Phys 322 Lecture 13 Chapter 5

Geometrical Example

f= 100 mm

Object is placed at 600 mm, 200 mm, 150 mm, 100 mm, 50 mm. Where would be the image? 1 1 1 s s Solution   o i 600 120 so si f 200 200 s f 150 300 s  o i 100  so  f 80 -400 Focal plane

second, or back focal plane Thin + paraxial approximation: All rays that go through center O do not bend

All bundles of parallel rays converge to focal points that lay on one plane: second, or back focal plane

Fo- lies on first, or front focal plane. Imaging with a lens

Each point in object plane is a point source of spherical waves and the lens will image them to respective points in the image plane. Converging lens: principal rays

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 l, 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 l (paraxial). Diverging lens: forming image

F o O.A. Object Fi Image

Assumptions: • paraxial monochromatic rays Principal rays: • thin lens

1) Rays parallel to principal appear to come from focal point Fi. 2) Rays through center of lens are not refracted.

3) Rays toward Fo emerge parallel to principal axis.

Image is virtual, upright and reduced. Lens

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 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 Thin lens equations s 1 1 1 M   i   T s s s f o o i f= 100 mm

Real image 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 (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  f1d  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

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 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 : 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: aberrations in optical systems Vignetting

The cone of rays that reaches image plane from 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: 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 the pupil is circular: the area varies as square of its diameter D. The image area varies as square of dimensions and is ~f2.

Flux density varies as (D/f)2. (D/f)  relative aperture 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

Change in neighboring numbers is 2 Intensity is ~1/(f/#)2: changing diaphragm 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 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

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.