Geometric Optics September 14, 2015

Areas of Optics Geometric Optics Light as a ray. Physical Optics Light as a wave. Quantum Optics Light as a particle.

Mirrors

Reflection Reflection occurs when light bounces off a surface. There are two types of reflection Regular reflection, off a smooth surface Diffuse reflection, off a rough surface

Types of Mirrors PlaneConcave (converging)Convex (diverging)

Ray Diagrams Ray tracing is a method of constructing an image using the model of light as a ray. We use ray tracing to construct optical images produced by mirrors and lenses. Ray tracing lets us describe what happens to the light as it interacts with a medium.

Light Rays Inherently, rays do not bend. However, if they encounter a different medium, they will react.

Plane Mirrors Incident ray Reflected ray Normal Line Plane Mirror A Normal Line is perpendicular to the mirror’s surface drawn at the point of contact.

Law of Reflection The angle of incidence of reflected light equals the angle of reflection. Note that angles are measured relative to a normal line.

Describing Images Nature real (converging rays) virtual (diverging rays) Orientation upright Inverted Size true enlarged reduced

Ray Tracing Identify the Image: Virtual, Upright, True Extend Reflected Rays Behind the Mirror Reflected Rays Diverge

Spherical Mirrors Positive Focal LengthNegative Focal Length Concave (Converging) Convex (Diverging)

Parts of a Spherical Mirror These are the main parts of a spherical concave mirror. The focal length is half of the radius of curvature. The focal length is positive for this type of mirror. R = 2f

Focus Rays parallel to the principal axis all pass through the focus for a spherical concave mirror.

Ray Tracing: Spherical Concave Mirrors The three “principal rays” to construct an image for a spherical concave mirror are the p-ray, which travels parallel to the principal axis, then reflects through focus. the f-ray, which travels through focus, then reflects back parallel to the principal axis. the c-ray, which travels through center, then reflects back through center. You must draw two of the three principal rays to construct an image.

Ray Tracing: Spherical Concave Mirrors Construct the image for an object located outside the center of curvature. It is only necessary to draw 2 of the three principal rays Identify the Image: Real, Inverted, Reduced

Ray Tracing: Spherical Concave Mirrors Construct the image for an object located at the center of curvature. Identify the Image: Real, Inverted, True

Ray Tracing: Spherical Concave Mirrors Construct the image for an object located between the center of curvature and the focus. Identify the Image: Real, Inverted, Enlarged

Ray Tracing: Spherical Concave Mirrors Construct the image for an object located at the focus. Identify the Image: No Image

Ray Tracing: Spherical Concave Mirrors Construct the image for an object located inside the focus. Identify the Image: Virtual, Upright, Enlarged

Mirror / Lens Equation s i : image distance s o : object distance f: focal length

Magnification Equation s i : image distance s o : object distance h i : image height h o : object height M: magnification

Sign Conventions Focal length (f) Positive for CONCAVE mirrors Negative for CONVEX mirrors Magnification (M) Positive for UPRIGHT images Negative for INVERTED images ENLARGED when M > 1 REDUCED when M < 1 Image Distance s i is POSITIVE for real images s i is NEGATIVE for virtual images

Sample Problem A spherical concave mirror, focal length 20 cm, has a 5-cm high object placed 30 cm from it. a) Draw a ray diagram and construct the image. b) Identify the image c) Mathematically verify your results

Parts of a Spherical Convex Mirror These are the main parts of a spherical convex mirror. The focal length is half of the radius of curvature, and both are on the dark side of the mirror. The focal length is negative for this type of mirror.

Spherical Convex Mirror Construct the image for an object located outside a spherical convex mirror. Identify the image: Virtual, Upright, Reduced All Diverging Mirrors (and Lenses) create an image with the same identity.

Sample Problem A spherical convex mirror, focal length 15 cm, has a 4-cm high object placed 10 cm from it. a) Draw a ray diagram and construct the image. b) Identify the image c) Mathematically verify your results

Mirror Summary Concave Image is real when object is outside focus Image is virtual when object is inside focus Focal length f is positive Convex Image is always virtual Focal length f is negative

Refraction

Refraction is the movement of light from one medium into another medium. Refraction cause a change in speed of light as it moves from one medium to another. Refraction can cause bending of the light at the interface between media.

Index of Refraction n: index of refraction c: speed of light (3 x 10 8 m/s) v: velocity of light in the medium

Snell’s Law n 1 : index of refraction of incident medium θ 1 : angle of incidence n 2 : index of refraction of refracting medium θ 2 : angle of refraction

Snell’s Law n1n1 n2n2 θ1θ1 θ2θ2 When the index of refraction increases, light bends toward the normal. n 2 > n 1

Snell’s Law n1n1 n2n2 θ1θ1 θ2θ2 When the index of refraction increases, light bends toward the normal. n 1 > n 2

Sample Problem Light enters an oil from the air at an angle of 50 ° with the normal, and the refracted beam makes an angle of 33 ° with the normal. a) Draw this situation. b) Calculate the index of refraction of the oil. c) Calculate the speed of light in the oil

Prism Problem Light in air enters a prism perpendicular to the long side and passes through the prism. If the refractive index of the glass is 1.55, calculate the angle of refraction when it leaves the prism.

Critical Angle

If light passes into a medium with a greater refractive index than the original medium, it bends away from the normal and the angle of refraction is greater than the angle of incidence. If the angle of refraction is > 90 °, the light cannot leave the medium. The smallest angle of incidence for which light cannot leave a medium is called the critical angle of incidence.

Calculating the Critical Angle

Sample Problem What is the critical angle of incidence for a gemstone with refractive index 2.45 if it is in air?

Lenses

Positive Focal LengthNegative Focal Length Converging Diverging

Focus All rays parallel to the principal axis refract through the focus of a converging lens.

Ray Tracing Ray tracing is also used for lenses. We use the same principal rays we used for mirrors. the p-ray, which travels parallel to the principal axis, then refracts through focus. the f-ray, which travels through focus, then refracts parallel to the principal axis. the c-ray, which travels through center and continues without bending. You must draw 2 of the 3 principal rays.

Principle Rays for Lenses Construct the image for an object located outside 2F. It is only necessary to draw 2 of the three principal rays

Mirror / Lens Equation s i : image distance s o : object distance f: focal length  s i : image distance  s o : object distance  h i : image height  h o : object height  M: magnification

Sample Problem A converging lens, focal length 20 cm, has a 5-cm high object placed 30 cm from it. a) Draw a ray diagram and construct the image. b) Mathematically verify your ray diagram. c) Identify the image

Sample Problem A diverging lens, focal length -15 cm, has a 4-cm high object placed 10 cm from it. a) Draw a ray diagram and construct the image. b) Mathematically verify your ray diagram. c) Identify the image

Summary Converging Lens f is positive s o is positive s i is positive for real images and negative for virtual images M is negative for real images and positive for virtual images h i is negative for real images and positive for virtual images Diverging Lens f is negative s o is positive s i is negative M is positive and < 1 h i is positive and < h o

Multiple Lenses / Mirrors When drawing ray diagrams for a combination of lenses/mirrors, use the image from the first lens/mirror as the object for the second. When appropriate, apply the p-ray, f-ray, and c-ray rules to the second lens/mirror. To Identify the image, the result is compared to the original object.

Sample Problem