Transcript 09Optics

OPTICS
• Visible Light: The visible region of the E&M spectrum.
• Optics:
– Study of the behavior of light.
– Principles extend to other E&M radiation.
– Many principles apply to sound and other waves.
• Geometrical Optics or “Ray Model”
– Describes waves in terms of lines called “rays”.
• Physical Optics or “Wave Model”
– Describes waves in terms of wavefronts and waves.
• Quantum Optics
– Describes E&M radiation in terms of packets or “quanta” or
photons.
Geometrical Optics
• Ray Model:
– Light travels in straight lines called rays.
– Rays point in direction of propagation of wave
• Medium:
– A material through which a wave propagates.
– Light can travel through a medium and it can travel
through a vacuum.
– Sound can only travel through a medium (since it
depends on vibrations of atoms).
– When a Ray hits an interface between two media it
can be reflected, transmitted, and/or absorbed.
Reflection
• Is actually an absorption and reemission of light (quantum model).
• Can be explained with use of rays
– Incident Ray from one medium hits
the surface of another medium at some
point.
– The Incident and Reflected Rays
define a plane.
– We draw an imaginary line normal to
the surface at the point of incidence and
in the plane of the two rays.
• Law of Reflection: θi = θr where
the angles of incidence and
reflection are measured relative to
the surface normal.
Specular and Diffuse Reflection
• Specular Reflection:
– light appears to obey the Law of Reflection from a smooth surface.
– A bright spot appears on the surface based on the viewer’s
orientation (Bright spot where eye is in line with reflected ray).
• Diffuse Reflection:
– Light reflecting off a rough surface is in different directions
because the surface normals are in different directions.
– Viewer sees an averaged brightness, with no bright spot.
– Surface’s appearance is view independent.
Refraction
• When a ray hits an interface between two media and is
transmitted
– It changes speed upon entering the new medium
– The change in speed causes the ray to bend or “refract”.
• Index of Refraction (n) for a material:
– n = c/v > 1
– where c = speed of light in vacuum
– and v = speed of light in medium
• Snell’s Law of Refraction: (again all angles are
measured relative to the surface normal.)
Or: n1
sinθ1 = n2 sinθ2
Refraction
(cont’d)
• When light travels from one medium into another:
–
–
–
–
–
Speed changes
Wavelength changes
Frequency (color) remains the same.
The relationship v=f λ still holds.
Light passing into medium of larger n bends towards
normal.
– Light passing into medium of smaller n bends away
from normal
• Total Internal Reflection: Beyond a “critical angle” of
incidence, the ray will be refracted back into the original
medium.
Example: Light passing through a
Rectangular Plate
• We have a glass plate with thickness: d=2cm
and ng=1.5. Air has na=1.0. Light reflecting off
an object is incident on the plate with an angle
θi=60 degrees. The light enters the plate and
emerges out the other side.
– Show that the incident ray is parallel to the
emergent ray, but displaced laterally.
– Find the lateral displacement (δ) of the emergent
ray from the incident ray.
Images formed by Plane Mirrors
Images Formed by Spherical Mirrors
• Concave Mirror
– Reflection takes place on the inner
spherical surface. Rays converge.
• Convex Mirror
– Reflection takes place on the outer
spherical surface. Rays diverge.
• Geometric Properties
– Principal Axis: straight line
perpendicular to the center of the
mirror.
– C, R = center and radius of
Curvature.
– F, f = focus and focal length
– f = R/2
Ray Tracing to Locate Images
•
•
Rays parallel to principal axis coming from ∞
when reflected, all pass very nearly through the
focus (F).
To map a point on an object to its image
location, we draw three rays:
1. Ray 1 is drawn parallel to the axis and passes through F
after reflection.
2. Ray 2 is drawn through F and its reflected ray is then
parallel to the axis.
3. Ray 3 is drawn perpendicular to the mirror (passing
through C) and its reflection comes back on itself.
Image Formed by a Concave Mirror
• Case 1: Object is further from mirror than focus:
– Image formed is REAL (light actually passes through the
image point).
– Image is inverted.
Image Formed by a Concave Mirror
(cont’d)
• Case 2: Object is closer to the mirror than focus:
– Image formed is VIRTUAL (appears behind mirror
where no light actually falls.)
– Image is upright.
Dimensions on a Ray Diagram
• Object Distance (so), Image Distance (si)
– Are (+) if on the reflecting side of the mirror.
– Are (-) if behind the mirror.
– Mirror Equation: (1/ so + 1/ si = 1/f)
• Object Height (ho), Image Height (hi)
– ho is always (+)
– hi is (+) if upright; (-) if inverted.
• Magnification: M = hi / ho = - si / so
• How can an object distance be negative?
– If it is an image of a real object and is the effective
object of a second mirror or lens.
Ray Diagram for a Convex Mirror
Image is Virtual and Upright.
Sign Conventions for Mirrors
• For use in The Mirror Equation and for
Magnification:
• 1/ so + 1/ si = 1/f ; M = hi / ho = - si / so
–
–
–
–
–
–
–
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so is (+) if the object is in front of the mirror.
so is (-) if the object is in back of the mirror.
si is (+) if the object is in front of the mirror.
si is (-) if the object is in back of the mirror.
f and R are (+) for a concave mirror.
f and R are (-) for a convex mirror.
If M is (+), image is upright
If M is (-), image is inverted.
Images Formed by Thin Lenses
• Images are formed by refracting rays.
• Rays are very nearly focused to a point if thickness
of lens is small compared to radius of curvature (R).
• Converging Lens
– Thicker in the middle than at the ends.
– Causes incoming parallel rays to converge
• Diverging Lens
– Thinner in the middle than at the ends.
– Causes incoming parallel rays to diverge.
Ray Tracing to Locate Images
(Thin Lenses)
•
•
Draw the thin lens as a line; incoming rays
refract (bend) as they pass the line.
To map a point on an object to its image
location, we draw three rays:
1. Ray 1 is drawn parallel to the axis and is refracted
by the lens to pass through a focal point (F) (far for
converging; near for diverging).
2. Ray 2 is drawn towards the very center of the lens
and passes straight through (thin lens).
3. Ray 3 is drawn on a line passing through the other
focal point (F′) and emerges parallel to the axis.
Image Formed by a Converging Lens
• Case 1: Object is further from lens than focal point:
– Image formed is REAL (light actually passes through the
image point).
– Image is inverted.
Image Formed by a Converging Lens (cont’d)
• Case 2: Object is closer to lens than focal point:
– Outgoing rays must be traced backward to find image.
– Image formed is VIRTUAL (light only appears to diverge from
the image point.)
– Image is upright.
Image Formed by a Diverging Lens (cont’d)
• Outgoing rays must be traced backward to find image.
• Image formed is virtual and upright.
The Lens Equation
• Relates image distance, object distance and focal
length (same as mirror equation):
• 1/ so + 1/ si = 1/f ; M = hi / ho = - si / so
• Sign Conventions:
– f is (+) for converging lens and (-) for diverging lens.
– so is (+) if it is on the side from which light is coming
(true except sometimes in combinations of lenses),
otherwise negative.
– si is (+) if it is on the opposite side of lens from where
light is coming, otherwise negative. (si is (+) for real
images and (-) for virtual images.)
– hi is (+) for an upright image; (-) if inverted
Lenses Immersed in a Media
• Suppose a converging lens is immersed in water.
What would be the effect on its focal length?
– Refracted rays bend less; the focal length increases.
• Suppose a converging lens is put in a vacuum
instead of in air?
– Refracted rays bend slightly more; focal length
decreases.
• Suppose a converging lens is immersed in a
medium that has a higher index of refraction than
the lens material.
– Refracted rays bend away from the normal; lens
becomes diverging.
Compound Systems of Lenses
• The image of the real object becomes the “object of the next
lens”. This image is called a “virtual object”
• Example : Image formation by a compound microscope
– AB is the actual (real) object seen by the “objective lens”.
– A´B´(image of AB) is the virtual object for the “eyepiece lens”.
– A´´B´´ is the final enlarged, virtual image.
Films at Hippocampus
Assignment: View these animations:
• After clicking Here, at Hippocampus.org Physics
APB II
– Scroll down to Geometric Optics
– Then click on Flat and Concave Mirrors (view
animation).
– Then click Lenses (view animation).
– Then click on Images (view animation).
Physical Optics (Wave Model)
• How can we predict the path of propagation of
a wave?
– Wavefront: the entire width of a crest or trough.
– For a spherical wave propagating from a point source
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Wavefronts are spherical surfaces
Circular in two dimensions
Radius of Curvature increases with distance from source
We have Plane Waves when R  ∞
Plane Waves are Forever.
• Huygens Principle
– provides us with a way to graphically construct
successive wavefronts from preceding wavefronts.
Huygen’s Principle
• To construct a subsequent wavefront from an existing
wavefront:
– Every point on an existing wavefront can be considered as a
point source of spherical wavelets spreading out at the wave
speed.
– The new wavefront is the surface that is tangent to these small
“wavelet fronts”.
(a) Spherical Waves:
(b) Plane Waves
Circular wavefronts
that get flatter.
Flat wavefronts
that stay flat.
How does a Plane Wave propagate
through a Small Slit?
(length ~ wavelength)
• Diffraction: Applying Huygens’ Principle Shows that the
wave bends around the corners of the slit (and other
obstacles).
• Do Particles Diffract?
Plane Waves propagate through
the slit as spherical waves.
Young’s Double Slit Experiment
• Light was passed through two closely spaced slits.
• Revealed the wave nature of light:
• Diffraction resulting in Interference.
• Sunlight (plane waves)
passes through single slit
creating spherical waves.
•Spherical waves (crests are
red, troughs are yellow) pass
through double slit and then
to the screen.
•Interference bands (or
“fringes”) result; alternating
bright and dark regions.
Young’s Experiment
(cont’d)
• Consider each wavelet
front drawn to be a crest.
•Midway between each
crest is a trough.
•When a crest from S1
meets a crest from S2,
we get a bright band.
•When a crest from S1
meets a trough from S2
(and vice-versa) we get a
dark band.
Young’s Experiment
(cont’d)
Animation Double Slit Experiment
Central Maximum
(m=0; zeroth fringe)
Young’s Experiment
(cont’d)
• How can we predict where on the screen we’ll find
bright and dark bands?
– Specify a Point P on the screen by θ
– Approximation: d << L, then I1 and I2 are nearly
parallel at slit
– I2 travels an extra distance (δ) to P than I1: δ = d sin θ
Young’s Experiment
(cont’d)
• δ = d sinθ is called the path difference between the two rays.
When would I1 and I2
interfere destructively at P?
When Path difference is an odd
number of half wavelengths or
d sinθ = (2m+1) (λ/2)
When would I1 and I2 interfere
constructively at P?
When Path difference is an even
number of half wavelengths or
d sinθ = (2m) (λ/2)
Where: m = 0,1,2, …. = the “order” of the fringe.
Young’s Experiment
(cont’d)
• Diffraction occurs when d ~ λ; (radio waves
can diffract around mountain tops).
• Example:
– Two slits d = .1 mm; L=1.20 m
– Light of wavelength λ = 500 nm passes through
– How far apart will the bright fringes be?
d sin θ = 2m (λ /2)
θ
d
xm
(xm/L) = tan θ ~ sin θ ~ θ
xm = (λL/d) m
L
Young’s Experiment
(cont’d)
• xm = (λL/d) m (positions of bright spots)
So:
• x0 = (λL/d) (0) = 0
• x1 = (λL/d) (1) = 6 mm
• x2 = (λL/d) (2) = 12 mm
• So fringes are spaced 6 mm apart.
• What happens if λ increases?
– Spacing increases
– Redder light is spaced more widely than blue light.
– White light: bright white at central max (m=0)
• THEN DISPERSION; SPECTROSCOPIC APPLICATIONS
– What happens to spacing if experiment is done in water?
Coherence
• When two waves have a distinct phase
relationship, they are coherent.
– Example: Light split from one source, then
recombined to form interference fringes.
• Incoherence: Light sources with random
phase differences. These will interfere
randomly and produce uniform brightness
instead of fringes.
The Diffraction Grating
• When coherent light is
passed through a series of
slits, the peaks become
sharper; more pronounced:
– Maxima are located by the
same geometry as used for
Young’s double slit:
sinθ = mλ/d; m=0, 1, 2, 3, …
– Edges of wider two-slit peaks
are removed by destructive
interference by light coming
through slits much further
away.
Top: 2 slit
Bottom: 5 slit
Diffraction Gratings (cont’d)
• Types:
– Transmission: arrays of slits allowing light to pass
through.
– Reflective: grooves that reflect light. Ex: CD’s can
act as reflective gratings.
• Applications:
– Spectroscopy: Can reveal the electronic structure
(energy levels) and composition of a material.
– Crystallography: Can reveal the atomic structure
(atomic positions) of a material. Spaces between
atoms act like slits.
Example
• For a diffraction grating containing N= 10,000
lines/cm calculate the first and second order
angles for 400 nm blue light and 700 nm red light.
• Note: d = (1/N) cm = 10-4 cm; sinθ = mλ/d
– First Order (m=1)
• Blue: sinθ = mλ/d = (1)(400nm)/ 10-4 cm = .400; θ=23.6o
• Red: sinθ = mλ/d = (1)(700nm)/ 10-4 cm = .700; θ=44.4o
– Second Order (m=2)
• Blue: sinθ = mλ/d = (2)(400nm)/ 10-4 cm = .800; θ=53.1o
• Red: sinθ = mλ/d = (2)(700nm)/ 10-4 cm = 1.40; θ = !!
• The second order red fringe does not appear.
Thin Film Interference
• When light reflects off a surface of higher index of
refraction, it gets a λ/2 phase shift. Find net path
difference between incident and reflected waves.
Oil Film on Water
Reflective Coating on Lens
Thickness of film plus λ/2 phase
shift cause constructive
interference of red light.
Thickness of coating plus two λ/2
phase shifts cause destructive
interference of green light.
Single Slit Interference
• Instead of treating a slit as a “point emitter” of light, we
view a slit as being an array of point emitters.
• As with the grating, there are mutual constructive and
destructive interferences giving maxima and minima.
• Maximum at θ = 0o and
• Minima found at: wsin θ = Nλ; N = 1, 2, 3, ….; N≠0
• Not usually quantitatively on AP B exam.
Single Slit Applets
• Single Slit Diffraction Applet: Can vary
wavelength and slit width.
• wsin θ = Nλ; N = 1, 2, 3, ….; N≠0
Optics
Reflection of Light and Mirrors