Gaussian Beam (Ch. 3) |
Oct. 16, 2017 |
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• Envelop equation: TEM
wave$\to $
no field in longitudinal dir (`z`); transverse
field `E_t = A(x,y,z) e^{-jkz}`
Nonparaxial wave
equation `grad_t^2 A + {partial^2 A} / {partial z^2} - 2 jk {partial A} / {partial z}=0`
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• Paraxial wave equation:
Envelop varies slowly along z
+ `grad_t^2 A - 2 jk {partial A} / {partial z}=0`
+ Normalized with `r_N = r /{w_o}`, `Z=z/{2 z_o}`
and
`psi = A / {A_o}`, `1 / {r_N} partial / {partial r_N} (r_N {partial psi} / {partial r_N})+1/{r_N^2} {partial^2 psi}/{partial phi^2} -2 j {partial psi} / {partial Z} =0`
where `w_o` is the initial (minimum) beam waist
(radius), `z_o` is the diffraction or Rayleigh length
and `2 z_o` is the depth of focus.
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• Particular soln: fundamental Gaussian
beam, `TEM_{0,0}` mode
`A = A_o {w_0}/{w(z)} e^{-(r/{w(z)})^2}e^{-j[{kr^2}/{2R(z)} - tan^{-1} (z/{z_o})]`
`w(z) = w_o sqrt{1+(z/{z_o})^2}`;
radius of
curvature `R(z) = z (1+({z_o}/z)^2)`
Rayleigh range or diffraction
length `z_o={pi w_o^2}/lambda`
where `w_o` is the min. spot size (radius) at 1/e relative amplitude of
the field
and `lambda = lambda_o / n`
the wavelength in the medium
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• Important parameters of Gaussian beam:
q parameter: `q=z+jz_o -> 1/q = 1/{R(z)} -j lambda / {pi w^2(z)}`
Divergence angle
(full) `2 theta_o = {2 lambda}/{pi w_o}`
Depth of
focus `2 z_o`
within which beam size expands
by `sqrt{2}` times
of `w_o`
Transverse
phase: `-{kr^2}/{2 R(z)}`,
min `R=2 z_o`, max `R -> oo` (plane wave) when `z=0`
and `z -> oo`
Longitudinal
phase: `tan^{-1} (z/{z_o})`; ranging
from `- pi / 2` to `pi/2` (Gouy effect).
Power: total `P_o = {I_o pi w_o^2}/2`
(indep of `z`)
where `I_o` is the peak intensity.
Fraction power within an
aperture `1 - e^{- 2{r^2}/{w^2 (z)}}`
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•
Consider far field,
i.e. `f \>\> z'_o`
The min. spot at the
focus `w'_o ~~ {lambda f}/{pi w(f)}`
where `w(f)` is the spot size at the lens.
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• Gaussian beam interacts with mirror: Gauss beam
inherits R from the radius of curvature of a mirror.
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• Focused Gaussian beam: at
focus `1/{R(z)} =0`,
i.e. `q = jz_o`; with this condition we can obtain the position and beam
size at focus.
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• Beam
quality `bbbM`: `bbbM^2 = {2w_m 2 theta_m}/{4 lambda / pi} >=1`
measures how close is the beam to Gaussian shape.
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• Exact description of Gaussian beam traversing a thin
lens: A thin lens modifies phase of Gaussian beam
by `e^{j{kr^2}/{2f}}` but not amplitude and beam size.
Relate plane 2 to plane
4 `1/ R-1/f=1/{R'}`.
Beam waist at plane
4 `w'_o = w/{sqrt{1+({pi w^2}/{lambda R'})^2}}`
and `z'={R'}/{1+ ({lambda R'}/{pi w^2})^2}`.
Relate plane 1 to plane 4 by
substitute `w=w_o sqrt{1+(z / {z_o})^2}`
and `R=z(1+({z_o}/z)^2)`.
Result in magnification from ray
optics `M_r = |f/{z-f}|`
while the precise magnification for Gaussian
beam `M={M_r}/{sqrt{1+r^2}}`
where `r={z_o}/{z-f}`.
More
equations `w'_o = Mw_o`, `2z'_o = M^2 2 z_o`, `2 theta'_o = {2 theta_o} /M`
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• Gaussian beam propagating through optical components:
e.g. `n(r) = n_o + f(r)`; `k = {2 pi n(r)}/ lambda = k_o +k_2(r)`;
i.e.
solving `grad_t^2 E + {partial^2 E} / {partial z^2} + k_o n^2 (r) E = 0`
with `E(x,y,z) = A_o psi(x,y,z) e^{-jk_o z}`,
A systematic
approach `->` combine ABCD matrix and Gaussian beam
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• ABCD Law: given a Gaussian beam with `q`
parameter, `q_{\i\n}` and an ABCD matrix for a component
`q_{\out} = {A q_{\i\n} +B} / {C q_{\i\n} + D}`
We can apply the ABCD law for an optical system by cascading
all optical elements and find the equivalent ABCD matrix
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• High order transverse modes:
Fundamental Gaussian mode
envelop `A_G`
Hermite
polynomial `H_m` obeys
`{d^2 H_m(u)}/{du^2} - 2 u {d H_m (u)}/{du} + 2m H_m(u)=0` and can be generated
recursively: `H_{m+1}(u) = 2uH_m(u) - 2 m H_{m-1}(u)` with
`H_0 (u)=1`, `H_1 (u) = 2u`, `H_2 (u) = 4u^2 - 2`
`m` -- `x` mode index; `n` -- `y` mode index
Fundamental Gaussian mode envelop `A_G`
Laguerre associated
polynomial `L_p^l`
obey `nu {d^2 L_p^l}/{d nu^2}+ (l+1-nu) {dL_p^l}/{dnu}+pL_p^l=0`
and can be generated by
`1/{p!} e^{nu} {d^pe^{-v} v^l v^p}/{d nu^p}` with
`L_0^l (x) =1`, `L_1^l(x)=l+1-x`, `L_2^l(x)=1/2(l+1)(l+2)-(l+2)x+1/2x^2`
`p` -- radial mode index; l -- angular mode index
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• : The transverse field profile assumes variation like Bessel function, i.e. `A_m=A_o J_m (k_t r) e^{jmphi}` Bessel beam satisfies the nonparaxial wave equation and is nondiffracting. However, it has infinite energy owing to its long tail, i.e. its RMS width is infinite.
(Ramo, Whinnery and Van Duzer, Fields and Waves in
Communication Electronics, 2nd Ed.
Wiley (1984).)
(Milonni and Eberlay,
Lasers, Wiley (1991).) (Notice that the
paraxial wave equation has opposite sign for j or i since
they assume that forward wave has propagation factor
of
`e^{j k r}`.