Gaussian Beam (Ch. 3) Oct. 16, 2017



• 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



• 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.



• 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})]
spot size (radius) 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



• 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)}}



Focusing Gaussian beam (approx. equation) by thin lens:
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.



• Gaussian beam interacts with mirror: Gauss beam inherits R from the radius of curvature of a mirror.



• 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.



• Beam quality bbbM: bbbM^2 = {2w_m 2 theta_m}/{4 lambda / pi} >=1 measures how close is the beam to Gaussian shape.



• 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



• 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



• 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



• High order transverse modes:
Rectangular (Hermite-Gaussian)

A_{m,n}=A_G H_m ( {sqrt{2} x}/w)H_n({sqrt{2} y}/w)e^{j(n+m)tan^{-1}(z/{z_o})

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

3D surface plot of Hermite Gaussian modes.
Cylindrical symmetric (Laguerre-Gaussian)

A_{p,l}=A_G L_p^l ({2r^2}/{w^2}) ({sqrt{2}r}/w)^l e^{+- jlphi} e^{j(2p+l) tan^{-1}(z/{z_o})

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



Bessel beam: 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.

Solving Helmholtz equations with separation of variables. (Ramo, Whinnery and Van Duzer, Fields and Waves in Communication Electronics, 2nd Ed. Wiley (1984).)
Obtaining HG mode solutions by solving the paraxial wave equation with separation of variables. (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}.