Physics

Chapter 2: Geometric Optics and Image Formation

Imron Rosyadi

Chapter 2: Geometric Optics and Image Formation

Overview of Topics

• 2.1 Images Formed by Plane Mirrors
• 2.2 Spherical Mirrors
• 2.3 Images Formed by Refraction
• 2.4 Thin Lenses
• 2.5 The Eye
• 2.6 The Camera
• 2.7 The Simple Magnifier
• 2.8 Microscopes and Telescopes

2.1 Images Formed by Plane Mirrors

Learning Objectives

By the end of this section, you will be able to:

• Describe how an image is formed by a plane mirror.
• Distinguish between real and virtual images.
• Find the location and characterize the orientation of an image created by a plane mirror.

Today, we’ll start our journey into geometric optics by looking at plane mirrors. We’ll cover how images are formed, what makes an image real or virtual, and how to determine an image’s location and orientation.

Plane Mirror Image Characteristics

Images in a plane mirror are:

• Same size as the object.
• Located behind the mirror.
• Upright (oriented in the same direction as the object).

To understand this, let’s look at a ray diagram.

When you look into a bathroom mirror, you’re seeing an image formed by a plane mirror. These images have distinct characteristics: they are the same size as the object, appear behind the mirror, and are upright.

How Plane Mirrors Form Images

• Light rays from point \(P\) reflect off the mirror.
• According to the law of reflection, the angle of incidence equals the angle of reflection.
• When reflected rays are extended backward behind the mirror (dashed lines), they appear to originate from point \(Q\).
• Point \(Q\) is the virtual image of point \(P\).
• Repeating this for all object points forms an upright image.

Figure 2.2: Image formation by a plane mirror.

Figure 2.2 illustrates how a plane mirror forms an image. Rays from point P on the object hit the mirror and reflect. Our eyes trace these reflected rays back, making them appear to come from point Q behind the mirror. This point Q is the virtual image. The image height is the same as the object height, and the image is upright.

Real vs. Virtual Images

Virtual Image

• Rays appear to originate from a point.
• Cannot be projected onto a screen.
• Example: Image in a plane mirror.

Real Image

• Rays physically go through the image point.
• Can be projected onto a screen.
• Example: Image formed by a movie projector.

This is a crucial distinction in optics. Virtual images are where rays seem to converge, but don’t actually pass through. Plane mirrors produce virtual images. Real images are where the light rays truly converge, allowing them to be projected onto a screen, like in a cinema.

Locating an Image in a Plane Mirror

• The law of reflection implies that triangles PAB and QAB in Figure 2.2 are congruent.
• This means the object distance (\(d_o\)) equals the image distance (\(d_i\)).
• For a plane mirror, the object and image are in opposite directions, so their distances have opposite signs:

\[d_i = -d_o\]

Note

The negative sign indicates the image is virtual (behind the mirror).

Using basic geometry and the law of reflection, we can prove that the distance from the object to the mirror is the same as the distance from the mirror to the image. This relationship is formalized by the equation \(d_i = -d_o\), where the negative sign signifies that the image is virtual and located behind the mirror.

Multiple Images

• When an object is placed between two mirrors, multiple images can form.
• Each image from one mirror can act as an object for the other.
• Parallel mirrors: Can produce an infinite number of images if the object is off-center.
• Mirrors at right angles: Produce three images (e.g., Image 1, Image 2, and Image 1,2 from reflections off both mirrors).

Figure 2.3: Images in parallel mirrors.

Figure 2.4: Images in mirrors at right angles.

Multiple images are a fascinating phenomenon. When you stand between two parallel mirrors, you see an endless reflection of yourself. Each image acts as a new object for the other mirror. With mirrors at right angles, you typically see three distinct images, as one ray can reflect off both surfaces to form a combined image.

2.2 Spherical Mirrors

Learning Objectives

By the end of this section, you will be able to:

• Describe image formation by spherical mirrors.
• Use ray diagrams and the mirror equation to calculate the properties of an image in a spherical mirror.

Now that we’ve covered plane mirrors, let’s delve into spherical mirrors. These mirrors, unlike plane mirrors, can create images of varying sizes and locations. We’ll learn about their types, key properties, and how to analyze them using both graphical (ray diagrams) and mathematical (mirror equation) methods.

Curved Mirrors: Concave and Convex

• Spherical mirrors are sections of a sphere.
• Concave Mirror: Reflecting surface is on the inside of the sphere (“cave-like”).
• Convex Mirror: Reflecting surface is on the outside of the sphere.
• Optical Axis (Principal Axis): Passes through the mirror’s center of curvature and vertex.

Figure 2.5: Concave and convex mirrors.

Spherical mirrors come in two main types: concave and convex. Imagine cutting a section from a shiny sphere. If you polish the inside, it’s a concave mirror. If you polish the outside, it’s a convex mirror. The optical axis is a line of symmetry crucial for understanding how these mirrors work.

Focal Point and Focal Length

• Focal Point (F): The point where parallel rays converge after reflecting from a concave mirror (real focus), or appear to diverge from after reflecting from a convex mirror (virtual focus).
• Focal Length (f): The distance along the optical axis from the mirror to the focal point.
• Spherical Aberration: Occurs in large spherical mirrors where parallel rays do not converge at a single focal point, leading to blurred images.
• Small spherical mirrors approximate parabolic mirrors well, reducing spherical aberration.

Figure 2.6: Focal point in (a) parabolic, (b) large spherical, and (c) small spherical mirrors.

The focal point is key to understanding spherical mirrors. For concave mirrors, parallel incoming light rays converge at a real focal point. For convex mirrors, they diverge as if coming from a virtual focal point behind the mirror. The focal length is the distance to this point. Large spherical mirrors can suffer from spherical aberration, where not all parallel rays converge perfectly. This is why smaller spherical mirrors or parabolic mirrors are preferred for precise focusing.

Focal Length and Radius of Curvature

In the small-angle (paraxial) approximation, the focal length (\(f\)) of a spherical mirror is half its radius of curvature (\(R\)):

\[f = \frac{R}{2}\]

Note

This approximation is valid when rays make small angles with the optical axis and are close to it.

This simple relationship, \(f = R/2\), is fundamental for spherical mirrors. It’s derived using geometry and the small-angle approximation, which assumes that light rays are close to the optical axis. This approximation is crucial for simplifying calculations in geometric optics.

Ray Tracing to Locate Images

To locate an image formed by a spherical mirror, draw at least two “principal” rays from a point on the object:

1. Parallel Ray: Enters parallel to the optical axis, reflects through the focal point (or appears to diverge from it for convex mirrors).
2. Focal Ray: Travels through the focal point (or towards it), reflects parallel to the optical axis.
3. Center of Curvature Ray: Travels towards the center of curvature, reflects back along the same path.
4. Vertex Ray: Strikes the vertex, reflects symmetrically about the optical axis.

Ray tracing is a powerful visual tool. We use these four principal rays because their paths after reflection are predictable and easy to draw, allowing us to accurately locate the image without complex calculations. You only need two to find the image, but drawing more can help verify your work.

Ray Tracing: Concave Mirror

Figure 2.9 (a): Principal rays for a concave mirror.

• Object beyond C: Real, inverted, smaller image between F and C.
• Object at C: Real, inverted, same size image at C.
• Object between C and F: Real, inverted, larger image beyond C.
• Object at F: Image at infinity.
• Object inside F: Virtual, upright, larger image behind the mirror.

This diagram shows how the principal rays work for a concave mirror. Notice how the reflected rays converge to form a real, inverted image in this case. The location and characteristics of the image depend heavily on the object’s position relative to the focal point (F) and center of curvature (C).

Ray Tracing: Convex Mirror

Figure 2.9 (b): Principal rays for a convex mirror.

• For any real object, a convex mirror always forms a:
  • Virtual image
  • Upright image
  • Smaller image
  • Located behind the mirror, between the vertex and the focal point.

For convex mirrors, the reflected rays diverge, making the focal point virtual. When extended backward, these rays appear to originate from a virtual image behind the mirror. This image is always virtual, upright, and smaller than the object, regardless of the object’s position. This property makes convex mirrors useful as passenger-side car mirrors or security mirrors.

The Mirror Equation

The mirror equation relates object distance (\(d_o\)), image distance (\(d_i\)), and focal length (\(f\)):

\[\frac{1}{d_o} + \frac{1}{d_i} = \frac{1}{f}\]

• For a plane mirror (\(R = \infty\), so \(f = \infty\)):

\(\frac{1}{d_o} + \frac{1}{d_i} = 0 \implies d_i = -d_o\).

This is a cornerstone of geometric optics for mirrors. The mirror equation, derived from geometry, allows us to quantitatively determine image locations. Notice how it simplifies to the plane mirror equation when the focal length is infinite.

Sign Convention for Spherical Mirrors

A consistent sign convention is crucial for using the mirror equation:

1. Focal Length (\(f\)):
   • Positive for concave mirrors.
   • Negative for convex mirrors.
2. Image Distance (\(d_i\)):
   • Positive for real images (on the same side as the object for a concave mirror, or on the opposite side for a lens).
   • Negative for virtual images (behind the mirror).
3. Object Distance (\(d_o\)):
   • Positive for real objects (in front of the mirror).
   • Negative for virtual objects (rare, occurs when image from another element acts as an object).

Mastering the sign convention is essential. It provides a standardized way to interpret the results of calculations. For example, a positive image distance means a real image, while a negative image distance means a virtual image.

Image Magnification

The linear magnification (\(m\)) of an image is defined as:

\[m = \frac{h_i}{h_o} = -\frac{d_i}{d_o}\]

• If \(m > 0\), the image is upright.
• If \(m < 0\), the image is inverted.
• If \(|m| > 1\), the image is larger than the object.
• If \(|m| < 1\), the image is smaller than the object.
• If \(|m| = 1\), the image is the same size as the object.

Magnification tells us about the image’s size and orientation relative to the object. A negative magnification always means an inverted image, and its absolute value indicates how much the image is enlarged or reduced.

Example 2.2: Image in a Convex Mirror

A keratometer measures the curvature of the cornea (acts like a convex mirror). If the light source is 12 cm from the cornea and the image magnification is 0.032, what is the radius of curvature of the cornea?

Strategy

1. Given \(d_o = 12 \text{ cm}\) and \(m = 0.032\).
2. Use the magnification equation to find \(d_i\).
3. Use the mirror equation to find \(f\).
4. Then, \(R = 2f\).

Let’s apply our knowledge to a practical example. Keratometers are crucial in optometry. We’re given the object distance and magnification, and need to find the radius of curvature of the cornea. We’ll use the magnification formula first, then the mirror equation, and finally the focal length-radius relationship.

Example 2.2: Solution

1. Find \(d_i\): \(m = -\frac{d_i}{d_o} \implies d_i = -m d_o\) \(d_i = -(0.032)(12 \text{ cm}) = -0.384 \text{ cm}\)

2. Find \(f\): \(\frac{1}{d_o} + \frac{1}{d_i} = \frac{1}{f}\) \(f = \left(\frac{1}{12 \text{ cm}} + \frac{1}{-0.384 \text{ cm}}\right)^{-1} = -0.40 \text{ cm}\)

3. Find \(R\): \(R = 2f = 2(-0.40 \text{ cm}) = -0.80 \text{ cm}\)

Note

The negative focal length confirms it’s a convex mirror, and the negative radius of curvature is consistent with our sign convention for convex mirrors.

Following the strategy, we first calculated the image distance, which came out negative, indicating a virtual image behind the mirror, as expected for a convex mirror. Then, we used the mirror equation to find the focal length, which was also negative, consistent with a convex mirror. Finally, the radius of curvature is simply twice the focal length. A typical cornea radius is indeed small, so this result is reasonable.

Aberrations in Spherical Mirrors

When the small-angle approximation breaks down, images created by spherical mirrors can become distorted. This is called aberration.

• Spherical Aberration: Rays far from the optical axis focus at different points than paraxial rays, blurring the image.

  

  Figure 2.12 (a): Spherical aberration.

  Figure 2.12 (b): Comatic aberration.

• Coma (Comatic Aberration): Occurs when incoming rays are not parallel to the optical axis. Rays focus at different heights and focal lengths, producing a comet-like “tail” in the image.

While the small-angle approximation simplifies calculations, real-world mirrors aren’t perfect. Aberrations are distortions that arise when this approximation fails. Spherical aberration blurs images when light hits the mirror far from the center, while coma causes a “comet-tail” effect for off-axis objects. These are important considerations in designing high-quality optical instruments.

2.3 Images Formed by Refraction

Learning Objectives

By the end of this section, you will be able to:

• Describe image formation by a single refracting surface.
• Determine the location of an image and calculate its properties by using a ray diagram.
• Determine the location of an image and calculate its properties by using the equation for a single refracting surface.

Moving from reflection, we now enter the world of refraction, where light bends as it passes from one medium to another. This section will explore how images are formed by these bending light rays, starting with simpler plane interfaces and then moving to spherical surfaces.

Refraction at a Plane Interface: Apparent Depth

• When light passes from one medium to another (e.g., water to air), it bends (refracts).
• This causes objects submerged in water to appear at a shallower depth than they actually are, known as apparent depth.
• For normal viewing (small angle approximation), the apparent depth (\(h_i\)) is:

\[h_i = \left(\frac{n_2}{n_1}\right) h_o\]

where \(h_o\) is the real depth, \(n_1\) is the refractive index of the object’s medium, and \(n_2\) is the refractive index of the observer’s medium.

Figure 2.14: Apparent depth due to refraction.

You’ve probably noticed a spoon appearing bent in a glass of water, or a fish looking closer than it actually is. This is due to refraction at a plane interface. The formula for apparent depth explains this phenomenon, showing that the apparent depth depends on the refractive indices of the two media involved. For example, a fish in water (n=1.33) viewed from air (n=1.0) appears at 3/4 of its actual depth.

Refraction at a Spherical Interface

• Spherical surfaces are common in optics.
• Consider a point source \(P\) in medium \(n_1\) in front of a convex spherical surface of radius \(R\) separating it from medium \(n_2\).
• Using Snell’s Law and the small-angle approximation, we derive the equation for a single refracting surface:

\[\frac{n_1}{d_o} + \frac{n_2}{d_i} = \frac{n_2 - n_1}{R}\]

where \(d_o\) is object distance, \(d_i\) is image distance.

Figure 2.15: Refraction at a convex surface.

Spherical refracting surfaces are the building blocks of lenses. This equation is derived from Snell’s law and geometry, and it allows us to calculate image locations for single curved interfaces. It’s a bit more complex than the mirror equation due to the two different refractive indices.

Sign Convention for Single Refracting Surfaces

1. Radius of Curvature (\(R\)):
   • \(R > 0\) if the surface is convex toward the object.
   • \(R < 0\) if the surface is concave toward the object.
2. Image Distance (\(d_i\)):
   • \(d_i > 0\) if the image is real and on the opposite side from the object.
   • \(d_i < 0\) if the image is virtual and on the same side as the object.

Just like with mirrors, a consistent sign convention is vital for refracting surfaces. Pay close attention to the curvature of the surface relative to the object and the location of the image when assigning signs.

2.4 Thin Lenses

Learning Objectives

By the end of this section, you will be able to:

• Use ray diagrams to locate and describe the image formed by a lens.
• Employ the thin-lens equation to describe and locate the image formed by a lens.

Lenses are ubiquitous, from eyeglasses to cameras to telescopes. We’ll now combine what we’ve learned about refraction at spherical surfaces to understand how thin lenses work. We’ll use both ray tracing and equations to analyze these crucial optical components.

Converging and Diverging Lenses

• Lenses manipulate light by refraction.
• Converging Lens (Convex): Thicker in the middle, causes parallel rays to converge at a real focal point.
• Diverging Lens (Concave): Thinner in the middle, causes parallel rays to diverge as if from a virtual focal point.
• Focal Length (\(f\)): Distance from the lens center to its focal point.

Figure 2.18: (a) Converging lens, (b) diverging lens.

Lenses come in various shapes, but their primary function is to converge or diverge light. Converging lenses are typically convex and focus light, while diverging lenses are concave and spread light out. The focal length is a defining characteristic for both types.

Thin Lens Approximation

• A lens is “thin” if its thickness (\(t\)) is much less than the radii of curvature of its surfaces.
• In this approximation, light rays are assumed to bend only once at the center of the lens.
• Rays passing through the center of a thin lens are undeviated.

Figure 2.19: Thin-lens approximation.

The thin-lens approximation greatly simplifies analysis. It allows us to treat the lens as a single plane where refraction occurs, rather than two separate curved surfaces. This simplification is valid for many common lenses. An important property is that rays going through the exact center of the lens pass straight through without deviation.

Ray Tracing for Thin Lenses

Similar to mirrors, ray tracing for thin lenses uses principal rays:

1. Parallel Ray: Enters parallel to the optical axis.
   • Converging: Exits through the focal point on the opposite side.
   • Diverging: Exits as if from the focal point on the same side.
2. Central Ray: Passes through the center of the lens, is not deviated.
3. Focal Ray:
   • Converging: Passes through the focal point on the same side, exits parallel.
   • Diverging: Approaches towards the focal point on the opposite side, exits parallel.

These three rules are your toolkit for visually determining image formation with thin lenses. They’re analogous to the mirror ray-tracing rules but adapted for refraction through a lens. Again, you only need two of these rays to locate the image.

Image Formation by a Converging Lens

Figure 2.22: Ray tracing for a converging lens.

• Rays from a point on the object converge at a corresponding point to form the image.
• Image can be real or virtual, inverted or upright, larger or smaller depending on object distance.
• In this example (\(d_o > f\)), a real, inverted image is formed.

This diagram shows how a converging lens forms a real, inverted image when the object is outside the focal length. The three principal rays converge to locate the image of the arrow’s tip. Note the definitions of object distance (\(d_o\)), image distance (\(d_i\)), object height (\(h_o\)), and image height (\(h_i\)).

The Thin-Lens Equation

The thin-lens equation relates object distance (\(d_o\)), image distance (\(d_i\)), and focal length (\(f\)):

\[\frac{1}{d_o} + \frac{1}{d_i} = \frac{1}{f}\]

• This equation is identical in form to the mirror equation.

The Lens Maker’s Equation relates focal length (\(f\)) to the lens’s properties:

\[\frac{1}{f} = \left(\frac{n_2}{n_1} - 1\right) \left(\frac{1}{R_1} - \frac{1}{R_2}\right)\]

• \(n_1\): Refractive index of the surrounding medium.
• \(n_2\): Refractive index of the lens material.
• \(R_1\), \(R_2\): Radii of curvature of the lens surfaces.

These two equations are the workhorses for quantitative lens analysis. The thin-lens equation is elegantly similar to the mirror equation, while the lens maker’s equation tells us how to design a lens with a specific focal length based on its material and geometry. Remember, it’s crucial to use the correct sign conventions for these equations.

Sign Conventions for Lenses

1. Image Distance (\(d_i\)):
   • Positive if the image is on the side opposite the object (real image).
   • Negative if the image is on the same side as the object (virtual image).
2. Focal Length (\(f\)):
   • Positive for a converging lens.
   • Negative for a diverging lens.
3. Radii of Curvature (\(R_1, R_2\)):
   • Positive for a surface convex toward the object.
   • Negative for a surface concave toward the object.

These sign conventions are critical for getting correct results when using the thin-lens and lens maker’s equations. Pay close attention to the definition of ‘positive’ for image distance and curvature, as it differs slightly from mirrors.

Magnification for Lenses

The linear magnification (\(m\)) for lenses is defined identically to mirrors:

\[m = \frac{h_i}{h_o} = -\frac{d_i}{d_o}\]

• If \(m > 0\), the image is upright.
• If \(m < 0\), the image is inverted.
• If \(|m| > 1\), the image is larger.
• If \(|m| < 1\), the image is smaller.

Magnification for lenses shares the same formula and interpretation as for mirrors. It provides immediate insight into whether an image is upright or inverted, and whether it’s enlarged or reduced.

Example 2.4: Converging Lens and Different Object Distances

A 3.0 cm high object is placed in front of a convex lens (\(f = 10.0 \text{ cm}\)). Find the location, orientation, and magnification for: (a) \(d_o = 50.0 \text{ cm}\) (b) \(d_o = 5.00 \text{ cm}\) (c) \(d_o = 20.0 \text{ cm}\)

Strategy

1. Use the thin-lens equation to find \(d_i\).
2. Use the magnification equation to find \(m\) and \(h_i\).
3. Interpret the signs of \(d_i\) and \(m\).

This example demonstrates the versatility of the thin-lens equation and how image characteristics change with object distance. We’ll work through three different scenarios for a converging lens.

Example 2.4: Solution (a) \(d_o = 50.0 \text{ cm}\)

1. Image Distance (\(d_i\)): \(\frac{1}{d_i} = \frac{1}{f} - \frac{1}{d_o} = \frac{1}{10.0 \text{ cm}} - \frac{1}{50.0 \text{ cm}} = \frac{5-1}{50.0 \text{ cm}} = \frac{4}{50.0 \text{ cm}}\) \(d_i = \frac{50.0}{4} \text{ cm} = 12.5 \text{ cm}\) (\(d_i > 0 \implies\) real image, opposite side of lens)

2. Magnification (\(m\)): \(m = -\frac{d_i}{d_o} = -\frac{12.5 \text{ cm}}{50.0 \text{ cm}} = -0.250\) (\(m < 0 \implies\) inverted image; \(|m| < 1 \implies\) smaller image)

3. Image Height (\(h_i\)): \(h_i = m h_o = (-0.250)(3.0 \text{ cm}) = -0.75 \text{ cm}\) (Height is 0.75 cm, inverted.)

For an object placed far from the converging lens, we get a real, inverted, and smaller image. This is a common setup for cameras, where a distant scene is focused onto a small sensor.

Example 2.4: Solution (b) \(d_o = 5.00 \text{ cm}\)

1. Image Distance (\(d_i\)): \(\frac{1}{d_i} = \frac{1}{f} - \frac{1}{d_o} = \frac{1}{10.0 \text{ cm}} - \frac{1}{5.00 \text{ cm}} = \frac{1-2}{10.0 \text{ cm}} = -\frac{1}{10.0 \text{ cm}}\) \(d_i = -10.0 \text{ cm}\) (\(d_i < 0 \implies\) virtual image, same side of lens)

2. Magnification (\(m\)): \(m = -\frac{d_i}{d_o} = -\frac{-10.0 \text{ cm}}{5.00 \text{ cm}} = +2.00\) (\(m > 0 \implies\) upright image; \(|m| > 1 \implies\) larger image)

3. Image Height (\(h_i\)): \(h_i = m h_o = (2.00)(3.0 \text{ cm}) = 6.0 \text{ cm}\) (Height is 6.0 cm, upright.)

Here, the object is placed inside the focal length. This configuration produces a virtual, upright, and magnified image, which is exactly how a simple magnifying glass works.

Example 2.4: Solution (c) \(d_o = 20.0 \text{ cm}\)

1. Image Distance (\(d_i\)): \(\frac{1}{d_i} = \frac{1}{f} - \frac{1}{d_o} = \frac{1}{10.0 \text{ cm}} - \frac{1}{20.0 \text{ cm}} = \frac{2-1}{20.0 \text{ cm}} = \frac{1}{20.0 \text{ cm}}\) \(d_i = 20.0 \text{ cm}\) (\(d_i > 0 \implies\) real image, opposite side of lens)

2. Magnification (\(m\)): \(m = -\frac{d_i}{d_o} = -\frac{20.0 \text{ cm}}{20.0 \text{ cm}} = -1.00\) (\(m < 0 \implies\) inverted image; \(|m| = 1 \implies\) same size image)

3. Image Height (\(h_i\)): \(h_i = m h_o = (-1.00)(3.0 \text{ cm}) = -3.0 \text{ cm}\) (Height is 3.0 cm, inverted.)

When the object is placed at twice the focal length (2F), the converging lens forms a real, inverted image of the same size, also at 2F on the other side. This is a special case that can be easily remembered.

2.5 The Eye

Learning Objectives

By the end of this section, you will be able to:

• Understand the basic physics of how images are formed by the human eye.
• Recognize several conditions of impaired vision as well as the optics principles for treating these conditions.

The human eye is an extraordinary optical instrument. We’ll explore its anatomy from an optics perspective, focusing on how it forms images and how common vision defects like nearsightedness and farsightedness can be corrected using lenses.

Anatomy of the Eye

• Cornea and Lens: Act as a single thin lens system.
• Retina: Light-sensitive surface where a real, inverted image is projected.
• Fovea: Center of the retina with highest receptor density and visual acuity.
• Pupil: Variable opening that controls the amount of light entering.
• Accommodation: The eye’s ability to adjust the focal length of the lens by changing its shape to focus on objects at different distances.

Figure 2.29: Basic anatomy of the eye.

The eye is a complex biological marvel. From an optics perspective, the cornea and lens are the primary focusing elements, working together to project an image onto the retina. Accommodation, the changing shape of the lens, is what allows us to focus on both near and far objects.

Optical Power of the Eye

• The cornea provides most of the eye’s focusing power (focal length \(\approx 2.3 \text{ cm}\)).
• The lens provides finer focus (focal length \(\approx 6.4 \text{ cm}\)).
• Optical Power (\(P\)): Defined as \(P = 1/f\) (where \(f\) is in meters). Units are diopters (D) (\(1 \text{ D} = 1 \text{ m}^{-1}\)).
• For multiple lenses close together, total power is the sum: \(P_{total} = P_1 + P_2 + \dots\)

Example 2.6: Effective Focal Length of the Eye

Given \(f_{cornea} = 2.3 \text{ cm}\) and \(f_{lens} = 6.4 \text{ cm}\). Find the net focal length and optical power of the eye.

Solution

\(\frac{1}{f_{eye}} = \frac{1}{f_{cornea}} + \frac{1}{f_{lens}} = \frac{1}{2.3 \text{ cm}} + \frac{1}{6.4 \text{ cm}}\)

\(f_{eye} = 1.69 \text{ cm} \approx 1.7 \text{ cm}\)

\(P_{eye} = \frac{1}{0.0169 \text{ m}} \approx 59 \text{ D}\)

Optical power is a convenient way to describe lens strength, especially for optometrists. The high optical power of the eye, mainly contributed by the cornea, highlights its strong focusing ability. The example calculation shows how individual powers add up to give the total optical power of the eye’s lens system.

Vision Defects: Nearsightedness (Myopia)

• Problem: Distant objects are blurry; eye over-converges light. Image forms in front of the retina.
• Far Point: Closest point from which objects are seen clearly (normally infinity). For myopic eye, far point is finite.
• Correction: Use a diverging (concave) lens to reduce the overall optical power. The diverging lens creates a virtual image of distant objects at the person’s far point, which the eye can then focus on.

Figure 2.31 (a): Myopic eye.

Figure 2.32: Correction for nearsightedness.

Nearsightedness is a common condition where the eye’s focusing power is too strong, or the eyeball is too long. The diverging corrective lens works by spreading out the light rays before they enter the eye, effectively “moving” the distant object’s image back onto the retina.

Example 2.8: Correcting Nearsightedness

A nearsighted person has a far point of 30.0 cm. What optical power of eyeglass lens is needed if the lens is 1.50 cm from the eye?

Strategy

1. Distant objects (\(d_o = \infty\)) must form a virtual image (\(d_i\)) at the person’s far point relative to the eye.
2. Calculate the required image distance from the eyeglass lens.
3. Use the thin-lens equation in terms of power: \(P = \frac{1}{d_o} + \frac{1}{d_i}\).

Let’s put the principles of vision correction into practice. The key here is that the corrective lens needs to create a virtual image of a distant object at a position the nearsighted eye can actually focus on, which is its far point. We must account for the distance between the eyeglass lens and the eye.

Example 2.8: Solution

1. Required image position for the eye: Virtual image at \(30.0 \text{ cm}\) from the eye.
2. Image distance (\(d_i\)) from eyeglass lens: \(d_i = -(30.0 \text{ cm} - 1.50 \text{ cm}) = -28.5 \text{ cm} = -0.285 \text{ m}\) (Negative because it’s a virtual image on the same side as the object.)
3. Optical power (\(P\)): \(P = \frac{1}{d_o} + \frac{1}{d_i} = \frac{1}{\infty} + \frac{1}{-0.285 \text{ m}}\) \(P = 0 - 3.51 \text{ D} = -3.51 \text{ D}\)

Note

The negative power indicates a diverging lens, as expected for nearsightedness.

The calculation confirms that a negative optical power is needed, meaning a diverging lens. This is consistent with the understanding that nearsightedness requires spreading out light rays before they reach the eye.

Vision Defects: Farsightedness (Hyperopia)

• Problem: Near objects are blurry; eye under-converges light. Image forms behind the retina.
• Near Point: Farthest point from which objects are seen clearly (normally 25 cm). For hyperopic eye, near point is greater than 25 cm.
• Correction: Use a converging (convex) lens to increase the overall optical power. The converging lens creates a virtual image of near objects at the person’s near point, which the eye can then focus on.

Figure 2.31 (b): Hyperopic eye.

Figure 2.33: Correction for farsightedness.

Farsightedness is the opposite of nearsightedness. The eye’s focusing power is too weak, or the eyeball is too short. A converging corrective lens adds power, bringing the image of near objects forward onto the retina.

2.6 The Camera

Learning Objectives

By the end of this section, you will be able to:

• Describe the optics of a camera.
• Characterize the image created by a camera.

From the human eye, we move to a man-made optical marvel: the camera. We’ll explore the basic principles behind how cameras capture images, focusing on the role of lenses and detectors.

Camera Optics

• A camera’s optics are similar to a single lens, where the object distance is significantly larger than the lens’s focal length.
• Lens: Forms a real, inverted image. In smartphones, it’s often a fixed wide-angle lens (focal length \(\approx 4-5 \text{ mm}\)).
• Detector (e.g., CCD): Records the image. In digital cameras, the CCD is a matrix of pixels.
• Fixed Lens-to-Detector Distance: Unlike the eye, many cameras (like smartphones) have a fixed distance between the lens and the detector.

Figure 2.35: Modern digital camera optics.

At its core, a camera is a light-tight box with a lens and a sensor. The lens projects a real, inverted image onto the sensor. The remarkable aspect of smartphone cameras is their ability to focus on both near (selfie) and far objects despite a fixed lens-to-sensor distance, thanks to their very short focal lengths.

Image Formation in a Smartphone Camera

Consider a smartphone camera with \(f = 5 \text{ mm}\).

• Selfie (\(d_o \approx 50 \text{ cm} = 500 \text{ mm}\)):

  \(\frac{1}{d_i} = \frac{1}{5 \text{ mm}} - \frac{1}{500 \text{ mm}}\)

  \(d_i = \left(\frac{100-1}{500 \text{ mm}}\right)^{-1} = \frac{500}{99} \text{ mm} \approx 5.05 \text{ mm}\)

• Distant Object (\(d_o \approx 5 \text{ m} = 5000 \text{ mm}\)):

  \(\frac{1}{d_i} = \frac{1}{5 \text{ mm}} - \frac{1}{5000 \text{ mm}}\)

  \(d_i \approx 5.005 \text{ mm}\)

Tip

For a short focal length lens, the image distance changes very little for objects ranging from a few tens of centimeters to infinity, making fixed-focus cameras feasible.

These calculations reveal why smartphone cameras can work with a fixed lens-to-detector distance. Because the focal length is so small, the image distance for both relatively close (selfie) and very distant objects is extremely close to the focal length itself. This tiny variation can be accommodated by the detector, often with the help of image processing.

2.7 The Simple Magnifier

Learning Objectives

By the end of this section, you will be able to:

• Understand the optics of a simple magnifier.
• Characterize the image created by a simple magnifier.

We’ve all used a magnifying glass. Now, let’s understand the physics behind how it works, focusing on the concept of angular magnification which describes how much larger an object appears to the eye.

Apparent Size and Angular Magnification

• The perceived size of an object depends on the angle it subtends from the eye (\(\theta_{object}\)).
• A simple magnifier (convex lens) is used to create a virtual, upright, and enlarged image.
• This magnified image subtends a larger angle (\(\theta_{image}\)) from the eye, making the object appear larger.
• Angular Magnification (\(M\)): Ratio of the angle subtended by the image to the angle subtended by the object (at the near point, 25 cm).

\[M = \frac{\theta_{image}}{\theta_{object}}\]

Figure 2.36: Apparent size.

Figure 2.37: Simple magnifier.

The concept of “apparent size” is key. Holding an object closer makes it look bigger because it subtends a larger angle at your eye. A magnifying glass takes advantage of this by creating a virtual image that is both larger and can be viewed closer to your eye than the actual object. Angular magnification quantifies this apparent increase in size.

Angular Magnification Formulas

For a simple magnifier, where the image is formed at the eye’s near point (\(L=25 \text{ cm}\)) and the lens is held close to the eye (\(\ell=0\)):

\[M = 1 + \frac{25 \text{ cm}}{f}\]

When the image is formed at infinity (\(L=\infty\)), providing relaxed viewing:

\[M = \frac{25 \text{ cm}}{f}\]

Important

Greater magnification is achieved with a shorter focal length lens.

These two formulas are the most commonly used for simple magnifiers. The first gives the maximum magnification, but requires the eye to accommodate for a near image. The second provides slightly less magnification but allows for comfortable, relaxed viewing as the eye is focused at infinity. Both show the inverse relationship between focal length and magnification.

2.8 Microscopes and Telescopes

Learning Objectives

By the end of this section, you will be able to:

• Explain the physics behind the operation of microscopes and telescopes.
• Describe the image created by these instruments and calculate their magnifications.

Finally, we arrive at the most sophisticated optical instruments: microscopes and telescopes. These devices extend our vision beyond the capabilities of the naked eye, allowing us to explore both the incredibly small and the incredibly distant. We’ll break down their components and how they achieve high magnification.

Compound Microscopes

• Uses two convex lenses:
  • Objective lens: Short focal length (\(f_{obj}\)), forms a real, inverted, magnified intermediate image.
  • Eyepiece (ocular): Longer focal length (\(f_{eye}\)), acts as a magnifier for the intermediate image, forming a final magnified virtual image.
• Total Magnification (\(M_{net}\)): Product of the linear magnification of the objective (\(m_{obj}\)) and the angular magnification of the eyepiece (\(M_{eye}\)).

\[M_{net} = m_{obj} M_{eye}\]

• Often, the intermediate image is at the focal point of the eyepiece for relaxed viewing.

Figure 2.39: Compound microscope optics.

Compound microscopes are essential for exploring the micro-world. They achieve much higher magnification than a simple magnifier by using two stages of magnification: the objective creates a magnified real image, and the eyepiece then further magnifies this image. The final image is typically virtual and inverted.

Magnifying Power of a Compound Microscope

When the final image is at infinity (for relaxed viewing):

\[M_{net} = -\frac{L}{f_{obj}} \left(\frac{25 \text{ cm}}{f_{eye}}\right)\]

• \(L\): Tube length (distance between objective and eyepiece focal points, often standardized at 16 cm).
• \(f_{obj}\): Focal length of the objective lens.
• \(f_{eye}\): Focal length of the eyepiece.
• The minus sign indicates an inverted final image.

This formula for the magnifying power of a compound microscope is very practical because it only depends on the focal lengths of the two lenses and the standardized tube length. A shorter focal length for both the objective and eyepiece contributes to higher magnification.

Telescopes

• Used for viewing distant objects; produce a larger image than the unaided eye.
• Objective lens/mirror: Gathers light from a distant object and forms a real, inverted intermediate image near its focal point.
• Eyepiece: Magnifies the intermediate image.
• Angular Magnification (\(M\)): Ratio of angles subtended by the image and the object.

\[M = -\frac{f_{obj}}{f_{eye}}\]

• Refracting Telescopes: Use lenses (Galileo’s design, modern designs).
• Reflecting Telescopes: Use mirrors (Newtonian, Cassegrain designs), eliminate chromatic aberration.

Figure 2.40 (b): Refracting telescope.

Figure 2.44 (b): Cassegrain reflecting telescope.

Telescopes, like microscopes, use multiple optical elements, but for very different purposes: to see distant objects. The objective is designed to gather as much light as possible and form an initial image, which the eyepiece then magnifies. To maximize magnification, you want a long focal length objective and a short focal length eyepiece. Reflecting telescopes overcome the limitations of refracting telescopes, such as chromatic aberration, by using mirrors instead of lenses.

Key Takeaways

• Plane mirrors form virtual, upright images that are the same size as the object, located behind the mirror (\(d_i = -d_o\)).
• Spherical mirrors (concave and convex) form varied images depending on object distance, described by ray tracing and the mirror equation (\(\frac{1}{d_o} + \frac{1}{d_i} = \frac{1}{f}\)).
• Refraction at interfaces causes apparent depth and image formation. A single refracting surface is described by \(\frac{n_1}{d_o} + \frac{n_2}{d_i} = \frac{n_2 - n_1}{R}\).
• Thin lenses (converging and diverging) form images via refraction, also described by the thin-lens equation (\(\frac{1}{d_o} + \frac{1}{d_i} = \frac{1}{f}\)). The lens maker’s equation relates focal length to lens material and curvature.
• Magnification (\(m = -d_i/d_o\)) describes image size and orientation for both mirrors and lenses.
• The human eye is a complex optical system, using accommodation to focus. Vision defects (myopia, hyperopia) are corrected with diverging or converging lenses, respectively.
• Cameras use lenses to project real, inverted images onto a sensor. Short focal lengths allow fixed-focus cameras to work for a wide range of object distances.
• Simple magnifiers increase apparent size via angular magnification (\(M = 1 + \frac{25 \text{ cm}}{f}\) or \(M = \frac{25 \text{ cm}}{f}\)).
• Microscopes and telescopes use multiple lenses to achieve high magnifications for small or distant objects, respectively.

Key Equations

Equation	Description
\(d_i = -d_o\)	Image distance for a plane mirror
\(f = R/2\)	Focal length of a spherical mirror (small-angle approximation)
\(\frac{1}{d_o} + \frac{1}{d_i} = \frac{1}{f}\)	Mirror Equation / Thin-Lens Equation
\(m = h_i/h_o = -d_i/d_o\)	Linear magnification (mirrors and lenses)
\(\frac{n_1}{d_o} + \frac{n_2}{d_i} = \frac{n_2 - n_1}{R}\)	Single refracting surface equation
\(\frac{1}{f} = \left(\frac{n_2}{n_1} - 1\right) \left(\frac{1}{R_1} - \frac{1}{R_2}\right)\)	Lens Maker’s Equation (for a thin lens in medium \(n_1\))
\(P = 1/f\)	Optical power (in diopters, \(f\) in meters)
\(M = 1 + \frac{25 \text{ cm}}{f}\)	Angular magnification of a simple magnifier (image at near point)
\(M = \frac{25 \text{ cm}}{f}\)	Angular magnification of a simple magnifier (image at infinity)
\(M_{net} = -\frac{L}{f_{obj}} \left(\frac{25 \text{ cm}}{f_{eye}}\right)\)	Net magnification of a compound microscope (image at infinity)
\(M = -f_{obj}/f_{eye}\)	Angular magnification of a telescope

Key Terms

Term	Definition
Real Image	An image formed when light rays actually converge at a point, able to be projected onto a screen.
Virtual Image	An image formed when light rays appear to diverge from a point, but do not actually pass through it; cannot be projected.
Focal Point (\(F\))	The point where parallel rays converge after reflection/refraction, or from where they appear to diverge.
Focal Length (\(f\))	The distance from the mirror/lens to its focal point.
Concave Mirror	A spherical mirror with the reflective surface on the inner side of the sphere.
Convex Mirror	A spherical mirror with the reflective surface on the outer side of the sphere.
Converging Lens	A lens that causes parallel light rays to converge to a single focal point (e.g., convex lens).
Diverging Lens	A lens that causes parallel light rays to spread out as if from a single focal point (e.g., concave lens).
Optical Power (\(P\))	The reciprocal of the focal length of a lens, measured in diopters (D).
Magnification (\(m\))	The ratio of image height to object height, indicating the relative size and orientation of an image (\(h_i/h_o\)).
Angular Magnification (\(M\))	The ratio of the angle subtended by the image to the angle subtended by the object, describing apparent size.
Myopia	Nearsightedness; the eye focuses distant objects in front of the retina, requiring a diverging lens for correction.
Hyperopia	Farsightedness; the eye focuses near objects behind the retina, requiring a converging lens for correction.
Accommodation	The process by which the eye’s lens changes shape to alter its focal length, allowing objects at various distances to be focused on the retina.
Spherical Aberration	A distortion in images formed by spherical mirrors or lenses, where rays far from the optical axis do not converge at the same point as paraxial rays.
Ray Tracing	A graphical method for locating and characterizing images by drawing the paths of light rays through optical systems.