Waves Electromagnetic Spectrum Geometric Optics: Mirrors Lenses

Overview  Waves  Electromagnetic

Spectrum  Geometric Optics

• Mirrors • Lenses

 Wave

Optics

Defining a Wave

Wavelength - distance from peak to peak, or trough to trough Frequency - cycles per second; how many peaks pass a given point in 1 second

Understanding Waves • Longitudinal waves - displacement is in same direction as the wave motion

• Example: sound waves • Obeys the equation λf = v, where λ is the wavelength, n is the frequency, and v is the velocity.

Understanding Waves • Transverse Waves - displacement is perpendicular to the direction of motion of the wave

• Example: Light • Obeys the equation λf = v, where λ is the wavelength, f is the frequency, and v is the velocity.

Special Things About a Light Wave • It does not need a medium through which to travel • It travels with its highest velocity in a vacuum • Its highest velocity is the speed of light, c, equal to 300,000 km/sec • The frequency (or wavelength) of the wave determines whether we call it radio, infrared, visible, ultraviolet, X-ray or gamma-ray.

EM Radiation Travels as a Wave

c = 3 x 108 m/s It’s not just a good idea, it’s the law!

Waves Bring Us Information About our Universe • Different energies/frequencies/wavelengths produced by different physical processes Waves are how we perceive the world around us - Sound waves = differences in pressure, hearing - Light waves = sight - Radio waves = communication

Electromagnetic Spectrum

ISNS 4371 Phenomena of Nature

      Law of Reflection -

Properties of Light Angle of Incidence = Angle of reflection

Law of Refraction - Light beam is bent towards the normal when passing into a medium of higher Index of Refraction. Light beam is bent away from the normal when passing into a medium of lower Index of Refraction. Index of Refraction -

Speed of light in vacuum n= Speed of light in a medium

Inverse square law - Light intensity diminishes with square of distance from source.

Law of Reflection

Normal

α β

Angle of incidence (α) = angle of reflection (β) The normal is the ray path perpendicular to the mirror’s surface.

Index of Refraction

As light passes from one medium (e.g., air) to another (e.g., glass, water, plexiglass, etc…), the speed of light changes. This causes to light to be “bent” or refracted. The amount of refraction is called the index of refraction.

Refraction – Snell’s Law •Snell’s law provides a way to determine the bending of light as it goes from one medium to another.

n 1 s in θ1 = n 2 s in θ 2 •Total internal relfection: occurs when the angle of refraction is greater than ninety degrees – all the light is reflected!

Geometric Optics  Flat

Mirrors  Spherical Mirrors  Images Formed by Refraction  Thin Lenses  Optical Instruments

Images - Terminology p: Object Distance q: Image Distance Real Images:

When light rays pass through and diverge from the image point. Virtual Images: When light rays do not pass through but appear to diverge from the image point.

Magnification

Image Height h′ M≡ = Object Height h

Images Formed by Flat Mirrors p=q The image is virtual For flat mirrors, M = 1

• • •

The image distance is equal to the object distance. The image is unmagnified, virtual and upright. The image has front-back reversal.

Concave Spherical Mirrors

Spherical Concave Mirror

A real image is formed by a concave mirror

Paraxial Approximation: Only consider rays making a small angle with the principal axis

Spherical Aberration

Image Formation

h h′ tan θ = = − p q

h′ q M = =− h p

tan α =

h h′ =− p−R R−q

Focal Point

h′ R−q =− h p−R

R−q q = p−R p

1 1 2 + = p q R

1 1 1 + = p q f

f =

R 2

Convex Spherical Mirrors

The image formed is upright and virtual

h′ q M = =− h p

1 1 1 + = p q f

Sign Conventions for Mirrors  

 

 

 

p is positive if object is in front of mirror (real object). p is negative if object is in back of mirror (virtual object). q is positive if image is in front of mirror (real image). q is negative if image is in back of mirror (virtual image). Both f and R are positive if center of curvature is in front of mirror (concave mirror). Both f and R are negative if center of curvature is in back of mirror (convex mirror). If M is positive, image is upright. If M is negative, image is inverted.

Ray Diagrams For Mirrors  Ray

1 is drawn from the top of the object parallel to the principal axis and is reflected through the focal point F.  Ray 2 is drawn from the top of the object through the focal point and is reflected parallel to the principal axis.  Ray 3 is drawn from the top of the object through the center of curvature C and is reflected back on itself.

Image is real, inverted and smaller than the object

Image is virtual, upright and larger than the object

Image is virtual, upright and smaller than the object

Image From a Mirror f = +10 cm

(a) p = 25 cm

1 1 1 + = p q f 1 1 1 + = 25 q 10

M=

q = 16.7cm

h′ q = − = −0.668 h p

Concave Mirror

(b) p = 10 cm

(c) p = 5 cm

1 1 1 + = 10 q 10

1 1 1 + = 5 q 10

q=∞

q = −10cm M=

h′ q =− =2 h p

Images Formed By Refraction

n1Sinθ1 = n2 Sinθ 2

n1α + n2γ = ( n2 − n1 ) β

n1θ1 ≈ n2θ 2

d tan α ≈ α ≈ p

θ1 = α + β

tan β ≈ β ≈

β = θ2 + γ

d R d tan γ ≈ γ ≈ q

d d d ( ) n1 + n2 = n2 − n1 p q R n1 n2 ( n2 − n1 ) + = p q R

Sign Conventions for Refracting Surfaces  

 

 

p is positive if object is in front of surface (real object). p is negative if object is in back of surface (virtual object). q is positive if image is in back of surface (real image). q is negative if image is in front of surface (virtual image). R is positive if center of curvature is in back of convex surface. R is negative if center of curvature is in front of concave surface.

Flat Refracting Surface R=∞ n1 n2 + =0 p q n2 q=− p n1 The image is on the same side of the surface as the object.

Apparent Depth p=d q=−

n2 p n1

1 q=− d = −0.752d 1.33 The image is virtual

Thin Lenses

The image formed by the first surface acts as the object for the second surface 1 n ( n − 1) + = p1 q1 R1

where, q1 < 0

n 1 (1 − n ) + = p2 q2 R2 p2 = − q1 + t ≈ − q1 −

n 1 (1 − n ) + = q1 q2 R2

1 1 1 1  + = ( n − 1)  −  p1 q2  R1 R2 

1 1 1 1  + = ( n − 1)  −  p q  R1 R2 

1 1 1  = ( n − 1)  −  f  R1 R2 

1 1 1 + = p q f Lens Makers’ Equation

h′ q M = =− h p

Lens Types

Converging Lenses

f1: object focal point f2: image focal point Diverging Lenses

Sign Conventions for Thin Lenses    

 

 

p is positive if object is in front of lens (real object). p is negative if object is in back of lens (virtual object). q is positive if image is in back of lens (real image). q is negative if image is in front of lens (virtual image). R1 and R2 are positive if center of curvature is in back of lens. R1 and R2 are negative if center of curvature is in front of lens. f is positive if the lens is converging. f is negative if the lens is diverging.

Ray Diagrams for a Converging Lens 





Ray 1 is drawn parallel to the principal axis. After being refracted, this ray passes through the focal point on the back side of the lens. Ray 2 is drawn through the center of the lens and continues in a straight line. Ray 3 is drawn through the focal point on the front side of the lens (or as if coming from the focal point if p < f) and emerges from the lens parallel to the principal axis.

The image is real and inverted

The image is virtual and upright

Ray Diagrams for a Diverging Lens 





Ray 1 is drawn parallel to the principal axis. After being refracted, this ray emerges such that it appears to have passed through the focal point on the front side of the lens. Ray 2 is drawn through the center of the lens and continues in a straight line. Ray 3 is drawn toward the focal point on the back side of the lens and emerges from the lens parallel to the principal axis.

The image is virtual and upright

Combination of Thin Lenses   

First find the image created by the first lens as if the second lens is not present. Then draw the ray diagram for the second lens with the image from the first lens as the object. The second image formed is the final image of the system.

I1 O

I2

f1

f2