Geometrical Optics Lenses Principal Rays Gaussian Lens Formula: 1 1 1 So Si F F I Optical Image Object Axis Fo
Phys 322 Lecture 13 Chapter 5
Geometrical Optics Lenses Principal rays Gaussian lens formula: 1 1 1 so si f F i Optical Image Object axis Fo
Principal rays:
1) Rays parallel to principal axis pass through focal point Fi. 2) Rays through center of lens are not refracted.
3) Rays through Fo emerge parallel to principal axis. In this case image is real, inverted and enlarged Assumptions: Since n is function of , in reality each color has different focal point: • monochromatic light chromatic aberration. Contrast to • thin lens. mirrors: angle of incidence/reflection • rays are all “near” the principal axis not a function of (paraxial). Converging lens: examples
This could be used in a camera. Big object on small film
so > 2F
This could be used as a projector. Small slide(object) on big F < so < 2F screen (image)
This is a magnifying glass 0 < si < F Lens magnification
so y Fi Image o optical axis Object f Fo yi 1 1 1 si so si f
Green and blue triangles are similar: Magnification equation:
yi si MT yo so
Example: f=10 cm, so=15 cm T = transverse 1 1 1 30cm si = 30 cm MT 2 15cm si 10cm 15cm Thin lens equations s 1 1 1 M i T s s s f o o i f= 100 mm
Real image formed formed by single lens is always inverted Longitudinal magnification 1 1 1 so si f
The 3D image of the horse is distorted: • transverse magnification changes along optical axis • longitudinal magnification is not linear Longitudinal magnification: 2 Negative: a horse looking dxi f 2 M L M towards the lens forms an image dx x2 T o o that looks away from the lens
dx d x x f 2 2 i 2 2 2 o i xi f / xo f / xo f / xo dxo dxo Thin lens combinations In most of applications several lenses are used Example: multiple lenses
Image from lens 1 becomes object for lens 2
1 2
Fi1 Fi2
Lens 1 creates a real, inverted and enlarged image of the object. Lens 2 creates a real, inverted and reduced image of the image from lens 1. The combination gives a real, upright, enlarged image of the object. Example: multiple lenses
so1 = 15 cm 1 2
Fi1 Fi2 f1 = 10 cm f2 = 5 cm si1 = 30 cm First find image from lens 1.
1 1 1 si1 = 30 cm 15cm si1 10cm Example: multiple lenses
so1 = 15 cm 1 2 d = 42 cm si2 = 8.6 cm
Fi1 Fi2 f1 = 10 cm f2 = 5 cm s = 30 cm i1 so2 =12 cm Now find image from lens 2.
1 1 1 si2 = 8.6 cm 12cm si2 5cm Example: multiple lenses, magnification
so1 = 15 cm 1 2 d = 42 cm si2 = 8.6 cm
Fi1 Fi2 f1 = 10 cm s f2 = 5 cm M i s = 30 cm T i1 so2=12 cm so 30cm Lens 1: M 2 Total magnification: T1 15cm M M M 1.44 8.6cm T T1 T 2 Lens 2: MT 2 .72 12cm +: image is not inverted Two lens equation
so1 1 d 2 si2
Fi1 Fi2 f1 f2 s i1 so2 f d f s f /s f For combination of two thin lenses: s 2 2 o1 1 o1 1 i2 d f2 so1 f1 / so1 f1
f2 d f1 distance from the last Back focal length (so1=): b.f.l. surface to the second d f1 f2 focal plane f d f Front focal length (s =): f.f.l. 1 2 distance from the first i2 d f f surface to the first 1 2 focal plane Multiple lenses: effective focal length
f2 d f1 f1d f2 b.f.l. f.f.l. d f1 f2 d f1 f2
If d 0 (lenses pushed close together), the effective focal length is: 1 1 1 1 ... If d 0 f f1 f2 f3 Fresnel lens Stops and apertures
Field stop - an element limiting the size, or angular breadth of the image (for example film edge in camera)
Aperture stop - an element that determines the amount of light reaching the image
Field stop determines the field of view Apperture determines amount of light only Entrance and exit pupils
Entrance pupil: the image of the aperture stop as seen from an axial point on the object through the elements preceding the stop Exit pupil: the image of the aperture stop as seen from an axial point on the image plane through the elements preceding the image no lenses between image and A.S. - exit pupil coincides with it. Chief and marginal rays
Marginal ray: the ray that comes from point on object and marginally passes the aperture stop
Chief ray: any ray from an object point that passes through the middle of the aperture stop It is effectively the central ray of the bundle emerging from a point on an object that can get through the aperture. Importance: aberations in optical systems Vignetting
The cone of rays that reaches image plane frome the top of the object is smaller than that from the middle. There will be less light on the periphery of the image - a process called vignetting Example: entrance pupil of the eye can be as big as 8 mm. Telescopes are designed to have exit pupil of 8 mm for maximum brightness of the image Collection Efficiency
Which lens collects more light?
f = 10 mm
f = 10 mm Relative aperture
Entrance pupil area determines the amount of light energy that reaches the image plane. Typically pupil is circular: the area varies as square of its diamater D. The image area varies as square of dimensions and is ~f2.
Flux density varies as (D/f)2. (D/f) relative apperture f-number Example: f=50 mm, D = 25 mm: f/# = 2 f f /# denoted as f/2 D F-number
The F-number, “f / #”, of a lens is the ratio of its focal length and its diameter. f / # = f / d f f
d1 f f d2
f / 1 f / 2 The F/#
f f /# D
•referred to as the “f-number” or speed •Exposure time is proportional to the square of the f-number •measure of the collection efficiency of a system •smaller f/# implies higher collected flux: • f or D decreases the flux area • f or D increases the flux area f-number of a camera lens
Change in neighbouring numbers is 2 Intensity is ~1/(f/#)2: changing diafragm from one label to another changes light intensity on film 2 times Numerical Aperture
Another measure of a lens size is the numerical aperture. It’s the product of the medium refractive index and the marginal ray angle.
NA = n sin() Why this definition? Because the magnification can be shown to f be the ratio of the NA on the two sides of the lens.
High-numerical-aperture lenses are bigger. Numerical Aperture
NA nsin
•describes light gathering capability for: lenses microscope objectives (where n may not be 1) optical fibers …
NA photons gathered Depth of Field
Only one plane is imaged (i.e., is in focus) at a time. But we’d like objects near this plane to at least be almost in focus. The range of distances in acceptable focus is called the depth of field.
It depends on how much of the lens is used, that is, the aperture.
Object Out-of-focus Size of blur in plane out-of-focus Image f plane
Focal Aperture plane
The smaller the aperture, the more the depth of field. Depth of field example A large depth of field isn’t always desirable.
f/32 (very small aperture; large depth of field)
f/5 (relatively large aperture; small depth of field)
A small depth of field is also desirable for portraits. Phys 322 Mirrors Lecture 13
Ancient bronze mirror
Liquid mercury mirror
Hubble telescope mirror Planar mirror also called plane, or flat mirrors
s = -s i r i o
Sign convention: s on the object side is positive, and negative on the opposite side Planar mirror
For a plane mirror, a point source and its image are at the same distance from the mirror on opposite sides; both lie on the same normal line.
Image is virtual, up-right, and life-size (MT = +1)
yi si The equation for lens works: MT yo so