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Lecture 3Lecture 3
Part 1: Finish Geometrical OpticsPart 1: Finish Geometrical OpticsPart 2: Physical OpticsPart 2: Physical Optics
Claire MaxUC Santa Cruz
January 15, 2012
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AberrationsAberrations
• In optical systems
• In atmosphere
• Description in terms of Zernike polynomials
• Based on slides by Brian Bauman, LLNL and UCSC, and Gary Chanan, UCI
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Zernike PolynomialsZernike Polynomials
• Convenient basis set for expressing wavefront aberrations over a circular pupil
• Zernike polynomials are orthogonal to each other
• A few different ways to normalize – always check definitions!
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Piston
Tip-tilt
From G. Chanan
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Astigmatism(3rd order)
Defocus
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Trefoil
Coma
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Spherical
“Ashtray”
Astigmatism(5th order)
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Units: Radians of phase / (D / r0)5/6
Reference: Noll
Tip-tilt is single biggest contributor
Focus, astigmatism, coma also big
High-order terms go on and on….
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References for Zernike References for Zernike PolynomialsPolynomials
•Pivotal Paper: Noll, R. J. 1976, “Zernike polynomials and atmospheric turbulence”, JOSA 66, page 207
• Hardy, Adaptive Optics, pages 95-96
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LetLet’’s get back to design of AO s get back to design of AO systems systems Why on earth does it look like Why on earth does it look like this ??this ??
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Considerations in the optical Considerations in the optical design of AO systems: pupil relaysdesign of AO systems: pupil relays
Pupil Pupil Pupil
Deformable mirror and Shack-Hartmann lenslet array should be ““optically conjugate to optically conjugate to the telescope pupil.the telescope pupil.””
What does this mean?mean?
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Define some termsDefine some terms
• “Optically conjugate” = “image of....”
• “Aperture stop” = the aperture that limits the bundle of rays accepted by the optical system
• “Pupil” = image of aperture stop
optical axis
object space image space
symbol for aperture stopsymbol for aperture stop
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So now we can translate:So now we can translate:
• ““The deformable mirror should be optically The deformable mirror should be optically conjugate to the telescope pupilconjugate to the telescope pupil””
meansmeans
• The surface of the deformable mirror is an The surface of the deformable mirror is an image of the telescope pupilimage of the telescope pupil
wherewhere
• The pupil is an image of the aperture stop The pupil is an image of the aperture stop – In practice, the pupil is usually the primary mirror of In practice, the pupil is usually the primary mirror of
the telescopethe telescope
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Considerations in the optical Considerations in the optical design of AO systems: design of AO systems: ““pupil pupil relaysrelays””
Pupil Pupil Pupil
‘PRIMARY MIRROR
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Typical optical design of AO Typical optical design of AO systemsystem
telescope primary mirror
Science camera
Pair of matched off-axis parabola mirrors
Wavefront sensor (plus
optics)Beamsplitter
Deformable mirror
collimated
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More about off-axis parabolasMore about off-axis parabolas
• Circular cut-out of a parabola, off optical axis
• Frequently used in matched pairs (each cancels out the off-axis aberrations of the other) to first collimate light and then refocus it
SORL
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Concept Question: what elementary optical Concept Question: what elementary optical calculations would you have to do, to lay out calculations would you have to do, to lay out this AO system? (Assume you know telescope this AO system? (Assume you know telescope parameters, DM size)parameters, DM size)
telescope primary mirror
Science camera
Pair of matched off-axis parabola mirrors
Wavefront sensor (plus
optics)Beamsplitter
Deformable mirror
collimated
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Levels of models in opticsLevels of models in optics
Geometric optics - rays, reflection, refraction
Physical optics (Fourier optics) - diffraction, scalar waves
Electromagnetics - vector waves, polarization
Quantum optics - photons, interaction with matter, lasers
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Part 2: Fourier (or Physical) OpticsPart 2: Fourier (or Physical) Optics
Wave description: diffraction, interference
Diffraction of light by a circular aperture
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MaxwellMaxwell’’s Equations: Light as an s Equations: Light as an electromagnetic wave (Vectors!)electromagnetic wave (Vectors!)
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Light as an EM waveLight as an EM wave
• Light is an electromagnetic wave phenomenon, E and B are perpendicular
• We detect its presence because the EM field interacts with matter (pigments in our eye, electrons in a CCD, …)
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Physical Optics is based upon the Physical Optics is based upon the scalar Helmholtz Equation (no scalar Helmholtz Equation (no polarization)polarization)
• In free space
• Traveling waves
• Plane waves
kHelmholtz Eqn., Fourier domain
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Dispersion and phase velocityDispersion and phase velocity
• In free space
– Dispersion relation k (ω) is linear function of ω– Phase velocity or propagation speed = ω/ k = c = const.
• In a medium– Plane waves have a phase velocity, and hence a
wavelength, that depends on frequency
– The “slow down” factor relative to c is the index of refraction, n (ω)
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Optical path – FermatOptical path – Fermat’’s s principleprinciple
• Huygens’ wavelets
• Optical distance to radiator:
• Wavefronts are iso-OPD surfaces
• Light ray paths are paths of least* time (least* OPD)
*in a local minimum sense
Page 28 Credit: James E. Harvey, Univ. Central Florida
What is Diffraction?What is Diffraction?
In diffraction, apertures of an optical system limit the spatial extent of the wavefront
Aperture
Light that has passed thru aperture, seen on screen downstream
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Diffraction TheoryDiffraction Theory
We know this
Wavefront UU
What is UU here?
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Diffraction as one consequence of Diffraction as one consequence of HuygensHuygens’’ Wavelets: Part 1 Wavelets: Part 1
Every point on a wave front acts as a source of tiny wavelets that move forward.
Huygens’ wavelets for an infinite plane wave
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Diffraction as one consequence of Diffraction as one consequence of HuygensHuygens’’ Wavelets: Part 2 Wavelets: Part 2
Every point on a wave front acts as a source of tiny wavelets that move forward.
Huygens’ wavelets when part of a plane wave is blocked
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Diffraction as one consequence of Diffraction as one consequence of HuygensHuygens’’ Wavelets: Part 3 Wavelets: Part 3
Every point on a wave front acts as a source of tiny wavelets that move forward.
Huygens’ wavelets for a slit
From Don Gavel
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The size of the slit (relative to a The size of the slit (relative to a wavelenth) matterswavelenth) matters
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Rayleigh rangeRayleigh range
• Distance where diffraction overcomes paraxial beam propagation
L
D λ / D)
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Fresnel NumberFresnel Number
• Number of Fresnel zones across the beam diameter
L
D λ/D)
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Fresnel vs. Fraunhofer Fresnel vs. Fraunhofer diffractiondiffraction
• Very far from a point source, wavefronts almost plane waves.
• Fraunhofer approximation valid when source, aperture, and detector are all very far apart (or when lenses are used to convert spherical waves into plane waves)
• Fresnel regime is the near-field regime: the wave fronts are curved, and their mathematical description is more involved.
S
P
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Regions of validity for diffraction Regions of validity for diffraction calculationscalculations
The farther you are from the slit, the easier it is to calculate the diffraction pattern
Near field Fresnel Fraunhofer (Far Field)
D
L
mask
Credit: Bill Molander, LLNL
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Fraunhofer diffraction equationFraunhofer diffraction equation
F is Fourier Transform
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Fraunhofer diffraction, continuedFraunhofer diffraction, continued
• In the “far field” (Fraunhofer limit) the diffracted field U2 can be computed from the incident field U1 by a phase factor times the Fourier transform of U1
• “Image plane is Fourier transform of pupil plane”
F is Fourier Transform
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Image plane is Fourier transform Image plane is Fourier transform of pupil planeof pupil plane
• Leads to principle of a “spatial filter”
• Say you have a beam with too many intensity fluctuations on small spatial scales
– Small spatial scales = high spatial frequencies
• If you focus the beam through a small pinhole, the high spatial frequencies will be focused at larger distances from the axis, and will be blocked by the pinhole
Page 43 Don Gavel
Depth of focusDepth of focus
D
fλ/ D
f
λλ 2
22
D
f
DfD
f
δ
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Details of diffraction from circular Details of diffraction from circular apertureaperture
1) Amplitude
2) Intensity
First zero at r = 1.22 λ/ D
FWHM λ/ D
Rayleigh resolution
limit:Θ = 1.22 λ/D
2 unresolved point sources
Resolved
Credit: Austin Roorda
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Diffraction pattern from Diffraction pattern from hexagonal Keck telescopehexagonal Keck telescope
Ghez: Keck laser guide star AO
Stars at Galactic Center
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Conclusions:Conclusions:In this lecture, you have In this lecture, you have learned …learned …
• Light behavior is modeled well as a wave phenomena (Huygens, Maxwell)
• Description of diffraction depends on how far you are from the source (Fresnel, Fraunhofer)
• Geometric and diffractive phenomena seen in the lab (Rayleigh range, diffraction limit, depth of focus…)
• Image formation with wave optics
