n Flashcards
What is the wave equation (in terms of E field)?
What is the Phase vector equal to?
What are the solutions to the wave equation?
(a) for Plane waves, (b) for Spherical waves
Can waves of light be observed directly?
NO! With detectors, we always observe INTENSITY rather than the waves directly.
To observe wave phenomena, all photons need to be _____ ?
All photons must be coherent(approximately the same).
State the Poynting Vector. What does the Poynting vector describe?
The Poynting vector describes the ENERGY FLOW of an EM wave.
What is the Intensity of an EM wave (eq) ?
Derivation of intensity equation
Time average, given by equation on image.
More info on steps on other side
- E_ x B_ = IE_I IB_I sin(a)
- Average of sin(theta) is 0 (sin wave oscillates between -1 and +1).
- IEI_ IB_I = EB (now not vector)
- cos^2 oscillates between 0 and +1, with an average of 1/2 (so replace cos^2 with 1/2 when pulling out avg brackets)
What is the most relevent property between two waves for superposition?
Relative Phase/Phase difference is the most important property between 2 waves.
What is the equation for constructive interference of a Spherical wave/for double slit?
What is the equation for destructive interference of a spherical wave/for double slit?
State the phase difference for both of the cases above in terms of path length difference.
State the general equation for the position on a screen (double slit). State equation for the postion of mininum and maximum on a screen.
L is the distance between the slit and screen.
To observe wave properties of light, waves must be coherent. State the two types of coherence.
Temporal Coherence, Spatial coherence.
What does Temporal and Spatial coherence mean?
Temporal coherence: All photons contributing to interference pattern must be of the same wavelength.
Spatial coherence: All photons must have same momentum/propagate in same direction.
What is a Temporal Source, a Coherent source?
Easy to work through, remember E(n) = sqrt(I(n))* e^(iphi))
Mod(E) **2 = (E x E(complex conjugate)
E = E1 + E2
Mod(E) 2 = (E x E) = (E1+E2)(E1+E2*)
What is the phase shift upon reflection with a mirror?
State both Stoke’s relations
There is also a phase shift of pi between a beam reflected from the top and one reflected from the bottom of the beam splitter (r = -r’) since there is a phase difference of pi, (e^(i*pi)) = -1
Which interfermometer produces spatially fully seperated beam paths?
The Mach-Zehdner interferometer.
What is the I + output of the Mach-Zehnder Interferometer? (derivation)
Hints:
1. What is the phase of light? Treat x as the distance of the path.
2. Using Phases Phi1, Phi 2, write the wave formula for the light going along path 1 and 2 (E1, E2)
3. Assume an equal beam splitter transmission and reflection (t = t’ = r).
3. I + =
mod(something E1 + something_elseE2)**2
Itot, I- and I+ derivation
State the TOTAL intensity equation for the Michelson Interferometer.
This is the same as the I(-) of the Mach-Zehdner interferometer except there are no t,t’,r,r’ contributions (so no 1/2 multiplier).
What is the condtition for constructive interference (bright fringe at centre) of the Michelson Interferometer?
What is the condtition for destructive interference (dark fringe at centre) of the Michelson Interferometer?
State the path length difference for many beams. (a) in terms of d sin(theta). (b) between rn and r1.
TOTAL Intensity derivation of diffraction
grating Interference pattern.
Important to note that this is the total.
Sketches help a lot!
State the TOTAL intensity equation for an interference pattern for a diffraction grating, where N is the total number of slits.(a) in terms of sin x1 / sin x2. (b) in terms of cosine and phase difference.
Instead of 4- the equation will have N^2.
State the diffraction angle for a diffration grating.
Note: Intensity increases as N (N of slits) increases.
What are two applications of Thin films?
AR-coating (anti-reflective coating) and dielectric mirrors.
Thin films intensity derivation
The path length difference- i.e. between the first and second reflection =
D2 - D1 = n2(IABI + IBCI) - n1(IADI)
Keep in mind it is the PATH DIFFERENCE reflected ray 1 has travelled an extra optical distance of n1(IADI), reflected ray 2 has an extra optical distance of n2(IABI + IBCI). Hence the difference in optical path lengths is these two subtracted from each other.
Use the image provided to find the intensity equation for thin films. (or just state final equation). Derivation is in Lecture notes PO5
Keep in mind it is the PATH DIFFERENCE reflected ray 1 has travelled an extra optical distance of n1(IADI), reflected ray 2 has an extra optical distance of n2(IABI + IBCI). Hence the difference in optical path lengths is these two subtracted from each other.
The path length difference- i.e. between the first and second reflection =
D2 - D1 = n2(IABI + IBCI) - n1(IADI)
May find useful
Sum[1-x^2+x^4-x^6+x^8…] = 1+x^2
Sum[1+x^2+x^4+x^6] = 1-x^2
State the equations for constructive and destructive interference for thin films in AIR.
(n1 = n3, n2 effects path difference)
State the conditions for constructive and destructive interferene for thin films on A MEDIUM i.e. n1 =/ n3. (yes n3 not n2, n2 effects path difference, shown in derivation)
Choose theta 2 = 0 as the equation is true for all theta.
What is the optical path length difference for a thin film?
(This is set equal to the path length differences for minimum or maximum reflection for thin films is air or medium - flashcards 32 and 33, needed for worked example 15/ card 34)
Note: d is the thickness of the medium between the two surfaces.
State conditions for constructive and destructive interference for thin films (a) in air, (b) in a medium and state eq for general path length difference
Most cases we have done involve assumption of plane waves (due to superposition of many spherical)
What is does G(x) represent in the Kirchoff-Fresnel Diffraction integral?
G(x) represents the trasnmission function of the aperture.
What is the near field approximation for r?
What is the total Electric field at a point x’ for the far field diffraction pattern?
Hint for Q2:
Use the near field approximation for r.
Near field: r = (p -x’x/p)
What is the far field diffraction intensity at a point x’ given by? Is this a fourier transform?
What is rho representing? What is kx from?
Rho represents the distance from a slit
Kx = k * x’/p, k = 2pi/lambda, x’/p = sin(theta)
What are the 2 diffraction regimes?
What are their properties?
Far field: Plane waves at long distance.
Near field: Waves are still curved at close distance to slit.
State the far field diffraction equation in 2D (extend 1 dimensional to 2D)
What is Huygens-Frenel Principle?
- Huygens principle + add all electric fields together and square to get an intensity interference pattern.
- To find the intensity of the interference pattern in one point r, we need
to sum up all spherical waves hitting this point and take the square
Can do this also by using equations from card 10.
The width of the central bright field:
- The edges of the central bright field are at the positions of the first minima, to find central width, find distance between these minina i.e. 2* the distance from the origin to one minima.
To find the width use the fact that
sin(theta) = x’/z
x’ = zsin(theta) = 2 * 0.01 = 0.02
x’ is the position of one maxima from the central maxima, width = 2x’ = 0.04m
How do you find the intensity equation at a position of any aperture?
Hint: Single slit aperture intensity equation.
Multiply the fourier transform of G(x) by the single slit aperture intensity equation.
Another/easier way to see what to do for simple cases is multiplying the intensity of a single slit by the intensity of the double slit, grating etc.
Derive the diffraction pattern of a single slit at position Kx. This is important as it is used as a base for all apertures(that are covered).
State the Intensity equation for a grating at a position Kx.
Slit spacing = a
State the intensity at a position Kx of at a screen due to double slit interference.
What is the pattern formed by the imaging of a spot?
When does the first minima occur for this type of pattern?
Airy pattern first minima u = 2.83
(when J1(u) =0, hence I(u) = 0)
In order to reconstruct an image of a point, what is required?
State equation of minimum resolution of an optical instrument, hint u = 3.83.
Image width must be equal to 2*Rmin to be able to resolve image.
Eq KRmin sin(theta) = 3.83
State Rayleigh criterion.
State xmin of a microscope.
Rmin = xmin as stated by Rayleigh criterion.
What is the Numerical Aperture equal to? (What does it replace in equations)
Why can’t you use the standard equation of minimum resolution for a telescope?
If you did know the distance of an object how could you find the spatial resolution from the angular resolution?
Because you do not know the distance of the object so must keep as sin theta.
If you found out the distances, rearrange
sin theta = r’/z.
Try to get the same answers as in the image.
What is the angular resolution of a telescope?
Michelson Interferometer
Given the number of fringes, n and a displacement of one mirror, give the equation to find the wavelength.
Attempt the question: 0.63 micrometres.
