no slide title - las positas collegelpc1.clpccd.cc.ca.us/lpc/molander/pdfs/thinlens.pdf• lens...

January 01 LASERS 51

Thin Lenses• Raytracing paraxial rays through lenses using the

parallel ray method– Can be done for any lens whether thick or thin– Idea of a focal plane

• Definition of a thin lens (how thin is thin)• Lens maker’s formula and lens shapes• Parallel ray method for thin lenses• Image formation with thin lenses

– Positive and negative lenses– Virtual images, virtual objects– Definitions of important rays, axial and chief ray

• Newtonian imaging formulas


Raytracing by parallel-ray method

• Object-space ray parallel to axis– intersects image-space ray at second principal plane– image-space ray passes through second focal point

• Object space ray passing through primary focal point– intersects image space ray at first principal plane– image space ray travels parallel to axis

• This locates the image, but what about other rays– According to our paraxial approximation, all the rays from the

object point pass through the image point, but what about angle

parallel to axisdeflected atprincipal plane

passes throughsecondary focal point

passes throughprimary focal pointparallel to axis

deflected atprincipal plane


A few more properties of principal planes

• The two principal planes are images of each other; m=1– Any ray intersecting the primary principal plane becomes after

refracting through the lens, a ray which intersects the secondary principal plane at the same height

– This will be given without proof • Another set of Gaussian planes is the nodal planes

– Plane with angular magnification of 1• A ray intersecting the first nodal plane turns into a ray from the second

nodal plane traveling at the same angle– If the index of refraction at the image and object are the same,

the nodal planes and the principal planes are identical– This is the basis for the nodal-slide

Θ

Θ

Object-space ray

Image-space ray

Object Image


Tracing an arbitrary ray

• Draw the desired ray up to primary principal plane– The extension of the ray past the surface is shown dotted to

indicate that it is not the actual light path inside the lens

• From intersection point, draw a horizontal line to second principal plane

• Image ray continues from this point to the image point

The image-space ray for any object-space ray is found from just the principal planes and focal points

Arbitrary object-space ray

Image-space ray

horizontalline


Off-axis objects and focal planes• A bundle of parallel rays

which are not parallel to the axis converge to a point in the focal plane– focal plane is perpendicular

to axis, passing through focal point

• Similar definition of primary focal plane– a source off the axis in

primary focal plane leads to a parallel set of rays at an angle to the axis

Secondaryfocalplane

F'F

Θf

fΘ

Field curvatureThe rays from an off axis object may come to a good focus, but not in the focal plane. This aberration is called field curvature.


Thin Lenses

• Spacing between principal planes is small, can be ignored– all the refraction appears to take place at one plane

• Several other ways of thinking of a thin lens– ray doesn’t change its height much on going through the lens– radius of curvature large compared to thickness

Principal planesFocalpoints

Note: Thin lens approximation is not the same as the paraxial approximation


Lens maker’s formula

• The effect of a thin lens depends only on its focal length and location

• The focal length is determined by the index and the two radii-Lens maker’s formula

R1R2

n( )

−−=

21

1111RR

nf

All quantities in the diagram are positive. Keep track of sign conventions!


Lens shapes• All positive lenses shown have

same focal length• All negative lenses shown have

same focal length• Positive lenses are thicker at the

center than at the edge– Negative lenses thinner at center

biconvexplanoconvex

positivemeniscus

biconcave planoconcave

negativemeniscus

• In paraxial optics for thin lenses, lens shape doesn’t matter, only focal length– aberrations do depend on shape– smaller aberrations if angles are not so steep– distribute “power”evenly among surfaces


Parallel-ray method for thin lens

• Three rays are easily traced1 chief ray not changed by lens (zero-thickness plane-parallel plate)2 object-space ray parallel to axis passes through 2nd focal point3 object-space ray through 1st focal point emerges parallel to axis

• Arbitrary rays also easily traced– all bending takes place at the lens– image-space ray goes through the image point

1

2

3FF'


Formation of a real image by a positive lens

• A positive lens can form a real image of a real object– A positive lens is one which is thicker at the center than the edge– Object must be farther away than one focal length from the lens


Magnifying glass

• When an object is closer than one focal length to a positive lens, the image is virtual, magnified, and non inverted


Formation of a virtual image by a negative lens

• A negative lens cannot form a real image of a real object, however it can form a virtual image– A negative lens in one which is thinner at the center

than at the edge– A negative lens can form a real image, but only of a

virtual object


Imaging using mathematical formulas

• Imaging equation can be derived either from graphical ray tracing, or from our previous formula for single surface refraction

• May be put in other forms, e.g. to find image distance when object distance, s, and focal length, f, are given

• By reversibility, s’ and s can be swapped to get a formula for s when s’ and f are known

• Be consistent with sign convention!!

y'

y

s s'

fss111

=′

+

fssfs

−×

=′


Imaging formula solved for any one of the three variables

fssfs

−×

=′• Two of the three variables must be

given in order to solve for the third– Make sure you understand which are the

givens• The sign conventions cover all the

cases– Make sure you understand what signs are

required– Real/virtual image means s’ is +/-– Real/virtual object means s is +/-

fssfs

−′′×

=

ssssf+′′×

=


Magnification (Lateral or transverse)

y'

y

s s'

• Magnification is defined as the ratio of the image height to the object height– since y’ is negative in this diagram, the

magnification is negative

• By similar triangles, being careful with sign convention (because s and s’ are both positive)

yym′

≡

ss

yym

′−=

′≡


Imaging formula with positive lens

• Either the object or the image must be real• Object in contact with lens coincides with its image• Object or image at negative infinity is the same as at

positive infinity (rays at lens are parallel)

Graph of Imaging formula

-4

-3

-2

-1

0

1

2

3

4

5

6

-5 -4 -3 -2 -1 0 1 2 3 4 5Object distance

Imag

e di

stan

ce

Distances measuredin units of focal length

Real objectReal imageVirtual object

Real image

Real objectVirtual image

Virtual objectVirtual image


Imaging formula with negative focal lengthImaging formula

-6

-5

-4

-3

-2

-1

0

1

2

3

4

-5 -4 -3 -2 -1 0 1 2 3 4 5

Object distance

Imag

e di

stan

ce

Distances measuredin units of focal length

Real objectReal image

Virtual objectReal image

Real objectVirtual image

Virtual objectVirtual image

• Either object or image must be virtual


• Works for positive and negative lenses• Works for different indices in object and image space

Imaging nomograph


Some important definitions

Conjugate points

• Axial ray goes from the base of the object to the limiting aperture of the system

• Any place the axial ray intersects the optical axis there is an image

• The chief ray goes from the top of the object through the center of the lens


Properties of the axial ray

• Base of object images in same plane as tip– Therefore the image plane is perpendicular to the axis at

the point at which the axial ray crosses the axis– This is true for more complicated systems: Image planes

are located everywhere the axial ray crosses the axis

• Magnification can be found from the ratio of slopes of the axial angle (see box above)– u’/u is called convergence ratio

u u'

s s'

hFor small anglesremember sign convention

shu =

shu′

−=′

therefore

uu

ssm

′=′

−=


Newtonian imaging equation

• Simpler to use sometimes– object 2 f from focal pt gives image 1/2 f from focal pt, etc.– one-one imaging occurs for z=z’=f

• Measure z from object to focal pt, z’ from focal pt to image– therefore z, z’ both positive in diagram– similar to the convention used for focal lengths

• Relationship with Gaussian equations

z

z'FF'

2fzz =′×

fz

zfm

′−=−=

fsz −=fsz −′=′


Power of a lens

• Definition• Units

– If f is in meters then Power is in diopters

• Importance– “stronger” lens has more power– used in raytracing– when combined with convergences, u and u’ gives

simplified imaging equation

fPower 1

=≡φ

huu φ−=′u u'

s s'

h


Sag of a surfaceR

s

h Rhs2

2

=

• Sag is short for sagittal distance• Formula is approximate

– accurate to the same order as the paraxial approximation

• Can also be used to find R

shR2

2

=

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