Gaussian Beam (Ch. 3)

Oct. 16, 2017

• Envelop equation: TEM wave no field in longitudinal dir (z); transverse fieldEt=A(x,y,z)ejkz
Nonparaxial wave equationt2A+2A/z22jkA/z=0
• Paraxial wave equation:
Envelop varies slowly along z
+t2Aj2kA/z=0
+ Normalized withrN=r/wo ,Z=z/(2zo) andψ=A/Ao ,1rNrN(rNψrN)+1rN22ψφ22jψZ=0
wherewo is the initial (minimum) beam waist (radius),zo is the diffraction or Rayleigh length and2zo is the depth of focus.
• Particular soln: fundamental Gaussian beamTEM0,0 modeA=Aoer2/w(z)2ej[kr2/2R(z)tan1(z/zo)]wo/w(z)
spot size (radius)w(z)=wo1+(z/zo)2 ;
radius of curvatureR(z)=z(1+(zo/z)2)
Rayleigh range or diffraction lengthzo=πwo2/λ
wherewo is the min. spot size (radius) at 1/e relative amplitude of the field andλ=λo/n the wavelength in the medium
• Important parameters of Gaussian beam:
q parameter:q=z+jzo1/q=1/R(z)jλ/(πw2(z))
Divergence angle (full)2θo=2λ/(πwo)
Depth of focus2zo within which beam size expands by2 times ofwo
Transverse phase:kr2/2R(z) , minR=2zo , maxR (plane wave) when z=0 andz>
Longitudinal phase:tan1(z/zo) ; ranging fromπ/2 toπ/2 (Gouy effect).
Power: totalPo=Ioπwo2/2 (indep of z) whereIo is the peak intensity.
Fraction power within an aperture1e-2r2/w2
Focusing Gaussian beam (approx. equation) by thin lens:
Consider far field, i.e.fzo
The min. spot at the focuswoλf/(πw(f)) wherew(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 focus1/R(z)=0 , i.e.q=jzo ; with this condition we can obtain the position and beam size at focus.
• Beam qualityMM :MM2=2wm2θm4λ/π1 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 byejkr2/(2f) but not amplitude and beam size.
Relate plane 2 to plane 4R1f1=R1 Beam waist at plane 4w0=w1+(πw2/(λR))2 andz=R1+(λR/(πw2)2
Relate plane 1 to plane 4 by substitutew=wo1+(z/zo)2 andR=z(1+(zo/z)2)
Result in magnification from ray opticsMr=|f/(zf)| while the precise magnification for Gaussian beamM=Mr1+r2 wherer=zo/(zf)
More equationswo=Mwo ,2zo=M22zo ,2θo=2θo/M
• Gaussian beam propagating through optical components: e.g.n(r)=no+f(r) ;k=2πn(r)/λ=ko+k2(r) ;
i.e. solvingt2E+2E/z2+kon2(r)E=0 withE(x,y,z)=Aoψ(x,y,z)ejk0z ,
A systematic approach combine ABCD matrix and Gaussian beam
• ABCD Law: given a Gaussian beam with q parameter,qin and an ABCD matrix for a componentqout=(Aqin+B)/(Cqin+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)

Am,p=AGHm(2x/w)Hp(2y/w)ej(p+m)tan1(z/zo)

Fundamental Gaussian mode envelopAG
Hermite polynomialHm obeysd2Hm(u)du22udHm(u)du+2mHm(u)=0 and can be generated recursively.Hm+1(u)=2uHm(u)2mHm1(u) withH0(u)=1 ,H1(u)=2u ,H2(u)=4u22
m -- x mode index; p -- y mode index

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

Ap,l=AGLpl(2r2/w2)(2r/w)le±jlφej(2p+l)tan1(z/zo)

Fundamental Gaussian mode envelopAG
Laguerre associated polynomialLpl obeyνd2Lpldν2+(l+1ν)dLpldν+pLpl=0 and can be generated by1p!eνdpevν1νpdνp withL0l(x)=1 ,L1l=l+1x ,L2l=0.5(l+1)(l+2)(l+2)x+0.5x2
p -- radial mode index; l -- angular mode index
Bessel beam: The transverse field profile assumes variation like Bessel function, i.e.Am=AoJm(ktr)ejmφ 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 amath e^{j k r}. endmath


Last Modified: Oct 16, 2017
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