Lenses and Imaging (Part I)

• Why is imaging necessary: Huygen’s principle – Spherical & parallel ray bundles, points at infinity• Refraction at spherical surfaces (paraxial approximation)• Optical power and imaging condition• Matrix formulation of geometrical optics• The thin lens• Surfaces of positive/negative power• Real and virtual images

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Parabloid mirror: perfect focusing(e.g. satellite dish)

f: focal length

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Lens: main instrument for imageformation

air

opticalaxis

airglass

Point source(object)

Point image

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.

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Why are focusing instrumentsnecessary?

• Ray bundles: spherical waves and plane waves• Point sources and point images• Huygens principle and why we can see around us• The role of classical imaging systems

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Ray bundles

pointsource

sphericalwave

(diverging)wave-front

⊥ rays

wave-front ⊥ rays

pointsource

very-veryfar away plane

wave

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Huygens principle

Each point on the wavefrontacts as a secondary light sourceemitting a spherical wave

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

opticalwavefronts

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Why are focusing instrumentsnecessary?

incident light

...is scattered by

the object

object

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Why are focusing instrumentsnecessary?

incident light

...is scattered by

the object

objectimage

⇒need optics to “undo”the effects of scattering,i.e. focus the light

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Ideal lens

air

opticalaxis

airglass

Point source(object)

Point image

Each point source from the object plane focuses onto a pointimage at the image plane; NOTE the image inversion

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Summary:Why are imaging systems

needed?• 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 ideal optical imaging system

opticalelements

object image

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Ideal imaging system:each point in the object is mapped

onto a single point in the image

Real imaging systems introduce blur ...

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Focus, defocus and blur

Perfect focus Defocus

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Focus, defocus and blur

Perfect focus Imperfect focus

“sphericalaberration”

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Why optical systems do not focusperfectly

• Diffraction• Aberrations

• However, in the paraxial approximation to Geometrical Optics that we are about to embark upon, optical systems do focus perfectly

• To deal with aberrations, we need non-paraxial Geometrical Optics (higher order approximations)• To deal with diffraction, we need Wave Optics

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Ideal lens

air

opticalaxis

airglass

Point source(object)

Point image

Each point source from the object plane focuses onto a pointimage at the image plane

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Refraction at single spherical surface

for each ray, must calculate• point of intersection with sphere• angle between ray and normal tosurface• apply Snell’s law to find direction ofpropagation of refracted ray

R: radius ofsphericalsurface

center ofsphericalsurface

medium 1index n,

e.g. air n=1

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Paraxial approximation /1

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

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” or“paraxial optics.”

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

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Paraxial approximation /2

Ignore the distancebetween the location

of the axial rayintersection and theactual off-axis ray

intersection

Apply Snell’s law as ifray bending occurred at

the intersection of theaxial ray with the lens Valid for small curvatures

& thin optical elements

axial ray

off-axis r

ay

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Refraction at spherical surface

Refraction at positive spherical surface:

Power x: ray heightα: ray direction

R: radiusof curvature

optical axis

off-axis r

ay

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Propagation in uniform space

Propagation through distance D:

x: ray heightα: ray direction

optical axis

off-axis ray

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Paraxial ray-tracing

air airglass

Free-spacepropagation

Refraction atair-glassinterface

Free-spacepropagation

Free-spacepropagation

Refraction atglass-airinterface

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Example: one spherical surface,translation + refraction + translation

Non-paraxial ray(approximation

gives large error) medium 1index n,

e.g. air n=1

center ofsphericalsurface

Paraxial rays(approximation

valid)

R: radius ofsphericalsurface

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Translation + refraction + translation /1

Starting ray: location x0 directionα0

Translation through distance D01 (+ direction):

Refraction at positive spherical surface:

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Translation + refraction + translation /2

Translation through distance D12 (+ direction):

Put together:

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Translation + refraction + translation /3

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Sign conventions for refraction

• 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

optical axis +z

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On-axis image formation

Pointobject

Pointimage

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On-axis image formation

All rays emanating at arrive at x2

irrespective of departure angleα0

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On-axis image formation

All rays emanating at arrive at x2

irrespective of departure angleα0

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

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Off-axis image formation

Pointobject

(off-axis)

Pointimage

opticalaxis

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Magnification: lateral (off-axis), angle

Lateral

Angle

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Object-image transformation

Ray-tracing transformation(paraxial) between

object and image points

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Image of point object at infinity

Pointimage

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Image of point object at infinity

objectat ∞

image

: image focal length

Note:

ambient refractive indexat space of point image

1/Power

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Point object imaged at infinity

Pointobject

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Point object imaged at infinity

object

Imageat ∞

: object focal length

Note:

ambient refractive indexat space of point image

1/Power

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Image / object focal lengths

Pointobject

Imageat ∞

objectat ∞

Pointimage

Power Power

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Matrix formulation /1

translation byDistance D 01

refraction bysurface with radius

of curvature R

ray-tracingobject-image

transformationform common to all

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Matrix formulation /2

Refraction by spherical surface Power

Translation through uniform medium

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Translation + refraction + translation

translation by D12

Refraction by r.curv. R

translation by D01

result…

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Thin lens in air

Objective: specify input-output relationship

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Thin lens in air

Model: refraction from first (positive) surface+ refraction from second (negative) surface

Ignore space in-between (thin lens approx.)

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Thin lens in air

Thin lens in air

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Thin lens in air

Power of thefirst surface

Power of thesecond surface

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Thin lens in air

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Lens-maker’sformula

thin lens

thin lens

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Thin lens in air

thin lens

thin lens Ray bending is proportionalto the distance from the axis

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The power of surfaces

• Positive power bends rays “inwards”

Simple sphericalrefractor (positive)

Plano-convexlens

Bi-convexlens

• Negative power bends rays “outwards”

Simple sphericalrefractor (negative)

Plano-convexlens

Bi-convexlens