Image of an extended object: spherical mirror

Physics 272

Fall 2014 http://www.phys.hawaii.edu/~philipvd/pvd_14_fall_272_uhm.html

Prof. Philip von Doetinchem philipvd@hawaii.edu

Phys272 - Fall 14 - von Doetinchem - 202

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Plane mirror produces image of same size and the same distance

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A plane mirror can be treated as a spherical mirror with very large radius

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Important: the observer sees an object at the image point, but no light rays go through this point

→ virtual image

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All reflected light rays actually go through the image point (unlike the plane mirror)

→ real image

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Focusing properties of spherical mirrors are, e.g., essential for photography and telescopes


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Relationship between angles:

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Image distance:


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Assume that angle  is small →  is also small:

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This object-image relationship does not depend on angles

→ all light rays meet in one point

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Object on the same side of center point of mirror is called concave mirror or converging mirror


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When the object is very far from the mirror:

● equation is exactly true for parabolic mirrors

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Therefore parabolic mirrors are preferred in technical applications (e.g., telescopes)

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If object placed at the focal point

→ trace rays in opposite direction

→ image is created at infinity


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An object placed further away from the mirror surface than the focal point appears inverted and can appear smaller, larger, equal in size depending on the position and the focal length:

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Source: http://en.wikipedia.org/wiki/Parabolic_mirror actual image away → it reduces the intensity (less energy is reflected)


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If an object is placed closer to a concave mirror than the focal point → image is virtual and magnified

→ example: makeup mirror

Source: http://en.wikipedia.org/wiki/Parabolic_mirror


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For a positive object distance a convex mirror always forms an erect, virtual, reduced, reversed image

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Virtual image of a convex mirror projects a larger field of view than a plane mirror

→ Objects in a convex mirror appear smaller

(“Objects in mirror are closer than they appear”)


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Light falls on a convex mirror → virtual image behind mirror

Object-image relation is valid as before if we respect the sign rules:

–

–

–

Object distance s is positive

Radius R is negative

Image distance s' is negative


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Principal light rays:

A ray parallel to the axis, after reflection passes through the focal point of a concave mirror or appears to come from the virtual focal point of a convex mirror

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A ray through (or proceeding toward) the focal point is reflected parallel to the axis

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A ray along the radius through or away from the center of curvature intersects the surface normally and is reflected back along its original path

A ray to the vertex is reflected forming equal angles with the axis


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Essential for understanding lenses

The same general laws for refraction as for a plane surfaces apply

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Relationship between angles:

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Refraction law and other conditions:


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Putting it all together:


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Object-image relationship for spherical refracting surface:

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Very similar structure compared to the reflection case, but modified with the index of refraction


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Magnification:

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Snell's law and small angle approximation:


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Thin lenses

What does thin mean?

–

–

–

Parallel light rays cross two spherical surfaces

Between surfaces material of different index of refraction

(typically higher)

After leaving the material: where do light rays cross the optic axis?

Surfaces are close to each other with respect to the length of the lens

→ thin lens: parallel light is focused in focal points

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Each side of the lens has one focal point

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For a thin lens the focal length on both sides is the same (even for different radii on both sides)

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Contacts or eye glasses are examples of thin lenses


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Construction:

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–

– sign rules from the discussion of spherical mirrors apply to lenses

Parallel light ray from object is refracted in thin lens through the focal point on the other side of the lens

– Light going through the middle of the lens passes straight through the thin lens (no change in direction)

For a 3-D object the two directions perpendicular to the optic axis are reversed, the arrow along the optic axis is not reversed


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Object-image relationship is the same as for spherical mirrors:


http://phet.colorado.edu/en/simulation/geometric-optics

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Object further away than focal point:

– light rays converge and form a real image on the other side of the lens

Object inside focal point:

– Light rays diverge and the image is virtual and larger than the object

Photography: having the sensor at the right focal point is essential for a sharp image


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The image of the first refracting surface is used as the object position for the second refracting surface

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The sketch shows a distance of d between the two spherical mirrors → we will set distance to zero


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Image position after the first surface:

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Image of first surface acts as object for second surface. In coming light on second surface is on the opposite side as the image from surface 1: s

2

=-s

1

'


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Combining both equations:

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For a lens in air (s

1

→s, s'

2

→s', n b

= n)


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Focal length on both sides of the object are the same (set object distance and image distance to infinity):

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Be careful: rays at larger distance from optic axis are not going to the same focus point → abberation


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Converging lens: thicker in the center than at the edges

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Diverging lens: thicker at the edges than at the center

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Parallel rays are diverged → virtual image in focal point

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For calculations: use negative focal length value for diverging lens


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The eye works very much like a camera

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Crystalline index of refraction has index of refraction of ~1.437

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Eye is filled with substance having similar optical properties as water (n=1.336)

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Muscles change the focal length of the eye by squeezing the lens → radius gets smaller

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Relaxed eye focuses on infinity

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Image is projected in retina → connects over optic nerve to brain



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How to correct the near point of a farsighted eye from 100cm to 25cm (standard value) using a contact lens?

→ form a virtual image of the object at 100cm:

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Prescriptions typically use the inverse of the focal length

→ a converging lens of +3.0 diopters would correct this eye




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Mirror radius and focal length:

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Magnification:

Source: http://en.wikipedia.org/wiki/Parabolic_mirror


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Small image in front of a cylindrical glass rod:

● image is inverted and reduced in size


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Immerse glass rod in water (n=1.33):

● the refracted rays do not converge

● and appear to diverge from a point 21.3cm to the left from the vertex

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The result is a virtual image

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Image is still erect and the virtual image appears magnified


Cameras: magnification

Focal length [mm]

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Converging lens in front of light detector

(film or chip)

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Lens forms an inverted image on the light detector

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Good lenses correct for paraxial approximation and dispersion

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Longer focal length → higher absolute magnification factor (still inverted) → image size on light detector increases


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Light intensity on the detector depends on field of view and the aperture opening

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Field of view scales roughly as

1/f 2

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Wide aperture allows more light to enter

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Adjusting is a typical function of cameras

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Eventually also the exposure time is adjusted


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Combination of movable converging and diverging lens make it possible to change the focal length

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Real zoom lenses are more complicated than that and use more than 10 lenses to correct for various aberrations


small aperture large aperture

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Introducing an adjustable lens aperture helps increasing the depth of focus:

–

– light rays coming from an object further away from a lens are less refracted than light rays from a closer object

Extended objects along the optic axis are in focus for a narrow (wide) region for large (small) apertures


large aperture short depth of field object

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Out of focus: image becomes blurry

The circle of confusion on the image side is the size when the objects starts to appear blurry

(typical: 1 pixel of the sensor) long depth of field object circle of confusion small aperture circle of confusion http://graphics.stanford.edu/courses/cs178/applets/dof.html


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Different autofocus techniques are available

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–

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Contrast detection

Assist lamp

Phase detection:

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Splits incoming light

Analyzes different parts projected on different sensors in the same plane

Chip compares intensity patterns

Sensor plane is in focus when patterns are the same

Source: http://en.wikipedia.org/wiki/Autofocus


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Size of an image depends on size on retina

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Moving object closer than near point does not help

→ eye cannot focus

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Converging lens:

–

–

–

Place object at focal point

Virtual image at infinity has a much larger angular size

Lateral magnification for virtual image at infinity is not a useful quantity


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Image of one lens can be used as an object for the second lens

→ greater magnification can be reached without making the lenses too big

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Objects are placed closely to the focal point

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Short focal length of the objective and eyepiece lens cause a greater magnification

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Overall magnification is composed of lateral and angular magnification


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Telescopes are used to magnify objects at large distances

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Objective lens forms a real, reduced image of the object on the sky

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Objects are very far → image nearly perfectly at focal point

● first image serves as object for the eyepiece lens

→ if at focal point of eyepiece

→ observer can see the magnified virtual object at infinity


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Large magnification requires a large focal length f1

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Large focal length → lower intensity

→ large collecting area needed

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Modern telescopes are reflecting telescopes


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Hubble Ultra deep field:

At this time the universe was only 800 Million years old

Large magnification requires a large focal length f1

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Large focal length → lower intensity

→ large collecting area needed

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Modern telescopes are reflecting telescopes



Image of an extended object: spherical mirror

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