introduction to diffraction - cornell universityib38/teaching/p330/... · 2018-11-16 · 7...
TRANSCRIPT
Introduction to Diffraction P3330 Exp Optics FA’20161
Introduction to Diffraction
Ivan BazarovCornell Physics Department / CLASSE
Outline• Gaussian beams in optical resonators (contd.)• He-Ne laser spectrum (contd.)• Introduction to diffraction• Fresnel vs. Fraunhofer diffraction
Introduction to Diffraction P3330 Exp Optics FA’20162
Two-mirror resonator
� =
A Gaussian beam must match wavefront curvature radii to those of the mirrors
Introduction to Diffraction P3330 Exp Optics FA’20163
Gaussian mode in 2-mirror cavity
R(z) = z +�⇤2
z
Recall Gaussian mode curvature:
z1-R1
z2R2
d
Introduction to Diffraction P3330 Exp Optics FA’20164
Solution: Rayleigh range & waist position in the cavity
Recall that the rms waist is given by:
Introduction to Diffraction P3330 Exp Optics FA’20165
Actual wave equation to solve
Or equivalently Helmholtz equation:
This is usually solved using the paraxial approximation
k = !/c
r =p
x
2 + y
2 + z
2 ⇡ z +x
2 + y
2
2z
Fouriertransform
Introduction to Diffraction P3330 Exp Optics FA’20166
Infinite number of solutions!
There are infinite solutions to the Helmholtz equation in free space.
E.g. Hermite-Gaussian TEMnm modes (E-field, not intensity!)
Introduction to Diffraction P3330 Exp Optics FA’20167
He-Ne laser energy levels
95% of laser power is in the TEM00 (Gaussian mode)
Width of the resonance (medium gain) ~1.5 GHz, i.e. only 2 axial modes are typically present
Can use Michelson’s interferometer to “see” the individual axial modes
10:1 He-Ne mixture
Introduction to Diffraction P3330 Exp Optics FA’20168
Diffraction
Geometrical optics…
I…light can’t turn a corner.
Introduction to Diffraction P3330 Exp Optics FA’20169
Diffraction
Physical optics…
…actually, it can.I
Francesco Maria Grimaldi(1618 - 1663)
Introduction to Diffraction P3330 Exp Optics FA’201610
Hyugens-Fresnel principle
every point on a wavefront may be regarded as a secondary source of spherical wavelets
The propagated wave follows the periphery of the wavelets.
Huygens, just add the wavelets considering interference!
Introduction to Diffraction P3330 Exp Optics FA’201611
Hyugens-Fresnel principle
If one perturbs a plane wavefront, the Huygens wavelets will no longerconstructively interfere at all points in space. Adding the wavelets byphysical optics explains why light can turn corners and create fringesaround images of objects.
Introduction to Diffraction P3330 Exp Optics FA’201612
Diffraction: a generic wave phenomenon
Key-holeIncoherent Coherent illumination Sea waves
Introduction to Diffraction P3330 Exp Optics FA’201613
Diffraction theory
We know this
Wavefront E
What is E here?
Note: the wavefront is considered to be perfectly coherent (i.e. fixed relationship between E phases for any 2 points)
Introduction to Diffraction P3330 Exp Optics FA’201614
Obliquity factor
- wavelets propagate isotropically—in forward and reversedirections
- to use the Huygens approach, modify amplitude of wavefrontas a function of q :
Introduction to Diffraction P3330 Exp Optics FA’201615
Calculating the diffracted wave amplitude
)( trkisO e
rEE w-¢
¢=
( ) dArr
eFeikEErrik
tiSP ¢
-=
¢+- òò
)(
2q
pw
2/piei -=-2cos1)( qq +
=Fobliquity factorphase shift
spherical waves:
Fresnel-Kirchhoff diffraction integral:
great for evaluation by a computer
Introduction to Diffraction P3330 Exp Optics FA’201616
Fresnel vs. Fraunhofer
Contemporaries, but not collaborators (nor competitors).
Introduction to Diffraction P3330 Exp Optics FA’201617
Fresnel vs. Fraunhofer diffraction
• 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
Introduction to Diffraction P3330 Exp Optics FA’201618
Aperture size matters
the amount of wavefront curvature decides whether to use Fresnel or the simplified Fraunhofer diffraction
Fresnel-like(or “near field”)
Fraunhofer-like(or “far field”)
Introduction to Diffraction P3330 Exp Optics FA’201619
Fresnel number definition
Δ ≅a2
2L
L
a
How many half-wavelength are contained in the curvature of the wavefront?
N =λ / 2
=a2
λLD
Fresnel number
screen-aperture distance
aperture size (radius)
Introduction to Diffraction P3330 Exp Optics FA’201620
Regions of validity for diffraction calculations
Math simplifies: Ep is simply a 2D FT of the Es at the aperture!
Introduction to Diffraction P3330 Exp Optics FA’201621
An example
mask
Credit: Bill Molander, LLNL
Introduction to Diffraction P3330 Exp Optics FA’201622
Links/references
http://www.optique-ingenieur.org/en/courses/OPI_ang_M01_C03/co/Grain_OPI_ang_M01_C03.html
http://edu.tnw.utwente.nl/inlopt/overhead_sheets/Herek2010/week7/13.Fresnel%20diffraction.ppt
http://www.ucolick.org/~max/289/Lectures%20Final%20Version/Lecture%203%20Physical%20Optics/Lecture3%20Physical%20Optics_v2.ppt