Lenses and imaging

MIT 2.71/2.71009/10/01 wk2-a-1

Huygens principle and why we need imaging instruments A simple imaging instrument: the pinhole camera Principle of image formation using lenses Quantifying lenses: paraxial approximation & matrix approach “Focusing” a lens: Imaging condition Magnification Analyzing more complicated (multi-element) optical systems: – Principal points/surfaces – Generalized imaging conditions from matrix formulae

The minimum path principle

MIT 2.71/2.71009/10/01 wk2-a-2

(aka Fermat’s principle) Consequences: law of reflection, law of refraction

∫Γ n(x, y, z) dl

Γ is chosen to minimize this“path” integral, compared to

alternative paths

The law of refraction

MIT 2.71/2.71009/10/01 wk2-a-3

nsinθ = n’sinθ’ Snell’s Law of Refraction

Ray bundles

MIT 2.71/2.71009/10/01 wk2-a-4

pointsource

planewave

pointsource

very-veryfar away

sphericalwave

(diverging)

Huygens principle

MIT 2.71/2.71009/10/01 wk2-a-5

Each point on the wavefrontacts as a secondary light source emitting a spherical wave

The wavefront after a shortpropagation distance is theresult of superimposing allthese spherical wavelets

opticalwavefronts

Why imaging systems are needed

MIT 2.71/2.71009/10/01 wk2-a-6

Each point in an object scatters the incident illumination into a spherical wave, according to the Huygens principle. A few microns away from the object surface, the rays emanating from all object points become entangled, delocalizing object details. To relocalize object details, a method must be found to reassign (“focus”) all the rays that emanated from a single point object into another point in space (the “image.”) The latter function is the topic of the discipline of Optical Imaging.

The pinhole camera

MIT 2.71/2.71009/10/01 wk2-a-7

The pinhole camera blocks all but one ray per object point from reaching the image space an image is formed (⇒ i.e., each point in image space corresponds to a single point from the object space). Unfortunately, most of the light is wasted in this instrument. Besides, light diffracts if it has to go through small pinholes as we will see later; diffraction introduces artifacts that we do not yet have the tools to quantify.

Lens: main instrument for imageformation

MIT 2.71/2.71009/10/01 wk2-a-8

The curved surface makes the rays bend proportionally to their distancefrom the “optical axis”, according to Snell’s law. Therefore, the divergent

wavefront becomes convergent at the right-hand (output) side.

Point source(object)

Point image

Analyzing lenses: paraxial ray-tracing

MIT 2.71/2.71009/10/01 wk2-a-9

Free-spacepropagation

Refraction atair-glassinterface

Free-spacepropagation

Free-spacepropagation

Refraction atglass-airinterface

Paraxial approximation /1

MIT 2.71/2.71009/10/01 wk2-a-10

In paraxial optics, we make heavy use of the following approximate (1st order Taylor) expressions:

sinε ε tanε cosε 1≒ ≒ ≒

where ε is the angle between a ray and the optical axis, and is a smallnumber (ε<<1 rad). The range of validity of this approximationtypically extends up to ~10-30 degrees, depending on the desireddegree of accuracy. This regime is also known as “Gaussian optics.”

Note the assumption of existence of an optical axis (i.e., perfectalignment!)

Paraxial approximation /2

MIT 2.71/2.71009/10/01 wk2-a-11

Valid for small curvatures& thin optical elements

Ignore the distancebetween the location

of the axial rayintersection and theactuall off-axis ray

intersection

off-axis ra

y

axial ray

Apply Snell’s law as ifray bending occurred atthe intersection of theaxial ray with the lens

Example: one spherical surface,translation+refraction+translation

MIT 2.71/2.71009/10/01 wk2-a-12

Translation+refraction+translation /1

MIT 2.71/2.71009/10/01 wk2-a-13

Starting ray: location x0 direction α0

Translation through distance D01 (+ direction):

--------------------------------------------------------

Refraction at positive spherical surface:

Translation+refraction+translation /2

MIT 2.71/2.71009/10/01 wk2-a-14

Translation through distance D12 (+ direction): ---------------------------------------------------------

Put together:

Translation+refraction+translation /3

MIT 2.71/2.71009/10/01 wk2-a-15

Sign conventions for refraction

MIT 2.71/2.71009/10/01 wk2-a-16

Light travels from left to right A radius of curvature is positive if the surface is convex towards the

left Longitudinal distances are positive if pointing to the right Lateral distances are positive if pointing up Ray angles are positive if the ray direction is obtained by rotating

the

+z axis counterclockwise through an acute angle

On-axis image formation

MIT 2.71/2.71009/10/01 wk2-a-17

All rays emanating at x0 arrive at x2

irrespective of departure angle α0

∂x2 / ∂α0

= 0

“Power” of the sphericalsurface [units: diopters, 1D=1m-1]

Magnification: lateral (off-axis), angle

MIT 2.71/2.71009/10/01 wk2-a-18

Lateral

Angle

Object-image transformation

MIT 2.71/2.71009/10/01 wk2-a-19

Ray-tracing transformation(paraxial) between

object and image points

Image of point object at infinity

MIT 2.71/2.71009/10/01 wk2-a-20

Point object imaged at infinity

MIT 2.71/2.71009/10/01 wk2-a-21

Matrix formulation /1

MIT 2.71/2.71009/10/01 wk2-a-22

translation bydistance D10

refraction bysurface with radius

of curvature R

form common to allray-tracing

object-imagetransformation

Matrix formulation /2

MIT 2.71/2.71009/10/01 wk2-a-23

PowerRefraction by spherical surface

Translation through uniform medium

Translation+refraction+translation

MIT 2.71/2.71009/10/01 wk2-a-24

Thin lens

MIT 2.71/2.71009/10/01 wk2-a-25

The power of surfaces

MIT 2.71/2.71009/10/01 wk2-a-26

Positive power bends rays “inwards”

Negative power bends rays “outwards”

Simple sphericalrefractor (positive)

Plano-convexlens

Bi-convexlens

Simple sphericalrefractor (negative)

Plano-concavelens

Bi-concavelens

The power in matrix formulation

MIT 2.71/2.71009/10/01 wk2-a-27

(Ray bending)= (Power)×(Lateral coordinate)

⇒ (Power) = −M12

Power and focal length

MIT 2.71/2.71009/10/01 wk2-a-28

Thick/compound elements:focal & principal points (surfaces)

MIT 2.71/2.71009/10/01 wk2-a-29

Note: in the paraxial approximation, the focal & principal surfaces are flat (i.e.,planar). In reality, they are curved (but not spherical!!).The exact calculation isvery complicated.

Focal Lengths for thick/compoundelements

MIT 2.71/2.71009/10/01 wk2-a-30

generalizedopticalsystem

EFL: Effective Focal Length (or simply “focal length”)FFL: Front Focal LengthBFL: Back Focal Length

PSs and FLs for thin lenses

MIT 2.71/2.71009/10/01 wk2-a-31

The principal planes coincide with the (collocated) glass surfaces The rays bend precisely at the thin lens plane (=collocated glass surfaces &

PP)

1/(EFL) ≡ P = P1 + P2 (BFL) = (EFL) = (FFL)

The significance of principal planes /1

MIT 2.71/2.71009/10/01 wk2-a-32

The significance of principal planes /2

MIT 2.71/2.71009/10/01 wk2-a-33

Imaging condition: ray-tracing

MIT 2.71/2.71009/10/01 wk2-a-34

Image point is located at the common intersection of all rays which

emanate from the corresponding object point The two rays passing through the two focal points and the chief ray

can be ray-traced directly

Imaging condition: matrix form /1

MIT 2.71/2.71009/10/01 wk2-a-35

Imaging condition: matrix form /2

MIT 2.71/2.71009/10/01 wk2-a-36

Imaging condition: 1Output coordinate x´ must notdepend on entrance angle γ

Imaging condition: matrix form /3

MIT 2.71/2.71009/10/01 wk2-a-37

Imaging condition: S’/n’ + S/n – PSS’/nn’ = 0

system immersed in air,n=n’=1;

power P=1/f

n/S + n’/S’ = P

1/S + 1/S’ = 1/f

Lateral magnification

MIT 2.71/2.71009/10/01 wk2-a-38

Angular magnification

MIT 2.71/2.71009/10/01 wk2-a-39

Generalized imaging conditions

MIT 2.71/2.71009/10/01 wk2-a-40

Power:

Imaging condition:

Lateral magnification:

Angular magnification:

image systemmatrix

object

P = -M12 ≠ 0

M21 = 0

mX = M22

Ma = n/n’M11