Week II:
REFLECTION AND REFRACTION AT SPHERICAL INTERFACES
II.A
Reflection and Focal Point of Concave Mirror (Report errors in measured quantities only)
Experiment: Place concave mirror into beam of parallel light rays, so that principal axis of mirror
coincides with central light ray. The incident parallel rays are reflected so that they
intersect at one point, the focal point of the mirror.
☛ Determine accurately the focal length f (distance between focal point and mirror
surface.
Experiment: For circular mirrors the focal length f is proportional to the radius of curvature R of the
mirror: f = A·R.
☛ Measure the radius of curvature R of the mirror (use the below hint), and determine the
constant A from the measured values of f and R.
Hint: R can be determined in two ways:
(1) By length measurements x and D on the millimeter paper using the illustrated geometric
relations between R, D and x
X
R
R–X
R
D
(2) Measure R by comparison with the circles on a polar graph paper.
II.B Reflection and Focal Point from a Convex Mirror (Report errors in measured quantities
only)
Experiment: Repeat same procedures from II.A for convex mirror. In this case, as the reflected light
beams don’t converge, trace the light beams on paper, extending them back to their point
of intersection (focal point).
☛ Determine focal length.
☛ What is focal length of a plane mirror?
II.C
Imaging by Convex Mirror (Report errors in measured quantities and do error propagation)
Experiment: Produce light beams diverging from a single “object point” by placing the circular plastic
element into a parallel light beam. Place the convex mirror with its vertex ~ 5 cm away
from this object point S, sketch arrangement and light rays on paper (starting from the
“object point”), and determine graphically the image point P conjugated to S.
☛ Test the validity of the spherical mirror equation,
1
1
2
1
,
by evaluating (1/so+1/si) and -2/R and 1/f separately with their proper sign convention!
Experiment: Turn mirror around (from convex to concave) leaving vertex at the same place.
☛ Observe qualitatively how the image point P changes when turning the mirror around.
☛ How are these changes reflected in the spherical mirror equation?
II.D
Refraction and Focal Distances of a Single Spherical Surface (Report errors in measured
quantities and do error propagation)
Experiment: Use the semicircular plastic element with R as its radius of curvature to determine the
refractive properties and the two focal distances of a single spherical surface between air
(n1=1) and plastic (n2).
a) To determine the focal point in the plastic: Let a parallel beam fall on circular surface
of the element. “Extend” the plastic by attaching the long plane-parallel element.
☛ Measure the distance of focal point in the plastic from the vertex.
b) To determine the second focal point outside the plastic: Let a parallel beam fall
normally onto the flat surface of semicircular element.
☛ Measure the distance of focal point from vertex outside the plastic.
Experiment: The two obtained focal distances in air (f1) and in the plastic (f2) obey the relation for a
single spherical surface.
☛ Evaluate n1/f1 and n2/f2 and (n2-n1)/R separately and check if they are equal within
experimental error (use the n2 value obtained in Lab. I which should be n2 = 1.50 ± 0.05) .
II.E
Refraction and Focal Distance of Thin Lenses (Report errors in measured quantities and do
error propagation)
Background:
A lens is a combination of two spherical interfaces, as treated in II.D. If the two interfaces are very
close, the thickness of the lens can be approximately neglected, and all relevant distances can be
measured from the center of the lens. The focal distance f of the lens with refractive index n (in air) is
determined by the radii of curvatures R1 and R2 of its two surfaces by the lens maker’s formula:
1
1
1
1
(Using the proper sign convention!)
Experiment: Observe qualitatively in a parallel beam of rays the focusing properties of the double
convex and double-concave lenses.
☛ What is the difference in the behavior of the two lenses, and how is this difference
reflected in the lens equation?
☛ Find focal points of both lenses. For the case of the concave lens, find the focal point
by ray-tracing.
☛ Check that the results are within experimental error of the ones predicted with the
above lens makers formula. Hint: Measure radii of curvature R1, R2 with the help of polar
coordinate paper.
II.F
Thick Lenses and Principal Planes (Report errors in measured quantities only)
Background:
For thick lenses with larger distance between the two spherical surfaces the “thin-lens-approximation”
does not work. The simples lens equation, however, can still be applied, if all distances are measured
from the so-called “principal planes” of the lens. These principal planes can be constructed easily as
shown in the figure on the next page in an experiment with parallel light incident on the lens by
extrapolating the incident and emerging beams and obtaining their focus of intersection. Doing this for
both directions for light passing through the lens yields the two principal planes (which may lie inside
or outside the lens, and for symmetric lenses must lie symmetric to the center of the lens). If focal-,
object-, and image-distances are measured from the principal planes, the simple lens equation for thin
lenses holds again. In graphic ray construction the light beams can be considered to be refracted only at
the two principal planes, with all rays running parallel to the axis between the two planes.
This concept of principal planes can be extended to any number and combination of (thick or thin)
lenses. It is always possible to determine two principal planes for any composite system, such that the
effective imaging process can be treated graphically or mathematically in the described way.
Primary Principal Plane H1
First Focal Point
V1
V2
H1
Front Focal Length : f.f.l.
Focal Length : f
d
h1
Secondary Principal Plane H2
Second Focal Point
V1
V2
H2
Back Focal Length : b.f.l.
Focal Length : f
h2
Experiment: Using the semicircular plastic piece as a thick lens, produce light beams like in the above
figures.
(a) Use the circular plastic piece right after the light box and the plano-convex lens to
produce a point object. Then place the semicircular piece behind it and move it until
the outgoing beams are parallel.
(b) Remove the circular piece but do not move the semicircle to obtain parallel incoming
beams.
☛ Determine in ray-diagrams both principal planes by observation, ray-tracing, and
extrapolation.
☛ Determine the focal distances measured from the principal planes and confirm that they
are identical within experimental error.
☛ Compare the measured results for the positions of the principal planes h1, h2 and focal
distances (f, b.f.l., f.f.l.) to the values calculated from the relations for thick lenses:
1
1
1
1
1
,
1
1
Experiment: Using the circular plastic piece as a thick lens,
☛ What do you expect for the distance between the principal planes? Confirm your
predictions with an experiment.
Sign Conventions (According to Hecht)
Sign Convention for Spherical Refracting Surfaces
and Thin Lenses (Light Entering from the Left)
so , fo
+
left of V
xo
+
left of Fo
si , fi
+
right of V
xi
+
right of Fi
R
+
if C is right of V
yo , yi
+
above the optical axis
Spherical Refracting Surface
Fo
V
fo
V
C
fi
F
Fi i
Fi
Thin Lens
yo
Fo
Fi
yi
xo
f
si
so
Meanings Associated with the Signs of Various
Thin Lens and Spherical Interface Parameters
Quantity
Sign
so
si
f
yo
yi
xi
f
+
-
Real object
Real image
Converging lens
Erect object
Erect image
Virtual object
Virtual image
Diverging lens
Inverted object
Inverted image
Sign Convention for Spherical Mirrors
Quantity
so
si
f
R
yo
yi
Sign
+
-
Left of V, real object
Left of V, real image
Concave mirror
C right of V, convex
Above axis, erect object
Above axis, erect image
Right of V, virtual object
Right of V, virtual image
Convex mirror
C left of V, concave
Below axis, inverted object
Below axis, inverted image
S
C
P
F
V
f
si
R
so