Optics, Waves, Lasers 2 Flashcards
displacement of wave - angular
y(x,t)=Acos{wt}
displacement of wave - linear
y(x,t)=Acos{2pift}
displacement of wave if wave travelling with speed v
y(x,t)=Acos{w(t-x/v)}
can be other way around since cos theta = cos -theta
can also change depending on what variables given (w=2pif, f=1/T)
wave number
k=2pi/lambda
phase of wave
{kx+/-wt}
radians
wave speed
v=dx/dt=w/k
expressions for transverse velocity and acceleration
differentiating equation for y (partial diff)
wave equation
consider d2y/dt2 / d2y/dx2 = v^2
rewrite as d2y/dx2=1/v^2 d2y/dt2
combining hookes law with acceleration in SHM
ax=Fx/m=-kx/m
energy in SHM
1/2mv^2+1/2kx^2=constant
maximum displacement in SHM
energy is entirely potential energy (no kinetic)
can rearrange this formula for v
total displacement
algebraic sum of displacements
*might need to use cos(a+/-b)=cosacosb-/+sinasinb
principle of superposition
when two or more waves overlap, the resultant displacement at any point and at any instant is found by adding the instantaneous displacements that would be produced at the point by the individual waves if each were present alone
standing wave
wave shape stays the same
zero points found when sin(kx)=0
coherent
two monochromatic sources of same frequency and constant phase relationship
constructive interference
if waves from two or more sources arrive in phases, then they reinforce each other
r2-r1=m lambda
destructive interference
meet exactly out of phase
“cancel out”
resulting amplitude is difference of individual As, ie if same = 0
r2-r1=(m+1/2)lambda
most reliable way to ensure light from two sources remains in phase
use of slits
slit experiment
light through one slit then either of two
shone onto screen
bright spots=constructive
approximate geometry used in slit experiment
treat rays as paraleel so r2-r1=dsintheta
can do this because distance R to screen»_space; distance d between slits
expression to find centres of bright bands on the screen
let ym be distance from centre to centre of mth bright band
theta m be corresponding angle for band
ym=Rtanthetam
since R»d, thetam very small so tantheta=sintheta
so sinthetam=ym/R
so dsintheta=dym/R=m lambda
hence ym=Rmlambda/d
assumption in interference in thin films
assume light comes in almost vertically
monochromatic light shining on air wedge
would expect pattern of C and D interference
in reality, pattern is reversed, dark regions at integer multiple and bright at 1/2 integer multiples. Phase shift is a result of Maxwell’s equations, EM nature of light
na>nb
slow in first
Ei and Er same sign, no phase shift between incident and reflected
na=nb
no refletion
na<nb
slow in 2nd
Ei and Er opposite signs
phase shift or reflected relative to incident is pi radians
let thickness of film bet t, if neither or both reflected waves have phase shift then
2t=mlambda CON
2t=(m+1/2)lambda DEST
let thickness of film bet t, if one of the reflected waves have phase shift then
2t=mlambda DEST
2t=(m+1/2)lambda CON
newtons rings
convex surface in contact with plane glass surface, creating a thin air film
circular interference pattern
to avoid glare, need
destructive interference
Huygen’s principle
from shape of wave at some instant, can work out shape at later time
new wave found by building a surface that is tangential to secondary wavelets
drawing diagram and zooming in shows low of refelction
refractive index
ration of speed of light in vacuum to material
diffraction
bending of light around obstacles
spontaneous emission
direction and phase of photons emitted are random
shining light to absorb photon. Excited atoms return to ground energy and emit photon
stimulated emission
each incident photon encounters a previously excited atom. Each excited atom emits a second photon with same frequency, direction and phases as incident photon.
two photons in phase - coherent
number of atoms in state i
ni=Aexp{-Ei/kbT
therefore less atoms exist in higher energy states - negative exponenet
population inversion
number of atoms in excited state > lower energy state
rate of energy radiation by stimulated emission can exceed rate of absorption, system will act as a net source of radiation
steps to create population inversion
- material pumped to excite atoms. (gas - electrodes used) High enough voltage, electric discharge occurs. Collision of atoms and electrons excite atoms
- spontaneously emit, states depopulated. Atoms pile up in E2 as long lifetime. E2 more populated than E1
- some E2 will transition to E1 and give off photons. Parallel mirrors send photons back and forth through gas, creating stimulated emission. Laser beam leaves as one mirror is partially trasnparent to desired wavelength
holography
technique for recording and reproducing an image of an object through the use of interference effects
how to make a hologram
illuminate object with monochromatic laser
place photographic film so that it is struck by scattering light and direct light
interference of the light forms interference pattern on film
to view: shine light through developed film creating virtual and real images
