3. Geometrical Optics

Geometric optics

—process of light ray through lenses and mirrors to determine the location and size of the image from a given object .

Reflection and Mirror

Law of reflection  i  i  r :   r : incident angle reflection angle

Image Formation by Reflection

Application of Double Reflection-Periscope

DIY Periscope

DIY Periscope (Cont’)

Law of reflection (Snell’s law)

n

1 sin  1 

n

2 sin  2

Types of Lenses

Ray Tracing through Thin Lenses

Image Formation by thin Lenses

Lens equation:

d

1

f

1 2  1

d

2 

f

1 (Gaussian form) 

z

1

z

2 (Newtonian form)

Magnificat ion M



h

2

h

1 

d

2

d

1

ABCD Matrix

ABCD Matrix (Cont’)

ABCD Matrix (Cont’)

ABCD Matrix (Cont’)

ABCD Matrix (Cont’)

ABCD Matrix (Cont’)

ABCD Matrix (Cont’)

Aberrations of Lenses

•

Primary Aberration



image deviate from the original picture/the first-order approximation Monochromatic aberrations

    

Spherical Aberration Coma Astigmatism Curvature of field Distortion Chromatic aberration

General Method of Reducing Aberration in Optical Systems-Multiple Lenses

United States Patent 6844972

General Method of Reducing Aberration in Optical Systems-Multiple Lenses (Cont’)

United States Patent 6995908

Chromatic Aberration

The focal lengths of lights with distinct wavelengths are different.

Solution of Chromatic Aberration-Using Doublet, Triplet, or Diffractive Lens

Spherical Aberration (SA)

Spherical Aberration for Different Lenses

(a) Simple biconvex lens (b) “Best-form” lens (c) Two lenses (d) Aspheric, almost plano-convex lens

Solutions of Spherical Aberration Using Aspherical Lens or Stop

Coma

Coma (Cont’)

(a) Negative coma (b) Postive coma

Astigmatism

Astigmatism (Cont’)

Solutions of Astigmatism-Using Multiple Lenses

Curvature of field

Solutions of Curvature of field-Using Multiple Lenses

Distortion

Picture taken by a wide-angle camera in front of graph paper with square grids

Solution of Distortion-Using Multiple Lenses

Nearsightedness (Myopia) and Farsightedness (Hyperopia)

Image Formation



Camera

Camera F-number

F



number



focal

length diameter of aperture

Exposure

E  BA f 2  B  d 2 4 f 2 Eg. 50 mm camera lens, aperture stop 6.25

mm

: F-number = 8 (f/8) E: energy collected by camera lens B: brightness of object A: area of aperture d: diameter of aperture stop For any given object E  1 (F number) 2

Camera Lenses

• Wide-angle Lenses the Aviogon and the Zeiss Orthometer lenses • Standard Lenses-the Tessar and the Biotar lenses • Lens of reducing the 3rd-order aberration the Cooke triplet lens

Depth of Field (DOF)

• The distance between the nearest and farthest objects in a scene that appear acceptably sharp in an image. • In cinematography, a large DOF is called deep focus, and a small DOF is often called shallow focus.

• For a given

F

-number, increasing the magnification decreases the DOF; decreasing magnification increases DOF. • For a given subject magnification, increasing the

F

-number increases the DOF; decreasing

F

-number decreases DOF.

Numerical Aperture (NA)

• The numerical aperture of an optical system is a dimensionless number that characterizes the range of angles over which the system can accept or emit light.

• Generally, • For a multi-mode optical fiber,

Telescope

Astronomical (Keplerian) Telescope

Magnification (magnifying power): General Keplerian telescope:

d

=

f

o +

f

e M    '  : angle subtended at input end in front of objective  ’: angle subtended at output end behind eyepiece

M

For small angle:    '  

f o f e

 0 (inverted image)

Galileo Telescope

M

   ' 

f o f e

 0 General Galileo telescope:

d

=

f

o -

f

e

Terrestrial Telescope

All images are erecting

Optical Microscope

Objective

Microscope Theory

Overall magnification: M  m o m e m o : linear magnification of objective m e : angular magnification of eyepiece Linear magnification: m o  y ' y   x ' f Numerical aperture (NA) NA  D f  1 F number (for objective)

Eyepiece

Microscope Theory (Cont’)

  tan   y 25 (if   1) Angular magnification: m e   '   25 f  1  25 f (usually, f  25 cm )  '  tan  '  y 25 '  y (if x  '  1) Overall magnification of microscope: M  m o m e   x f o ' 25 f e f o : focal length of objective f e : focal length of eyepiece (normal reading distance)

Simple Projection System

Fresnel Lens and Plates

focusing point (in phase) • Radius of the concentric circular:

r n

= [(

n

=0, 1, 2,….

n

 ) 2 +2

fn

 ] ½ , • Sapce between two adjacent circular • zone: 

r n

=

r n

+1 

r n