Practice Questions Flashcards
M(obj) =
I(obj)(max)-I(obj)(min) / I(obj)(max)+I(obj)(min)
I(obj)(max)
b0 + b1
I(obj)(min) =
b0 - b1
M(img) =
I(img)(max) - I(img)(min) / I(img)(max) + I(img)(min)
I(img)(max) =
c0 + c1
I(img)(min) =
c0 - c1
M =
M(img)/M(obj)
What happens to the image contrast and the modulation transfer function if the value of c1 is halved
contrast is halved
harder to distinguish edge of image
the peak of the PSF is reduced
the image is dimmer, less contrast
<ε(iso)^2> =
(θ/θ(0))^5/3
θ(0) =
r(0)/h
Seeing =
β = 0.98 λ/r(0)
Strehl ratio =
S = (r(0)/D)^2
<ε(fit)^2> =
k/N^(5/6) (D/r(0))^(5/3)
Marechal approximation
S ~ e^(<ε^2>)
fewer actuators means
design and operation of AO system is easier
angular resolution
α = 1.22 λ/D
cut off frequency of the modulation transfer function
f(c) = D/λF
most cases F = 1
phase structure function is
a measure of the phase variance between two locations on the phase front separated by r
phase structure function =
D(φ)(r) = <|φ(x) - φ(x+r)|^2>
coherence function is
an indication of the correlation of the phase front between two locations on the phase front
coherence function =
B(φ)(r) = <|φ(x) φ(x+r)|>
qualitatively explain this expression
D(φ)(r) = 2[B(φ)(0) - B(φ)(r)]
phase structure function eq and definition
coherence function eq and definition
B(φ)(0) is the correlation of the phase value with itself at the same location (since the shift parameter r is 0).
2[B(φ)(0) - B(φ)(r)] is then equivalent to the evaluation of the phase structure function.
Show that
D(φ)(r) = k^2δh (∞ ∫ -∞) [D(N)(r,z) - D(N)(0,z)] dz
B(φ)(r) = k^2δh (∞ ∫ -∞) B(N)(r,z)dz
substitue into
D(φ)(r) = 2[B(φ)(0) - B(φ)(r)]
inserting B(N)(0,0)-B(N)(0,0)
re-express the fried parameter in terms of temperature T and its corresponding structure constant C(T)
r(0) ∝ [ ∫ C^2(N)(h) dh]^(-3/5)
n - 1 ∝ P/T
dn/dT ∝ -P/T^2
C^2(N) = C^2(T) |dn/dT| ∝ C^2(T) P/T^2
=> r(0) ∝ [ ∫ C^2(T)(h) P/T^2dh]^(-3/5)
Should the Fried parameter vary with zenith angle
r(0) ∝ [0.423k^2(cosγ)^-2 ∫ C^2(N)(h) dh]^(-3/5)
and r(0) ∝ (cosγ)^(3/5)
where γ is the zenith angle. We see that at large zenith angles, cosγ goes to 0. So the fried paramter is smaller which is expected for stronger turbulence
Show that the phase structure function can be expressed as
D(φ)(r) = 6.88 (τ/τ(0))^(5/3)
D(φ)(r) = 6.88 (r/r(0))^(5/3)
r = v(bar)τ
and τ(0) = r(0)/v(bar)
substituting into D(φ)(r)
<ε(servo)^2> =
(τ(0) f(3dB)^(-5/3)
τ(3dB) =
1/f(3dB)
where f(3dB) is the bandwidth of the servo
Littrow configuration
θ(i) = θ(B)
θ(m) = -θ(B)
θ(m) = -θ(i)
how should the FSR be considered when designing a spectrograph
can adjust the free spectral range by changing the spacing between grating rules
groove separation
a measured in m
optical path difference
OPD = Δnl
refractive index =
n = 1+7.8x10^-7 P/T
turbulence occurs around the
tropopause at an altitude of 10km
with pressure 100mbar
and temperature 200K
number of grooves
N = 2π/Δφ
where Δφ = Δl k
in terms of 1 rad <ε(fit)^2> =
(N(1 rad)/N)^(5/6)
<ε(total)^2> =
sum of different <ε^2> =
for a loseless system
G(s) = G(r)
Kolmogorov Model
describes how the Kelvin-Helmholtz instabilities are broken up into smaller eddies
The main points of the Kolmogorov Model
- The kinetic energy is added to the system at the outer scale
- The energy then cascades down to the smaller scales
- when the scale is sufficiently small, the air viscosity dissipates the kinetic energy and the formation of new eddies stops. The average diameter at the smaller eddies is 1mm.
Shack-Hartmann wavefront sensor scheme
- an array of lenslets to focus the incoming wavefront onto an array of CCDs.
- Senses the position by measuring the gradient of wavefronts and is therefore, achromatic
- a plane wavefront would be focused to a grid of equally spaced images
- any distortion in wavefront will cause image to be focused away from central position
Shack-Hartmann WS diagram
see notes
Re-express the Fourier transform U in polar coordinates and evaluate the Fourier transform.
X = pcosθ and Y = psinθ
dXdY= |J|dpdθ = pdpdθ
x = ωcosψ and y = ωsinψ
substitute into U
Bessel function
identify Bessel function in U
n=0 , sigma = (θ-ψ) and x = kpω
use Bessel function identities
U(P) = U(0) = [2J(1)(ka)/(ka)]
How does the fourier transform U change with a
the intensity takes the form of an Airy function -> resulting image an Airy disc
for an ideal telescope the PSF is described by an airy function
as a changes the height of the central peak of the airy function will increase and its width decreases
a is the radius of the primary mirror.
as a increases the width of the central spot of the Airy disc will decrease and the central spot will be brighter
Fried parameter describes
the strength of the wave front distortion due to atmospheric turbulence
the value of the Fried parameter is
the equivalent diameter of an ideal telescope for observations limited by random inhomogeneities in the atmosphere’s refractive index
Strehl ratio
the ratio between the intensity of the central spot of the point spread function with that expected in the ideal case
the Strehl ratio compares
the observed intensity peak to that for a telescope working at the diffraction limit
For telescopes with large primary mirrors, the observed image will be limited by
the length scale of atmospheric turbulence which is characterized by the Fried parameter
for a wavefront sensing scheme that uses the same wavelength as the observation
α(SA) = λ/d
The fried parameter is a measure
of the characteristic coherence length of the distorted wavefront
isoplanatic angle is
the greatest angular distance between the guide star and the observing target. Where light from both pass through the same turbulent region.
Why does the isoplanatic angle vary with zenith
The isoplanatic angle is smaller with increasing zenith angle because light passes through more turbulent atmosphere at larger zenith angles.
Sodium Laser Guide Stars
There is a layer of sodium at altitudes of about 90km above sea level.
Laser light is used to excite an atomic resonance line in this layer of sodium to produce an artificial guide star.
How do laser guide stars improve the sky coverage
laser guide stars improve the sky coverage of AO systems because observations no longer need to be close to natural guide stars
The cone effect is
the result of laser guide stars being much closer to the observer than their targets.
As a result, light from the laser guide star probe all the turbulent air that distorts the observed wavefront for light from a distant star
cone effect diagram
see notes
As a increases the PSF
becomes narrower tending towards a delta function
S =
I(0)PSF/I(0)airy
as a increases the height
of the PSF increases thus the Strehl function decreases as a increases
how to obtain an image from the point spread function
I(x,y) = O(x,y) * P(x,y)
DΦ(r) = <|Φ(x) - Φ(x+r)|^2> =
(∞ ∫ -∞) |Φ(x) - Φ(x+r)|^2 dx
<ε(iso)^2> =
(∞ ∫ h(0)) DΦ(r) dh
PSF tells us
how the light from a source spreads out in an image
Modulation transfer function
measure of how contrast is lost in the image
Optical transfer function can be expressed
in terms of a phase structure function
turbulence generates
temperature cells
the long exposure OTF is given by
the phase structure parameter C^2(N) with height
the seeing is
the FWHM of the atmospheric PSF
atmospheric time constant
is the timescale over which a turbulence cell moves by its own size
adaptive optics compensate
the wavefront by sensing the wavefront then actuating on a deformable mirror
fitting error comes from
finite actuator spacing
bandwidth error comes from
finite servo bandwidth
free spectral range
is the wavelength difference at which orders start to overlap
a blazed grating
concentrates light into the desired order and makes efficient use of power
wavefront correction
is carried out by segmented or deformable mirrors
Curvature WFS is
a simple alternative to SHWS
doppler shift can be resolved by
using a grating in High order
downside is that orders overlap contaminating the spectrum
an echelle spectrograph overcomes this by using echelle grating followed by a cross disperser to create a 2D echellogram
the etendue of a system is set by
the stop
the bandpass of a spectrometer is
the minimum wavelength resolution achieved by matching diffraction from the entrance slit with that from the grating
