Paraxial Optics Less than Meets the Eye

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Paraxial Optics Less than Meets the Eye
Dr. Rod Fullard
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Goals

• Make sense of image formation by a lens using
  simplified (paraxial) optics and a point object
• Decide if we need to go to the trouble of using
  wave optics to explain real world image formation
  (including retinal image formation)

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Chapter 1 Paraxial Optics Less than Meets the
Eye
Page 1
Figure 1 Light ray emanating from a point
source and traveling toward a plus lens.
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Paraxial Optics Makes Retinal Image Formation
Really Easy
2?
16.67 mm
b
Object point to image point
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Paraxial Optics Makes Retinal Image Formation
Really Easy
Positivelens
AIR
AIR
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Wave Optics Makes it More Challenging
Ocular Surface
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Wave Optics Makes it More Challenging
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• Can anyone explain how this happens?
• Why is the image point to the right?
• What determines the exact position of the
  image point?

Positivelens
AIR
AIR
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What well cover today

• Light waves
• The principle of vergence of light
• Lenses, vergence and power

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Light Waves
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Drop a stone in a pond
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Drop a stone in a pond
Get a series of waves propagating in all
directions
In any specific direction the wave profile is a
series of crests and troughs
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Taking all directions, the advancing wavefront is
circular Here, successive crests are shown
Wave profile traveling to right
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Propagation of Light Waves
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Wavelength
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Amplitude Wave Height
Light wave amplitude relates to brightness
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Typically in optical systems, we restrict the
passage of light from a source with apertures or
other means
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2 D water waves vs. 3 D light waves
Water waves are 2-dimensional ? circular
wavefront Light waves propagate in 3 dimensions ?
spherical wavefront circular aperture produces a
conic section
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Optical wave diagrams simplify to 2 D
appearance
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Summary so far Light Waves

• Light waves behave like water waves (water 2D
  light 3D)
• A point source of light produces a spherical
  advancing wavefront
• The wave profile in any direction is a series of
  crests and troughs
• Characterize light (and water) waves by
  wavelength and amplitude

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What well cover today

• Light waves
• The principle of vergence of light
• Lenses, vergence and power

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Concept 1 Vergence the Key to Optics
Definition vergence wavefront curvature Which
wavefront below has the greatest vergence?
Place them together
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Vergence Wavefront Curvature (R)
Vergence is higher at A____ or B_____ ?
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Quantifying Vergence
Wavefront radius of curvature (r) measured FROM
wavefront TO center of curvature (point
source). Cartesian system distance measured from
right to left is negative
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Quantifying Vergence
Wavefront radius of curvature (r) measured FROM
wavefront TO center of curvature (point
source). Cartesian system distance measured from
right to left is negative
?0.20 m
A
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Vergence is very high near a point (source) and
decreases with distance
R2 ?2.5 D
R1 ?50 D
High vergence
Lower vergence
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What happens to the wave as it continues to
DIVERGE?
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What happens to the wave as it continues to
DIVERGE?
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Vergence and Distance
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Vergence and Distance
At a great enough distance from the source,
wavefronts are plane
Plane waves have zero vergence (radius of
curvature ?)
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A few more Wave Properties

• Refractive Index. What happens when light waves
  move from air to a refractive medium (e.g.
  water, glass)?

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Light Waves and Refractive Index

• A light wave traveling through air maintains
  constant wavelength
• What happens if the wave enters e.g. glass?
• Wave speed? ___________________________
• Wavelength? ___________________________

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Glass
Air
Water
? 587.6 nm
? 440.8 nm
? 385.8 nm
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Light Waves and Refractive Index

• What happens if the wave enters e.g. glass?
• Wave speed? ___________________________
• Wavelength? ___________________________

Slows down
Decreases
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ophthalmic crown glass
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Recap - Wave Properties Vergence

• Light waves emanate from a point source in all
  directions producing spherical wavefronts
• The wave profile (cross-section) in any given
  direction is a series of crests and troughs
• Vergence wavefront curvature
• Wavefronts become flatter with distance from a
  point (source). Therefore vergence decreases
  with distance from the point (source)
• Wavelength and velocity of light passing from air
  into a refractive medium decrease in direction
  proportion to the refractive index of the medium

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Wave Properties Vergence

• With the properties covered so far, we can
  explain (NOT remember) 90 of Geometrical Optics.

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What well cover today

• Light waves
• The principle of vergence of light
• Lenses, vergence and power

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Plane Waves Incident at a Slab of Flat Glass
(Optical Flat)

• Waves slow down in glass
• Wavelength decreases in glass
• Net effect when re-emerging into air
• still plane waves
• ? same as pre-entry
• speed same as pre-entry
• glass has delayed wavefront exit a little

Plane waves (zero vergence) and wave profile
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Plane Waves Incident at a Slab of Flat Glass
(Optical Flat)
Look at specific locations on the wavefronts
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Plane Waves Incident at a Slab of Flat Glass
(Optical Flat)
Marching band analogy Each dot is a member of a
marching band Band marches from a smooth surface
onto a muddy surface Mud slows them down
Look at specific locations on the wavefronts
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Plane waves incident at an angle to the normal
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Analogy Marching Band Traveling from concrete
to Mud
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Refraction at a Plane Surface Snells Law
Figure 1.10 Proving Snells Law of refraction
based on the fact that light waves slow down in a
denser medium. When light is incident from a
medium of lower index at an angle above the
normal, upper parts of the wavefronts are
retarded first. This causes the fully refracted
wavefronts to change direction and travel closer
to the normal in the denser medium.
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Figure 1.11 Magnified version of the previous
figure showing that the upper parts of wavefronts
are retarded first.
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Refraction at a Lens
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Positive (converging) Lens (nglass 1.523)
Air, n 1.00
Air, n 1.00
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Positive (converging) Lens (nglass 1.523)
Air, n 1.00
Air, n 1.00
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Positive lens converges waves to a point image
What happens with a thicker positive lens?
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Positive lens converges waves to a point image
What happens with a thicker positive
lens? ____________________________________________
________________________
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Positive lens converges waves to a point image
R _______ D
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Positive lens converges waves to a point image
R _______ D
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Paraxial Optics and Vergence

• Paraxial optics uses object and image distances
  and vergences, not wavefront curvature and radius
  of curvature
• Substitute object and image distance (? and ??)
  for wavefront radius of curvature
• Use corresponding vergence symbolsL vergence
  in object space L? vergence in image space

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Quantifying Vergence (Paraxial Optics)
Wavefront curvature (R) and radius (r) are
simple reciprocals Distance (? or ??) and
vergence (L, L?) are reciprocally related through
refractive index
In air, n 1.000, so vergence and distance are
effectively reciprocals
?0.20 m
A
?A ?0.20 m
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Positive lens converges light to a point image
Air n 1.000
L? 2.50 D
?? 40 cm
R 2.50 D
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Light converging in water (n? 1.333)
Water n? 1.333
?? 40 cm
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Vergence and Power
Definition Power change in vergence
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Power Change in vergence

• Can we calculate the power of this lens?
• Vergence leaving the lens (L?) 2.50 D
• What is the vergence entering the lens (L)?
• __________________________

Air n 1.000
L? 2.50 D
?? 40 cm
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Power Change in vergence

• Plane waves enter the lens (L 0.00 D)
• Vergence leaving the lens (L?) 2.50 D
• Vergence change 2.50 D lens power (F)

Air n 1.000
L? 2.50 D
?? 40 cm
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Vergence and Power
Definition Power change in vergence
The Vergence Relation L? L F L? image
vergence (refracted vergence) L object
vergence (incident vergence) F lens power
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Power Change in vergence

• Object vergence, L 0.00 D
• Image vergence (L?) 2.50 D
• L? L F ? F L? ? L 2.50 0.00 2.50 D

L0.00 D
Air n 1.000
L? 2.50 D
?? 40 cm
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Vergence Relation L? L F
F4.00 D
L0.00 D
L? L F 0.00 4.00 D 4.00 D
L?4.00 D
AIR, n? 1.000
?? _______ cm
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Recap Vergence Power

• Wavefronts slow down in a refractive medium.
• A positive lens is thicker in the center, causing
  wavefronts to lag behind in the center. This
  results (for a parallel incident wavefront) in a
  convergent emergent wavefront
• Power (F) change in vergence
• A thicker positive lens has greater power because
  wavefronts lag even further behind in the center
• Vergence relation L? L F

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Vergence Relation
Non-parallel incident light
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Vergence Relation Divergent Incident Light
F9.00 D
AIR
AIR
L?4.00 D
L?5.00 D
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Recap - Goal Make sense of image formation by a
lens using simplified (paraxial) optics and a
point object
Positivelens
AIR
AIR
Image point
??
?
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• Paraxial simplifications
• 1. Instead of showing actual wavefronts, show
  their path
• Light Rays describe the path of an advancing
  wavefront
• Light Rays are perpendicular to the wavefront

Positivelens
AIR
AIR
Image point
??
?
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• Paraxial simplifications
• 2. Represent the lens as a thin lens (zero
  thickness)
• Simplifies distance calculations
• Allows us to avoid dealing with lens thickness
  issues

Positivelens
AIR
AIR
Image point
??
?
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• The downside of these paraxial simplifications
• Vergence (wavefront curvature) cannot be
  appreciated
• Contribution of lens shape to power (vergence
  change) is lost
• Image formation becomes more abstract

Positivelens
AIR
AIR
Image point
??
?
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Recap Vergence Relation Paraxial
Simplifications

• For non-parallel incident light waves, lens power
  is still equal to change in vergence
• Image vergence (L?) differs from object vergence
  (L) by lens power (F)
• Paraxial simplifications
• Light waves are replaced by light rays. These
  are imaginary lines showing the wave path and are
  always perpendicular to the advancing wavefront
• Finite thickness lenses are replaced by thin
  lenses (zero thickness) to keep the math simple

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Lens Foci (Focal Planes) Second Principal Focus
The physical image located at F? is defined as a
real image
Second principal focus F?
Parallel incident light L 0 D
n? 1.000
?? f ?
Second (principal) focal length (f?)
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Lens Foci (Focal Planes) First Principal Focus
First principal focus F
Parallel refracted light, L? 0 D
n 1.000
? f
First (principal) focal length (f )
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Significance of Principal Foci
Second principal focus (F?) parallel incident
light produces an image at the second
focus Second focal length (f ? ) is a visible
measure of lens power. Shorter focal length
greater power First principal focus (F) light
incident from the first principal focus produces
parallel refracted light waves Collimation a
collimator is an optical device that produces
parallel (refracted) light. Placing a light
source at the first principal focus of a positive
lens collimates the light
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Collimator
A physical object is defined as a real object
Light source at first principal focus F
Parallel refracted light, L? 0 D
n 1.000
? f
First (principal) focal length (f )
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Negative (Diverging) Lenses
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Outer part of wave enters lens first and leaves
lens last ? greatest retardation at the edge,
producing divergent light
Where will the image waves focus?
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Image plane not readily defined because no
physical image forms (light keeps diverging to
the right)
Define the virtual image plane mathematically It
is the location from which the divergent
wavefronts appear to be originating
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The divergence in image space defined by a
virtual image plane is optically equivalent to
that produced by a real object (and no lens) in
the same location
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Recap Negative Lenses

• Negative lenses diverge parallel incident light
• Divergent light continues to diverge so no REAL
  image plane exists
• The center of curvature of the diverging
  refracted wavefront defines the virtual image
  plane
• No physical image exists at the virtual image
  plane

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Try some finite object ray-tracing
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Q1. In which of the locations below is vergence
greatest in magnitude (positive or negative)?
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Q1. In which of the locations below is vergence
greatest in magnitude (positive or negative)?
Vergence is lowest at ? and ?
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Q2. What is the nature of vergence in the
indicated location beyond the image plane? (A)
Convergence(B) , (C) Zero,
(D) Undefined.
Divergence
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Can we Justify using Wave Optics?
This seems nice and simple Two rays locate the
image
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Can we Justify using Wave Optics?
Is this really necessary? Waves are more
complicated
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Can we Justify using Wave Optics?
Answer yes, because object points rarely
produce image points. In the real world, a
point spread function is produced Retinal
images of points are point spread functions The
point spread depends on pupil size, ocular
surfaces, etc. etc. Light waves can define the
amount of point spread. Rays cannot