Geometric Optics Unit 10. Optics We have already seen that light can be thought of as an EM wave. We...

Geometric Optics

Unit 10

Optics

• We have already seen that light can be thought of as an EM wave.

• We perceive light using our eyes, and depend heavily on sight.

• For this reason, the study of light and how it behaves is an important one.

• This field is called optics.

Optics• We can perceive an object by sight in two ways:– The object may be a source of light (like a light

bulb, fire, or a star).– We see the light that is reflected off of the object.

• The subject of how objects emit light is rather complicated.

• We will look at how light is reflected (and transmitted).

Geometric Optics

• Although light is an EM wave, many of its most important properties can be understood without worrying about this fact.

• These include reflection, refraction, and the formation of images using mirrors and lenses.

• This area of optics is called geometric optics.

The Ray Model of Light

• When the wave characteristics of light can be ignored, we employ the ray model to describe the propagation of light.

• This model assumes that light travels in straight lines.

• These lines are represented with rays.


• The ray itself is actually an idealization.

• It represents an extremely narrow beam of light.

• The ray points along the direction the EM wave is moving.

• Therefore, it is perpendicular to both the E and B fields of the wave.


• In the ray model, rays are reflected off of each point on the object is many different directions.

• Some of these rays reach our eyes.


• When we see an object, light is reaching out eyes from every point on the object.

• This also explains why we can’t see the far side of the object: the light rays can’t reach our eye since it only travels in straight lines.

Thoughts on Geometric Optics

• All phenomena covered in geometric optics can be described using light rays.

• In our pictures, we will generally only draw the rays that reach our eyes.

• Because these explanations involve straight lines at various angles, these are all essentially geometry problems.

Reflection

Reflection

• When light strikes the surface of an object, at least some of that light is reflected off of the object.

• The rest of the light is either transmitted through the object, or absorbed by the object (and transformed into heat).

• For very shiny objects (such as mirrors) over 95% of the light is reflected.

Reflection

• Let’s consider a narrow beam of light striking a flat surface.

• The beam is represented by a light ray, called the incident ray.

Reflection

• The ray makes an angle with the line perpendicular to the surface.

• This angle is called the angle of incidence and is represented by θi.

Reflection

• The incident ray is reflected off the surface and travels off in a new direction.

• This ray is called the reflected ray.

Reflection

• The reflected ray also makes an angle with the line normal to the surface.

• This angle is called the angle of reflection and is represented by θr.

Reflection

• How are these two angles related?

• The answer is given by the law of reflection.

• This is the same law we saw for 2D and 3D waves last quarter.

Law of Reflection

• The law of reflection states

The angle of reflection equals the angle of incidence.

θr = θi

Law of Reflection

• Furthermore, both the incident and reflected rays lie in the same plane.

• It is also the same plane as the line normal to the surface.

Surface Types

• What happens if the surface is not flat, but is rough like the surface of water (or even a piece of paper)?

• The law of reflection still applies at each point along the surface.

• However, since the surface is rough, it is tilted different ways at different points.

Diffuse Reflections• As a result, when a beam of light hits a rough

surface, the light is reflected in many different directions depending on where it hits the surface.

• This is called diffuse reflection.

Diffuse Reflections• When you look at light reflected off of a rough

surface, it is possible to see the image from many different angles.

• This is because your eye is picking up different reflected rays at different points.

Diffuse Reflections• When you read a book or look at other

everyday objects, you are seeing diffuse reflections.

• You can see these objects from many different angles because light is reflected at many different angles off of the rough surfaces.

Specular Reflections• Reflections off of flat, reflective surfaces are

called specular reflections.

• This comes from the Latin word for mirror, “speculum.”

Specular Reflections• When you shine a beam of light on a mirror,

you will only be able to see the reflection if your eye is at the correct location.

• This position is determined by the law of reflection.

Plane Mirrors

Plane Mirrors

• We are most familiar with plane mirrors.

• A plane mirror is a surface that is essentially flat and highly reflective.

• Generally, a plane mirror is made by putting a metallic coating on one side of a flat piece of glass.

Plane Mirrors• When you look in a mirror, you see yourself

and the objects around you.

• It appears as if what you are seeing is in front of you, beyond the plane of the mirror.

• This is not really the case.

• What you are seeing is called an image of the objects that are in front of the mirror.

Example: Plane Mirror

• Let’s consider the case of viewing a simple object in a mirror.

• When light strikes the bottle, light rays are reflected off in all directions.

• However, only some of these rays enter your eye.


• This bundle of rays are bordered by the four rays shown in the diagram.

• These light rays reflect off the mirror and enter the eye.


• However, the eye interprets these rays to have traveled in a straight line from a single point in space.

• This point is called the image point.


• Let’s focus on the two rays that originate from point A.

• These rays hit the mirror at B and B’.

• From the geometry of the problem, we can see that angles ADB and CDB are right angles.


• Lastly, we know that, by the law of reflection, θi = θr.

• Based on this, angles ABD and CBD must be equal.

• Thereforce, AD = CD.


• In other words, the image appears as far as behind the mirror as the object is in front.

• The distance from the object to the mirror is called the object distance, do.


• The distance from the image to the mirror is called the image distance, di.

• Then, for a plane mirror


• Notice also that the height of the image is the same as the height of the object.

• Also notice that no actual light rays pass through the image location (it’s behind the mirror).

• Images of this type are called virtual images.


• In other situations (such as with lenses), real light rays will pass through the image location.

• These images are called real images.

• Our eyes can see both types of images. It is up to use the geometry of the system to identify the image type.

Homework

• Read 23-1 and 23-2.

• Do problems 1, 2, and 4 on page 658.

Spherical Mirrors

Spherical Mirrors

• More often than not, reflecting surfaces are not flat.

• Often, spherical mirrors are used to magnify objects, or to obtain a wider field of view.

Spherical Mirrors• There are two types of spherical

mirrors.

• A convex mirror has the reflective surface on the the outside of the sphere.

• A concave mirror has the reflective surface on the inside of the mirror.

Spherical Mirrors• There are two types of spherical

mirrors.

• A convex mirror has the reflective surface on the the outside of the sphere.

• A concave mirror has the reflective surface on the inside of the mirror.

Spherical Mirrors• In both cases, the law of

reflection is obeyed at each point on the surface.

• Since the surface is curving, not all incoming rays are reflected at the same angle.

• However, if the curvature is known, we can predict how the rays will be reflected.

Properties of Spherical Mirrors

• To figure out how images are formed with spherical mirrors, we need to define some characteristics of the mirror.

• Let’s consider rays from a distant object striking a concave mirror.


• If the object is very far away, the rays arriving at the mirror are nearly parallel to one another.

• For our purposes, we will consider these rays to be exactly parallel (far away objects like the sun approach this).


• As the parallel rays arrive at the mirror, they are reflected according to the law of reflection at each point on the surface.

• Since the mirror is curved, the rays do not come together at a single point (recall that the rays must come together to form a sharp image).


• However, if the size of the mirror is small compared to its curvature, then all the rays will come together at almost the same point.

• This point is known as the focus.


• The line going through the center of the mirror is known as the principal axis of the mirror.

• The mirror is symmetric about this axis.

• All rays entering the mirror parallel to the principal axis are reflected through the focus.


• For this reason, this point F is also called the focal point of the mirror.

• The distance between F and the center of the mirror (A) is called the focal length, f, of the mirror.

• The focal point is also the image point for an object infinitely far away from the mirror.


• For any spherical mirror whose size is small compared to its radius of curvature, the focal length is given by:

• Where f is the focal length and r is the radius of the mirror.


• One last comment: this analysis is an approximation.

• The rays don’t all converge exactly at F.

• For more curved mirrors, this can lead to a blurred image.


• This feature of mirrors (and lenses) is called spherical aberration.

• We will discuss this effect later in the semester.

• For right now, you can assume these are “ideal” mirrors and all parallel rays converge at F.

Image Formation

Ray Diagrams

Ray Diagrams

• We are now going to learn how to locate a reflected image off of a spherical mirror.

• The image is formed at the point where the reflected rays from an object come back together.

• But remember, the location of the image is dictated only by geometry.

• So, we just have to use some geometry to locate the image.

Ray Diagrams

• Some comments:

– You MUST use a ruler when drawing ray diagrams. Any ray diagram drawn without a ruler is automatically a zero.

– Pay careful attention during this section. We will use ray diagrams to locate images formed by mirrors. However, they will also be used with lenses.

Ray Diagrams

• As we already know, any point on an object reflects rays in all directions.

• Only some of these rays strike the mirror.

• Furthermore, there are three rays that behave in an easily predictable manner.

• These are called the three principal rays.

Ray Diagrams• Let’s consider an object placed near a concave

spherical mirror.

• The object is at point O (which is outside the focus), and has a height given by the arrow (up to O’).

Ray Diagrams• We locate the image by drawing the three

principal rays.

• Ray 1 starts at the top of the object and travels parallel to the principal axis.

Ray Diagrams• The ray strikes the mirror and then passes

through the focus (point F).

• Extend the ray a ways, as the image will likely be behind the object.

Ray Diagrams• Ray 2 starts at the top of the object and passes

through focus.

• The ray then reflects off of the mirror and continues parallel to the principal axis.

Ray Diagrams• Ray 3 starts at the top on a line that passes

through the center of curvature (point c).

• The ray strikes the mirror and is reflected directly back.

Ray Diagrams• The image is formed at point I, where all three

rays converge.

• The height of the image is I’.

Ray Diagrams• When you look in the mirror, you see a

reflected image that looks as if the rays are all coming from point I’.

Ray Diagrams• Notice that real light rays are passing through

the reflected image at point I.

• This means that the image is a real image.

Ray Diagrams

• Let’s review our procedure for drawing ray diagrams:1. Start by drawing the mirror and principal axis.

2. Label the center of curvature and the focus.

3. Represent the object with an arrow. Make sure the arrow has the correct height.

4. Locate the reflected image using the principal rays.

Example: Inside the Focus

Let’s redo the last problem with the object located inside the focal length.

Convex Mirrors

• Let’s consider a convex mirror.

• If the mirror is struck by parallel rays, the rays diverge.

Convex Mirrors

• However, the rays all appear to originate from a single point F behind the mirror.

• This is called the focal point.

• As with concave mirrors, f = r/2.

Convex Mirrors

• We can use the same ray diagram technique to locate the image.

• Ray 1 starts parallel to the principal axis and is reflected along the line pointing to the focus.

Convex Mirrors

• Ray 2 starts toward the focus, then is reflected parallel to the axis.

• Ray 3 is directed toward the center, and is reflected directly back.

Convex Mirrors

• Notice that the rays seem to diverge.

• To locate the image, extend the rays to the other side of the mirror until they converge.

Convex Mirrors

• From this analysis, it is clear that the image is virtual. This is true no matter where the object is placed.

• The image is also always upright.

Conclusions

• For a concave mirror,

– If the object is located outside the focus, the image is real and inverted.

– If the object is located inside the focus, the image is virtual and upright.

Conclusions

• For a convex mirror,

– No matter where the object is placed, the image is virtual and upright.

– We will also see in a moment that the virtual image is also smaller than the actual object.

The Mirror Equation

The Mirror Equation

• As fun as drawing ray diagrams is, the process can be tedious.

• We would like to be able to locate the image without drawing the ray diagram if necessary.

The Mirror Equation

• Consider an object being imaged through the mirror shown.

• The image formed is real and inverted, but the following analysis applies to all spherical mirrors.

The Mirror Equation

• Let’s begin by drawing two rays coming from O’.

• The first ray passes through the focus and is reflected parallel to the axis.

The Mirror Equation

• The second strikes the center of the mirror, forming an angle θ with the axis.

• By the law of reflection, the reflected ray also forms an angle θ with the axis.

The Mirror Equation

• Notice that we have created two similar triangles: O’AO and I’AI.

• Based on this fact, we can set up a proportion:

(1)

The Mirror Equation

• If the mirror is small compared to its radius, then we can say AB ≈ hi.

• As a result, triangles O’FO and AFB are also similar.

The Mirror Equation

• So

• Notice that FA is the focal length and OF is the object distance minus the focal length.

The Mirror Equation

• So(2)

The Mirror Equation

• Combining proportions 1 and 2, we get

• If we divide both sides by do and rearrange, we get the mirror equation…

The Mirror Equation

Object distance

Image distance

Focal Length

The Mirror Equation

• This equation lets us determine the location of the image if we know the object distance and focal length.

The Mirror Equation• Using this analysis, we can define the

magnification of the mirror:

Sign Conventions

• The minus sign in the last equation is intended to give both the correct location and height of any image.

• To make this happen, we employ several sign conventions:– hi is positive if the image is upright and negative if it is

inverted.

– do is positive if the object is in front of the mirror.

– di is positive if the image is in front of the mirror and negative if the image is behind the mirror.

Example: Concave Mirror

A 1.5 cm tall diamond ring is placed 20.0 cm from a concave mirror. The mirror has a radius of curvature of 30 cm.

a) What is the focal length of the mirror?b) What is the position of the image?c) What is the size of the image?

You Try: Concave Mirror Again

Redo the last example of the 1.5 cm tall ring, but place the ring 10 cm from the mirror. The ROC of the mirror is still 30 cm.

Example: Convex Mirror

Let’s redo the last problem with a convex mirror. The ROC is still 30 cm.

Homework

• Reread 23-3.

• Do problems 7, 9, 10, and 11 on page 659.

Problem Day


• We will whiteboard at the end of class.

Homework

• Do problems 15 and 16 on page 659.

• Also, answer question 19 on page 658. Look carefully at the picture!

Refraction

Straw and Water Demo

• When a straw is placed in a glass of water, what appears to happen to the straw?

• Is the straw really bending or breaking?

• Why do you think it looks as if it is bent?

Index of Refraction

• We learned last quarter that light travels at a constant speed of 3 x 108 m/s.

• However, this the speed light travels if it is traveling in a vacuum.

• When light travels through other mediums, it travels at different speeds (much like sound waves).

Index of Refraction

• We find it useful in physics to set up a ratio comparing the speed of light in a vacuum (c) to the speed of light in a given material.

• The quantity n is known as the index of refraction for the given material.

Index of Refraction

• Notice that the index of refraction is never less than 1. This is because light will never travel faster than c.

• We will also learn in the next unit that n is also somewhat dependent on the wavelength of the light.

• For now, you can consider n to be a constant. Also, we will approximate nair as 1.

Example: Speed of Light in Plexiglass

What is the speed of light in plexiglass?

Snell’s Law

Refraction

• Consider a light wave approaching the surface of a piece of glass.

• Since the glass has an index of refraction greater than 1, the light will travel more slowly in the glass.

• ANIMATION!

http://webphysics.davidson.edu/physlet_resources/bu_semester2/index.html

Refraction• If the light is incident at an angle, part of the

wave front will strike the glass first.

• This part of the wave travels at a slower speed.

Refraction• Since the rest of the wave is moving faster, the

wave begins to bend.

• This causes the refracted ray to form a new angle with the vertical.

Refraction• The change in direction of

the light ray is called refraction.

• In the real world, when light strikes the surface of a material, part of the light rays is reflected and part is refracted.

Refraction• When describing the

geometry of the refraction effect, it is useful to define some angles.

• The incident ray makes an angle with the line perpendicular to the surface.

• This is the angle of incidence (θ1).

Refraction• The bent ray makes a new

angle with the perpendicular.

• This is known as the angle of refraction (θ2).

• Notice that in this case, the n for water is greater than the n for air.

Refraction• When n1 > n2, the ray

always bends toward the perpendicular.

• However, if the light ray were coming from the other side, and n2 > n1, then the light ray would bend away from the perpendicular.

Snell’s Law• The angle of incidence, θ1,

and the angle of refraction, θ2, are related to one another according to

• This is known as Snell’s Law and is the law of refraction.

Snell’s Law

• Snell’s Law allows us to determine the amount a light ray will be bent when traveling into a new medium.

• However, it can also be used to determine the index of refraction of an unknown material.

Snell’s Law

• Notice that if n1 < n2, then θ1 > θ2. This means the light bends toward the perpendicular, which is what we saw in the picture.

• Also, if n1 > n2, then θ1 < θ2 (meaning the ray bends away from the perpendicular). This also matches our observations.

Example: Refraction through Glass

Light traveling through air strikes a flat piece of glass (n = 1.5). The angle of incidence is 60°.

a) What is the angle of refraction in the glass?

b) What is the angle of refraction when the ray emerges from the glass?

Announcements

• The final draft of your paper is now due Friday, April 22.

• We will start presentations on Monday, May 2.

• Our test on geometric optics will be this Friday, April 8.

Thin Lenses

Ray diagrams again…

Demo Time!

Questions

• What happened to the image on the screen when the piece of glass was put on the track?

• Was the piece of glass flat?

Thin Lenses

• The changes in the demo image were the result of the light passing through a thin lens.

• Lenses have been used in optical devices since the 16th centuries.

• Today lenses have many diverse uses, from cameras, to eyeglasses, to telescopes.

Thin Lenses

• A thin lens is a piece of glass (or plastic) whose two faces are portions of a sphere.

• The two faces can be convex, concave, or planar (flat).

• Like mirrors, lenses are used to form images of objects.

Thin Lenses

• To begin our investigation, let’s consider parallel rays passing through a double convex lens.

• As with a mirror, we define the axis of lens as the line passing through the center of the lens perpendicular to the surfaces.

Thin Lenses

• When these rays strike the lens, they bend according to Snell’s Law.

• Buts since the surface of the lens is curved, each ray is refracted at a different angle.

Thin Lenses

• On the other side of the lens, all the rays converge at a single point.

• This is known as the focal point, F, of the lens.

• This is also the image point for a far away object.

Thin Lenses

• Note that, like mirrors, the rays only converge to a point under certain conditions.

• The converge only if the lens is very thin compared to its diameter.

• Lenses of this type are known as thin lenses.

Thin Lenses

• If the parallel light comes in at an angle, the rays still converge the at the same horizontal distance from the lens.

• We can then define the focal plane of the lens.

Thin Lenses

• Any lens that is thicker at the center than at the edges is called a converging lens.

• A lens that is thicker at the edges than at the center is called diverging lens.

Thin Lenses

• When describing lenses, optometrists do not use the focal length to describe the strength of the lens.

• Instead they use the reciprocal of focal length, called the power of the lens.

Thin Lenses

• The power is defined

• The unit of power is the diopter (D).

Ray Diagrams

Ray Diagrams• Lenses behave much like spherical mirrors in

that certain rays entering the lens behave in a predictable manner.

• For example, a ray entering the lens parallel to the axis is refracted through the focal point.

• As with mirrors, it is useful to use the three principal rays to draw a ray diagram to locate an image.

Ray Diagrams

• Let’s first consider an object placed outside the focal length of a converging lens.

• As with the mirror example, draw an axis through the lens and represent the object with an arrow.

Ray Diagrams

• On your axis mark the focal point of the lens.

• Remember that lenses have a focal point on either side of the lens.

• Also, draw a center line through the lens.

Ray Diagrams

• The first principal ray leaves the object parallel to the axis.

• It then refracts at the center line through the point F on the other side.

Ray Diagrams

• The second ray starts by passing through F’.

• It then strikes the center line and is refracted parallel to the axis.

Ray Diagrams

• The third ray passes through the center of the lens.

• The ray passes through the lens without being refracted.

Ray Diagrams

• As with the mirror, the image is located where the three principal rays converge.

• The point I is known as the image point.

Questions

• What type of image is formed by this lens? Is it real or virtual?

• Is the image upright or inverted?


Let’s diagram an object placed inside the focal length of the lens in the last example.

What type of image is formed in this case?

Conclusions

• For a converging lens:– If the object is located outside the focus, the

image formed is real and inverted. The image is located on the opposite side of the lens from the object.

– If the object is located inside the focus, the image formed is virtual and upright. The image is located on the same side of the lens as the object.

Diverging Lens• A diverging lens is similar to a converging lens.

• However, note that F and F’ have been switched.

• This will allow us to keep the same conventions when drawing diagrams.

Diverging Lens• As before, the first ray goes out parallel to the

axis.

• When it strikes the center line, it is refracted along the line connecting that point to F.

Diverging Lens• The second ray starts along the line pointing to F’.

• It is then refracted parallel to the axis.

• The third ray passes through the center of the lens and is not refracted.

Diverging Lens• Since the refracted rays diverge, we must

trace them back to the other side of the lens to find the image.

• Notice that the image is virtual, upright, and smaller than the object.


Let’s diagram an object placed inside the focal length of the lens in the last example.

What type of image is formed in this case?

Conclusions

• For a diverging lens:

– Regardless of the location of the object, the image formed is virtual, upright, and smaller than the original object.

– The image is also located closer to the lens than the object.

Homework

• Read 23-7 and 23-8.

• Work on your papers and presentations.

The Thin Lens Equation

The Thin Lens Equation

• As with mirrors, ray diagrams are useful tools for locating images formed by lenses.

• However, we would like to be able to locate the image mathematically, without having to draw a diagram.

• To do this, we use the Thin Lens Equation.

The Thin Lens Equation• Let’s again consider an object outside the

focus of a converging lens.

• Drawing rays 1 and 3, we can quickly locate the image.

The Thin Lens Equation• Notice that we have formed two triangles

(shaded in the picture).

• These are right triangles, and they are similar.

The Thin Lens Equation• Based on this, we can set up a proportion:

The Thin Lens Equation• But notice that triangles OAO’ and IAI’ are similar

as well.

• Therefore:

The Thin Lens Equation• Combining these two relationships, we get

The Thin Lens Equation• Combining these two relationships, we get

Object distance

Image distance

Focal Length

The Thin Lens Equation• Interestingly, this is exactly the same as the

mirror equation.

• However, the sign conventions are different (more on that in a moment).

The Thin Lens Equation• Notice if we have a diverging lens, then the

proportion becomes:

The Thin Lens Equation• If we rearrange this, the mirror equation is

• But, this is is just the original equation if we let di and f be negative. This gives us a hint about sign conventions…

Sign Conventions

• The thin lens equation is valid for both converging and diverging lenses if the following sign conventions are used:

1. The focal length is positive for converging lenses and negative for diverging lenses.

2. The object distance is positive if the object is on the same side of the lens from where the light is coming from (always true if there is only one lens).

Sign Conventions

• The thin lens equation is valid for both converging and diverging lenses if the following sign conventions are used:

3. The image distance is positive if the the image distance is on the opposite side of the lens from where the light is coming from. If the image is on the same side, it is negative.

4. The image height is positive if the image is upright, and negative if is inverted.

Magnification

• As with mirrors, we can define the magnification of a lens:

• If the sign conventions are followed, the magnification will always give you the correct sign on the image height.

Example: Close to a Converging Lens

An object is placed 10 cm from a converging lens with a focal length of 15 cm.

a) Where is the image located?b) What is the magnification of the image?c) What type of image is this (real/virtual, upright/inverted)?

You Try: Diverging Lens

Where should an object be placed in relation to a diverging lens with a focal length of 25 cm if the lens is to form a virtual image 20 cm in from the lens?

Homework

• Reread 23-7 and 23-8.

• Do problems 43 and 45 on page 661.

Problem Day

• Do problems 46, 47, 48 and 50 on pages 660-661.

• We will whiteboard at the end of the day.

Homework


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