1. Lens Equation
( < 0 )

2. Signs in the Lens Equation for Thin Lenses
p is positive for real object
q is positive for real image
q is negative for virtual image
m is positive if image is upright
m is negative if image is inverted
f is positive if converging lens
f is negative if diverging lens
p is negative for virtual object

3. Geometric Optics vs Wave Optics
Geometric optics is a limit of the general optics where wave effects such as interference and diffraction are negligible.
Geometric optics applies when objects and apertures involved are much larger than the wavelength of light.
In geometric optics, the propagation of light can be analyzed using rays alone.
Wave optics (sometimes also called physical optics) - wave effects play important roles.
Wave optics applies when objects and apertures are comparable to or smaller than the wavelength of light.
In wave optics, we must use the concepts relevant to waves such as phases, coherence, and interference.

4. Coherence
When the difference in phase between two (or more) waves remains constant (i.e., time-independent), the waves are said to be perfectly coherent.
- laser light and light transmitted through a small aperture are coherent.
- light from a light bulb and sun light over some area are incoherent.
A single light wave is said to be coherent if any two points along the propagation path maintains a constant phase difference.
Only coherent waves can produce interference fringes!
Coherence length: the spatial extent over which light waves remain coherent.

5. Interference of Two Coherent Waves
Snapshot of wave fronts at a given instant
Constructive interference (in phase)
Destructive interference (completely out of phase) A
B,C

6. Intensity of Interference Fringes
Let the electric field components of the two coherent electromagnetic waves be
The resulting electric field component point P is then
I=0 when Φ = (2m+1)π , i.e. half cycle + any number of cycle.

7. Thin Films
Phase change by π
Thin here means that the thickness is comparable to the wavelength of the light.
The reflected light waves from the two sides of a thin film interfere.
Destructive interference eliminates (or minimizes) the reflected light!
e.g., non-reflecting lens coating
Phase difference could come from:
reflection, path length difference,
different indices of refraction
If the incident light propagates from a medium of lower index of refraction toward one of higher index of refraction, the phase of the reflected wave shifts by π.

8. Thin-Film Interference-Cont’d
Path length difference:
(Assume near-normal incidence.)
ray-one got a phase change of 180o due to reflection from air to glass.
the phase difference due to path length is:
then total phase difference: f = f’+180.

9. 7B-11 OIL FILM INTERFERENCE
DOCCAM 2
7B OIL FILM INTERFERENCE

10. Thin-Film Continued
The previous discussion was for the situation in which n2 > n1 and n2 > n3 , i.e., the index of refraction of the film is larger than those of the surrounding media, but they are also valid if the index of refraction of the film is smaller than those of the surrounding media (n2 < n1 and n2 < n3 ).
The equations “fail” for some of the following situations. Which one(s)?
For other cases, the conditions for maxima and minima are simply reversed.
If the film has an intermediate index of refraction
Conditions for maxima/minima will reverse!

11. Thin-Film Wedge
Newton’s Rings: The air between the glass plates acts like a thin film.
For a small strip of the wedge, the thin-film equations can be used to determine whether constructive or destructive interference of the reflected light occurs.
Since the thickness of the film changes over the length of the wedge, alternating bright and dark fringes form, when the wedge is illuminated.
A sensitive technique for checking
imperfections in a material!
“optically flat”

12. DOCCAM 2
7B NEWTONS RINGS

13. Newton’s Rings Summary
The air between the glass plates acts like a thin film.
Since the thickness of the film changes over the radius of the plates, alternating bright and dark fringes form, when the plates are illuminated. Because of the curvature of the upper piece, the film thickness varies more rapidly at larger radius. Thus the fringe separation is smaller toward the outside.

14. Two (narrow) slit Interference
According to Huygens’s principle,
each slit acts like a wavelet. The
the secondary wave fronts are
cylindrical surfaces.
Young’s double-slit experiment
Upon reaching the screen C, the
two wave interact to produce an
interference pattern consisting of
alternating bright and dark bands
(or fringes), depending on their
phase difference.
Constructive vs. destructive
interference

15. A bright fringe is produced if the path
Interference Fringes
For D >> d, the difference in path lengths
between the two waves is
A bright fringe is produced if the path
lengths differ by an integer number of
wavelengths,
A dark fringe is produced if the path
lengths differ by an odd multiple of
half a wavelength,
m is referred to as the order of the fringes:
m = 0, central maximum or first minimum
m = +/- 1, the first maximum or second minimum ……

16. Intensity of Interference Fringes
Let the electric field components of the two coherent electromagnetic waves be
The resulting electric field component point P is then
I=0 when f = (2m+1)p , i.e. half cycle + any number of cycle.

17. Intensity of Interference Fringes-Cont’d

18. Demo 7B-17 Laser and slits (small apertures)
DOCCAM 2
Demo 7B-17 Laser and slits (small apertures)