Geometric Optics and Image Formation

• Homework 3 peer reviews are due on Tuesday, April 8.
• Corrected submissions for Homework 3 are due on Thursday, April 10.
• Homework 4 will be issued on Thursday.
• The reading assignment for Thursday is Unit Q, Chapter Q1.
• A reading quiz on Unit Q, Chapter Q1 is due on Wednesday, April 9, at 12 noon for full credit.
• There was a typo in Problem 3 (T10M.7) giving the power as 100 MW instead of 1000 MW.
• Solutions using either value for power are considered correct in peer reviews and corrections.
• The mean score for the midterm was 65.5%.
• The standard deviation for the midterm scores was 20%.
• This lecture covers geometric optics and image formation.
• The topics covered are:
• Plane Mirrors
• Spherical Mirrors
• Refraction
• Thin Lenses
• The Eye
• Telescopes
• Images formed using mirrors and lenses seem to originate straight from the object
• The image may have a different position, size, or shape than the actual object.
• The brain thinks the ray came from the image, but it is a virtual image in reality.
• One can locate each point on an image with two rays.

Locating Points on an Image

• A ray normal to the mirror
• A ray that reaches the observer's eye
• The image is reversed from front to back.
• By congruent triangles, the distance from the image to the mirror equals the distance from the object to the mirror, given by the equation d = d'.
• The height of the image is the same as the height of the object.

Properties of Plane Mirror Images

• The distance from the image to the mirror is the same as the distance from the object to the mirror: d = d'
• The mirror image is upright but reversed front to back.
• The mirror image has the same size as the object.

Parabolic Mirrors

• These mirrors are shaped into a parabola of rotation, with a cross-section of y = x².
• Parallel light entering this mirror along the axis of rotation reflects through a common point called the focus.
• The focal point (or focus) is where an optical element brings parallel rays together.
• Parallel rays from distant objects converge at the focal point after reflection in the parabolic mirror.
• Fabrication of parabolas are difficult, spherical optics are easier to make. Spherical optics will be examined next.

Spherical Mirrors

• Spherical mirrors lack a focus, but with small angles between light hitting the mirror, the paraxial approximation is used. This implies:
• sin θ ≈ tan θ ≈ 0
• cos θ ≈ 1
• Approximations are assumed for the rest of the lecture.
• CF cos θ = R/2
• CF ≈ R/2
• CF + FP = R
• R/2 + FP ≈ R
• FP = f ≈ R/2

Convex mirror

• Rays incident parallel to the axis reflects as if from the focus
• The focal distance f is given by f = -R/2

Concave Mirror

• Rays incident parallel to the axis reflects through the focus.
• The focal distance f is given by f = R/2

Analyzing Spherical Mirrors

• We draw special rays and apply the law of reflection where they hit the spherical surface
• A ray parallel to the mirror axis reflects through the focal point f
• A ray passing through the focus reflects parallel to the axis
• A ray that strikes the center of the mirror reflects symmetrically
• A ray passing through the center of curvature c, returns on itself
• One traces one ray incident parallel to the axis.
• Then trace a second ray incident through the focus.
• The image is located at the intersection of the two rays.
• Repeat for every point in the image.
• Practical applications might only need the 'head' and 'tail' of the image.

Spherical Mirrors - Concave

• When the object is beyond c, the image is real, reduced, and inverted.
• When the object is between c and f, the image is real and inverted.
• When the object is between f and the mirror, the image is virtual, upright, and magnified.
• In refraction at a planar interface, the image of an object underwater appears closer than its actual position due to bending of light.

Refraction at a Planar Interface According to Snell's Law

hi = (n2/n1) * h0

• n1: refractive index of air
• n2: refractive index of water.

Refraction at a Spherical Interface According to Snell's Law

n1/do + n2/di = (n2 - n1)/R

• n1/f1 + n2/∞ = (n2 - n1)/R
• f1 = n1R/(n2 - n1)
• n1/∞ + n2/f2 = (n2 - n1)/R
• f2 = n2R/(n2 - n1)

Thin Lenses

• A thin lens is an optical component using small angle approximation that concentrates parallel light rays at one point.

Converging verse Diverging Lenses

• Converging lenses can be bi-convex, plano-convex, or meniscus convex.
• Diverging lenses can be bi-concave, plano-concave, or meniscus concave.

Characteristics of Ray Tracing

• A ray leaving the object is parallel to the axis, refracted to pass through the focal point
• A ray passing through the lens's center is undeflected
• A ray passing through the focal point (on object side) is refracted to end up parallel to the axis

Lens Example

• Object between 2f and f results in an image that is inverted, real, enlarged
• Object between f and lens results in an image that is upright, virtual, and enlarged.

Simple Examples using Diverging Lens

• For the object beyond 2f, the image is upright, virtual, reduced.
• For the object between f and lens, the image is upright, virtual, reduced.
• A parallel ray to the axis diverges so the extension passes through the focal point.

Sign Conventions for Lenses

• Use ray tracing

• Converging/convex lens:

• Focal length is positive

• Image distance is positive when on the other side of lens

• Upright height is positive, inverted is negative

• Diverging/concave lens:

• Focal length is negative

• The image is always virtual, and the distance is always negative

• Upright height is positive, inverted is negative

Thin Lens Equation Summary:

• M = h'/h = -i/p
• This is the magnification
• 1/p + 1/i = 1/f
• "p" is the object distance
• "i" is the image distance
• "f" is the focal length

Properties of the Thin Lens Equation with 1/p + 1/i = 1/f

• When p = f, i = ∞
• When p = 2f, i = 2f and magnification is 1.
• When f > p > 0, i < 0 (image is virtual and on the same side)
• If f < 0, i is always negative
• A diverging lens cannot produce a virtual image, it always produces a virtual image

The Lensmaker's Formula

• The lens equation gives the image distance as a function of the object distance and the focal distance: 1/f = 1/p - 1/i
• The lensmaker's formula gives the focal length f as a function of R₁ and R₂: 1/f = (n - 1) * (1/R₁ - 1/R₂)

The Eye

• The thin lens focuses light onto the retina
• Retina sends signals to the bain about illuminated sensor
• Brain interprets the signal

Correcting vision

• Nearsightedness is corrected with diverging lens
• Farsightedness is corrected using converging lens

Telescopes

• Used to magnify very distant objects

Two types of telescopes

• Reflecting telescopes: requires a 3rd "erecting" lens so the image is not upside-down
• These require very long tubes that need to be inside the telescope
• Reflecting telescopes are the more common telescopes

Formula:

• Angular magnification M = θimage divided by θobject
• Derive M = fobj divided by feye For an inverting refracting telescope:
• M = -fobj/feye(is negative since image is inverted) For a reflecting telescope:
• M = +fobj/feye and note that fobj = R/2
• larger mirror = more magnification