clinical optometry, understanding how vergence changes as light travels between surfaces is essential for precise lens prescriptions and adjustment of optical devices. The concept of vergence at a distance and effectivity of a refracting surface is particularly important in high prescriptions, contact lens fitting, and vertex distance corrections.

1. Recap: What is Vergence?

Vergence (L) is a measure of the degree of convergence or divergence of a bundle of light rays and is given by:

L = n / r

Where:

L = Vergence in diopters (D)
n = Refractive index of the medium
r = Distance to the point of focus (in meters, with sign)

Converging rays: Positive vergence (rays move toward a point)
Diverging rays: Negative vergence (rays spread from a point)

2. Vergence at a Distance Formula

As light travels through space without passing through any lens or refractive surface, its vergence changes depending on the distance traveled. This change in vergence is governed by the formula:

L₂ = L₁ / (1 – (d × L₁))

Where:

L₁ = Vergence of the light at the starting point (initial position)
L₂ = Vergence of the light at a distance d (after traveling a certain distance)
d = Distance between the two planes (in meters, with sign)

Signs and Units:

d is positive if light travels in the direction of propagation (e.g., from spectacle lens to eye)
d is negative if light travels against the direction of propagation
Distances must be in meters, vergences in diopters

3. Derivation of the Formula

Starting with the basic vergence definition:

L = n / r

Suppose light has vergence L₁ at point A. When the light travels a distance d to point B, the new vergence becomes:

L₂ = n / (r – d)

But since L₁ = n / r → r = n / L₁, substituting into the above:

L₂ = n / ((n / L₁) – d)

Simplify:

L₂ = L₁ / (1 – d × L₁ / n)

If n = 1 (in air), it becomes:

L₂ = L₁ / (1 – d × L₁)

4. Clinical Interpretation and Application

This formula is widely used in clinical practice, especially for:

Converting spectacle prescriptions to contact lens power (vertex distance adjustment)
Predicting the effect of shifting a lens forward or backward
Adjusting refractive outcomes post-surgery

Vertex Distance Example:

A patient wears a –10.00 D spectacle lens positioned 12 mm (0.012 m) from the eye. What power of contact lens is needed at the cornea?

L₂ = –10 / (1 – (0.012 × –10)) = –10 / (1 + 0.12) = –10 / 1.12 ≈ –8.93 D

So, the contact lens needed is approximately –8.93 D (less negative than spectacle lens).

5. Effectivity of a Refracting Surface

Effectivity refers to how the optical power of a lens or surface changes when its position is altered relative to the eye.

Why it matters:

High-powered lenses change their effective power significantly with small shifts in position
Contact lens prescriptions differ from spectacle prescriptions because the lens sits directly on the cornea
In post-surgical or aphakic eyes, vertex adjustment is critical for clear vision

Effectivity formula (refractive power shift):

F₂ = F₁ / (1 – d × F₁)

This is the same formula as vergence change, just interpreted as power shift when lens is moved.

Signs:

d > 0: Lens moved closer to the cornea (toward eye)
d < 0: Lens moved away from cornea (e.g., phoropter settings)

Example:

If a +12.00 D lens is moved 10 mm (0.01 m) closer to the eye:

F₂ = +12 / (1 – (0.01 × 12)) = 12 / (1 – 0.12) = 12 / 0.88 ≈ +13.64 D

The lens becomes effectively more powerful when moved closer to the eye.

6. Practical Applications in Optometry

✔ Contact Lens Fitting:

Contact lenses sit at 0 mm vertex distance, unlike spectacles at 10–14 mm. Vertex correction must be applied when converting prescriptions above ±4.00 D.

✔ Refraction Adjustments:

Phoropters often sit ~12–13 mm in front of the eye. Final prescriptions need to be recalculated based on spectacle position.

✔ IOL Calculations:

Intraocular lens (IOL) power estimation for cataract surgery must consider effective lens position (ELP) and post-op vergence changes.

✔ High Refractive Errors:

In high myopia or hypermetropia, incorrect vertex adjustments can lead to over- or under-correction. Accurate vergence tracking is essential.

7. Diagram Suggestions

Vergence change illustration: Diverging ray bundle becoming more parallel as distance increases
Vertex distance comparison: Spectacle lens vs. contact lens positioning
Graph: Lens power vs. effective power at different distances

8. Summary

Vergence at a distance helps calculate how light changes as it travels through space.
The formula L₂ = L₁ / (1 – d × L₁) accounts for vergence change due to distance (especially in air).
Effectivity is crucial for converting prescriptions and understanding lens behavior in different positions.
High-power lenses require accurate vertex distance adjustments for optimal visual correction.

Mastery of these concepts allows optometrists to prescribe more accurately and ensure better visual outcomes, especially in complex clinical scenarios involving high ametropia, presbyopia, or post-surgical corrections.

Definition of Lens as a Combination of Two Surfaces; Different Types of Lens Shapes

In geometrical optics, a lens is defined as a transparent optical component made by joining two refracting surfaces, at least one of which is curved. These surfaces cause light to bend (refract) in such a way that images are formed — either real or virtual, magnified or diminished, depending on the shape of the lens and the object's position.

This topic explains how lenses work as a combination of two surfaces and the classification of lenses based on their curvature and shape. This understanding is foundational in optometry, where lens shapes are selected based on visual needs and refractive error correction.

1. Definition of a Lens

A lens is an optical device bounded by two surfaces, at least one of which is curved. It refracts light rays to converge (in convex lenses) or diverge (in concave lenses) to form images. Lenses are typically made from transparent materials like glass or plastic with a higher refractive index than air.

Lens surfaces can be:

Convex: Bulges outward
Concave: Curves inward
Plane: Flat surface

Formation:

A lens is formed by combining two spherical surfaces:

Lens = First refracting surface + Second refracting surface

Depending on the curvature and direction of the surfaces, the lens may be converging (positive power) or diverging (negative power).

Power of a lens (F):

F = (n – 1) × [(1/R₁) – (1/R₂)]

Where:

n = Refractive index of the lens material
R₁ = Radius of curvature of the first surface
R₂ = Radius of curvature of the second surface

2. Classification of Lenses Based on Shape

Lenses are broadly classified into two groups based on their optical power:

1. Convex Lenses (Converging Lenses)

These lenses are thicker at the center and thinner at the edges. They converge parallel rays to a real focus.

Types:

Biconvex (Double Convex): Both surfaces are convex
Plano-convex: One flat surface, one convex surface
Concavo-convex (Meniscus, positive): One concave and one convex surface; convex surface has stronger curvature

Uses: Correction of hypermetropia, presbyopia, magnifying lenses, focusing lenses in instruments

2. Concave Lenses (Diverging Lenses)

These lenses are thinner at the center and thicker at the edges. They diverge parallel rays as if coming from a virtual focus.

Types:

Biconcave (Double Concave): Both surfaces are concave
Plano-concave: One flat surface, one concave surface
Convexo-concave (Meniscus, negative): One convex and one concave surface; concave surface has stronger curvature

Uses: Correction of myopia, beam spreading in laser optics

3. Shape and Power Relationship

The optical power of the lens depends on:

The curvature of both surfaces (R₁ and R₂)
The refractive index of the material (n)
The lens thickness (for thick lens systems)

General rules:

More curvature → More power
Convex surfaces → Positive power
Concave surfaces → Negative power

4. Visual Representation of Lens Shapes

Lens Type	Shape	Power	Effect on Rays
Biconvex	Both sides bulge outward	Positive	Converges rays
Plano-convex	One flat, one convex	Positive	Converges rays
Concavo-convex	Concave + weaker convex	Positive	Converges rays
Biconcave	Both sides curve inward	Negative	Diverges rays
Plano-concave	One flat, one concave	Negative	Diverges rays
Convexo-concave	Convex + stronger concave	Negative	Diverges rays

5. Lens Design in Clinical Practice

In optometry, lens shape selection influences:

Optical performance – minimizing aberrations
Cosmetic appearance – especially in high prescriptions
Weight and thickness – lighter materials preferred
Comfort and fitting – especially in spectacles and contact lenses

Examples:

High hyperopes: Meniscus or aspheric biconvex lenses to reduce thickness
High myopes: Plano-concave or meniscus lenses with minimal edge thickness
Contact lenses: Base curve chosen based on corneal curvature; often meniscus design

6. Advantages of Using Specific Shapes

Biconvex: High convergence, used for strong plus corrections

Plano-convex: Simple magnifiers and focusing elements

Meniscus (convex): Best balance of optics and cosmesis in spectacles

Biconcave: Strong divergence for high myopia

Plano-concave: Cosmetic advantage with mild divergence

Meniscus (concave): Modern myopia correction lenses

7. Clinical Relevance in Optometry

Refraction: Choice of lens depends on the power required and desired image properties
Spectacle Dispensing: Shape determines lens aesthetics and weight
Contact Lens Design: Curvature and thickness influence comfort and fit
Low Vision Aids: Magnifiers use plano-convex or biconvex lenses
Surgical IOLs: Intraocular lens implants use specific curved profiles for image clarity

8. Summary

A lens is a combination of two refracting surfaces, at least one of which is curved.
Convex lenses converge light and have positive power, used in hyperopia.
Concave lenses diverge light and have negative power, used in myopia.
Lenses are classified by their surface shapes: biconvex, plano-convex, concavo-convex, etc.
Lens shape selection influences optics, comfort, aesthetics, and clinical success in optometry.

Understanding lens shapes is foundational for prescribing accurate optical corrections, designing comfortable eyewear, and ensuring optimal patient vision outcomes.

Image Formation by a Lens Using Vergence at a Distance Formula

In geometrical optics, especially in optometry, understanding how lenses form images involves applying the vergence at a distance formula and analyzing the role of key optical parameters like vertex powers, equivalent power, focal points, and principal planes. These concepts are essential for accurate lens design, prescription conversion, and understanding optical systems such as the human eye or instruments.

1. Image Formation Using Vergence at a Distance Formula

The vergence formula allows us to predict how light behaves as it travels through a lens:

L' = L + F

Where:

L = Vergence of incoming rays at the first surface
F = Power of the lens (in diopters)
L' = Vergence of outgoing rays (image vergence)

To use this formula for a real lens that has two surfaces and physical thickness, we apply it in two steps:

Step 1: Refraction at Front Surface

Use: L₁' = L₁ + F₁ (Light enters lens, vergence changes based on front surface power F₁)

Step 2: Transfer vergence through lens thickness

Use vergence transfer formula: L₂ = L₁' / (1 – (t / n) × L₁') (Where t is lens thickness and n is lens refractive index)

Step 3: Refraction at Back Surface

Use: L' = L₂ + F₂

This method gives a complete analysis of how a thick lens forms an image.

2. Front and Back Vertex Powers

Vertex power is the effective power of a lens measured from one surface to the opposite focal point. It’s essential in converting spectacle to contact lens powers.

Front Vertex Power (Fv)

Measured when light enters from the back and exits from the front:

Fv = F₂ + (t/n) × F₁ × F₂

Back Vertex Power (Bv)

Measured when light enters from the front (normal case for vision):

Bv = F₁ + (t/n) × F₁ × F₂

Note: Bv is most commonly used for spectacle prescriptions as the light typically enters from the front.

3. Equivalent Power of a Lens

Equivalent power (Feq) is the single power that can replace a thick lens system (with two surfaces and thickness) and produce the same image result.

Feq = F₁ + F₂ – (t/n) × F₁ × F₂

This value is useful for analyzing thick lenses like intraocular lenses (IOLs) or special spectacle designs.

Clinical Relevance: Equivalent power is helpful in calculating the net effect of progressive lenses or in low vision where thick magnifying lenses are used.

4. First and Second Principal Planes/Points

Principal planes are imaginary planes within or outside a lens system where all refraction can be considered to occur.

First Principal Plane (H):

The plane from which object distances are measured to form the same image as the actual lens system.

Second Principal Plane (H’):

The plane from which image distances are measured as if all refraction occurred at a single point.

Key Features:

If both surfaces of the lens are symmetric and thin, H and H’ may lie within the lens
In complex lens systems, they may lie outside the physical boundary of the lens

Importance: Simplifies ray tracing and allows for accurate image prediction without tracking each refracting surface separately.

5. Primary and Secondary Focal Points (Planes)

Focal points are specific locations where parallel rays either converge (convex lens) or appear to diverge from (concave lens).

Primary Focal Point (F):

The point on the object side from which light must originate to become parallel after refraction through the lens.

Secondary Focal Point (F’):

The point where incoming parallel rays converge after refraction.

Focal planes are planes perpendicular to the optical axis passing through the focal points. All rays parallel to the optical axis and entering the lens converge (or diverge) to/from the focal plane.

Location:

For convex (positive) lenses, both focal lengths are real and on opposite sides
For concave (negative) lenses, both focal lengths are virtual and on the same side

6. Primary and Secondary Focal Lengths

Focal length (f) is the distance between the principal point and the corresponding focal point.

For thin lenses:

f = 1 / F

Where F is the lens power in diopters, and f is in meters.

Positive f: For converging (convex) lenses
Negative f: For diverging (concave) lenses

In Thick Lenses:

Primary focal length (f): From first principal plane to F
Secondary focal length (f’): From second principal plane to F’

7. Clinical Applications in Optometry

Vertex correction: Required when converting spectacle prescriptions to contact lenses using vertex formulas and vertex powers
Design of IOLs and high-index lenses: Uses equivalent power and principal planes
Refraction and visual correction: Knowing exact location of focal points helps predict vision clarity and blur circles
Low vision aids: Magnifying lens design uses focal length calculations extensively

8. Suggested Diagrams

Ray tracing showing vergence change through a thick lens
Location of first and second principal planes (H and H’)
Focal point and focal plane diagrams for convex and concave lenses
Comparison between vertex power and equivalent power in a lens

9. Summary

Image formation through lenses can be understood using the vergence formula: L' = L + F
Vertex powers and equivalent power help understand how lenses perform in different positions
Principal planes and focal points simplify complex ray paths and image predictions
Focal lengths determine where rays will focus and how sharp an image will be

Mastery of these concepts ensures accurate optical correction, efficient lens fitting, and precise visual outcomes in clinical optometry.

Newton’s Formula; Linear Magnification; Angular Magnification

In optical systems such as lenses and optical instruments, understanding how an object is magnified or imaged is essential. Newton’s formula offers an alternate approach to image formation, while linear and angular magnification describe how much an image’s size or angular spread increases compared to the original object.

This topic covers all three concepts, their derivations, and clinical significance in optometry — especially for low vision aids and magnifying devices.

1. Newton’s Formula for Lenses

Newton’s formula provides a relationship between the distances of the object and image from their respective focal points:

x × x' = f²

Where:

x = Distance of the object from the primary focal point (F)
x' = Distance of the image from the secondary focal point (F')
f = Focal length of the lens (positive for convex, negative for concave)

Derivation:

We know from the lens formula: 1/v – 1/u = 1/f

Let:

x = u – f
x' = v – f

Substituting and manipulating gives:

x × x' = f²

This simplifies ray tracing, especially when using conjugate points with known focal distances.

Example:

A convex lens has f = 10 cm. If the object is placed 5 cm in front of F, x = –5 cm.

x' = f² / x = 100 / (–5) = –20 cm → Image is 20 cm behind F'

2. Linear Magnification (M)

Linear magnification describes the ratio of the image height to the object height. It helps determine whether the image is:

Bigger or smaller than the object
Erect or inverted

Formula:

M = h'/h = v/u

Where:

h = Object height
h' = Image height
u = Object distance (from lens)
v = Image distance (from lens)

Sign convention:

Positive M: Erect image
Negative M: Inverted image

Special Cases:

Object at 2f: M = 1 → same size image
Object at f: No image (image at infinity)
Object closer than f: Virtual, erect, magnified image

Clinical Example:

A spectacle magnifier is placed 25 cm from the eye. An object 5 cm tall is imaged 50 cm away. Then:

M = v / u = 50 / 25 = 2 → Image is twice the size of the object

3. Angular Magnification (MA)

Angular magnification describes how much larger an object appears when viewed through an optical device compared to viewing it with the naked eye (at near point, usually 25 cm).

Formula:

MA = θ' / θ

Where:

θ = Angle subtended by the object without lens (naked eye)
θ' = Angle subtended by the image seen through the lens

For a simple magnifier (lens held close to eye):

MA = D / f

Where:

D = Near point distance (usually 25 cm = 0.25 m)
f = Focal length of the lens (in meters)

Alternatively:

MA = 1 + (D / f) if the image is formed at the near point

Example:

Magnifier with focal length 5 cm = 0.05 m:

MA = D / f = 0.25 / 0.05 = 5x magnification

Clinical Examples:

Hand magnifiers: Used for near reading in low vision patients
Telescopic aids: Combine angular and linear magnification
Stand magnifiers: Placed on reading materials, focus fixed, used with or without spectacles

4. Comparison of Magnifications

Aspect	Linear Magnification	Angular Magnification
Definition	Size of image compared to object	Apparent size vs. actual angle subtended
Formula	v / u	θ’ / θ or D / f
Common in	Image formation problems	Low vision magnifiers, telescopes
Measured in	Ratio (unitless)	x (times magnification)

5. Clinical Applications in Optometry

✔ Low Vision Aids:

Angular magnification is the most relevant in helping patients with central vision loss or macular degeneration read printed material.

✔ Instrument Design:

Ophthalmoscopes, retinoscopes, and telescopes depend on Newtonian ray tracing and angular magnification principles.

✔ Spectacle Magnifiers:

Simple plus lenses used for near magnification rely on understanding linear magnification and working distance.

✔ Visual Acuity Testing:

Optotypes subtend standard angles — angular magnification helps estimate real-world vision capabilities under magnifiers.

6. Suggested Diagrams

Ray diagram showing Newton’s formula x × x′ = f²
Object and image comparison showing linear magnification (h and h′)
Eye viewing through magnifier to demonstrate angular magnification
Real-world example: reading a newspaper with a 5x magnifier

7. Summary

Newton’s formula (x × x′ = f²) is a useful lens equation based on distances from focal points.
Linear magnification = v / u, describes image size compared to object size.
Angular magnification = θ′ / θ, describes perceived size enhancement.
All three are essential in lens selection, optical instrument use, and low vision rehabilitation.

Understanding these principles enables optometrists to evaluate and prescribe appropriate magnifying solutions, enhancing patients’ quality of life through better vision.

For more geometrical optics unit click below 👇

👉 Unit 1

👉 Unit 2

👉 Unit 3

👉 Unit 5