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Chapter 2 The

2.1 The Gaussian Beam

2.2 The TEM00 mode 2.3 Beam quality: M2 factor 2.4 Transmission of Gaussian Beam through thin

1 2.1-The Gaussian Beam

! The Gaussian Beam is an important solution of the Helmholtz (Maxwell) paraxial wave equation(s). ! The Gaussian Beam solutions are the modes of the spherical optical resonator (See III. Optical resonator). ! The of a beam is essentially that of the Gaussian beam

2 2.1-The Gaussian Beam ! ! U(r ) = A(r )e−ikz ! A(r ) variation with position is very small over a distance of one λ. It is still approximately planar. ! The in the Paraxial Approximation becomes: € ∂A ∇2 A −i2k = 0 T ∂z ∂ 2 ∂ 2 ∇2 = + ≡ Transverse Laplace operator T ∂x 2 ∂y 2 3

€ 2.1-The Gaussian Beam ! The Gaussian beam solution of Maxwell’s equations for the electric vector E is given by: ! ! x ! E ( r ) = E (−ˆx + ˆz )U( r ) 0 z iz + 0 ! xˆ and zˆ are units vectors in the 0x and 0z directions respectively and U(r) is the €complex amplitude of the scalar Gaussian beam.

€ € 3 2.1-The Gaussian Beam •U(r) is written in the form [magnitude X exp(- iphase)] ! ! U( r ) = A( r )e−ikz $ 2 ' $ 2 ' ! W0 ρ ρ A( r ) = A0 exp&− 2 ) exp&− ik +iζ (z )) W( z ) % W ( z )( % 2R( z ) ( The beam parameters are: W( z ): Beam width = radius (!) € W0 : Beam waist R( z ): Radius of curvature of wavefronts ζ ( z ): Phase factor 4

€ 2.1-The Gaussian Beam r, ρ(radial position) ! Wavefronts of scalar Gaussian z = 0 z Beam (cylindrical coordinates)

! Electric field lines in (x-z) plane

5 2.1-The Gaussian Beam

z = 0 z = z0 z = 2z0

! ! 2 I( r ) = U(r ) • At z = z0, the on-axis ! is halved I( r ) = I( ρ,z ) • z0 is called the Rayleigh # & 2 2 W0 2ρ range = I 0% ( exp(− 2 ) 6 $ W( z )' W ( z )

€ 2.1-The Gaussian Beam

1 2 ( " z % + 2 W( z ) = W0*1 +$ ' - z )* # 0 & ,-

1 2 $ λz0 ' •Measurement of the beam W0 = & ) : Beam Waist € % π ( waist provides z0. z : the Rayleigh range 0 •e.g: 266 nm (quadrupled Nd-

W YAG laser), W0 = 2.5 mm: W( z ) ≈ 0 z =θ z z 0 0 ! z = 75 m (depth of focus) € 0 λ θ 0 = : !θ0 = 0.01 mrad πW0 7

€ 2.1-The Gaussian Beam ! Wavefronts and their radius of curvature ( " z % 2* R( z ) = z(1 +$ 0 ' * ) # z & +

•Wavefronts Radius of curvature is curvature of minimum at z . spherical € 0 wavefronts •Same as spherical wavefronts at large z R( z ) ≈ z 8

€ 2.1-The Gaussian Beam ! The phase of the Gaussian Beam:

kρ 2 ϕ(R,z) = kz −ζ (z) + 2R(z) ϕ(0,z) = kz −ζ (z): on -axis & z ) ζ (z) = arctan( + : phase retardation ' z 0 * with respect to plane wave 9

€ 2.1-The Gaussian Beam ! Most general solutions are Hermite- Gaussian functions (higher order Gaussian beams). Intensity distribution of Hermite-Gaussian modes: " % 2 ( + ( + 2 W0 2 2x 2 2y I l ,m(x,y,z) = Al ,m $ ' Gl * -G m* - # W(z)& ) W(z), ) W(z),

TEMlm modes: Gl , Gm Hermite -Gaussian function of order l, m

Al ,m = constant (l, m )

TEM 00 = Gaussian Beam

€

TEM00 TEM01 TEM02 TEM11 TEM22 10 2.1-The Gaussian Beam Hermite Gaussian Functions - Cartesian Coordinates https://www.rp-photonics.com/hermite_gaussian_modes.html

https://en.wikipedia.org/wiki/Hermite_polynomials

12 https://www.rp-photonics.com/hermite_gaussian_modes.html 2.2 - TEM00 mode ! Normal or Gaussian (non-normalised)

function: x2 − G( x) = e 2σ 2 σ rms (root mean square) width ! Generally:

2 €g( x) = e−α x G(0) = g(0) = 1 limG(x) → 0 x→±∞ ! If G(x) represents light intensity, how € does one define the “edge” of the beam?

€ 11 2.2 - TEM00 mode ! Need to truncate G(x) at certain

values of x (xt measured from Imax)

! 2xt defines the corresponding width

Imax × attenuation Truncation @ 1/√2 = 0.707×1 0.83σ (3 dB) 1/√e = 0.606×1 σwidth) (rms width) 1/2 = 0.5×1 1.18σ (FHWM) 1/e = 0.368×1 1.414σ (1/e) 1/ e2 = 0.13×1 2width)σ (1/e2 width) (see graph on next page) 12 2.2 - TEM00 mode

! The intensity can be written as a function of P: total optical power carried by the beam. P is measured directly with a power meter. ∞ 1 2 P = ∫ I(ρ,z)2πρdρ = I 0(πW0 ) 0 2 2P 2ρ 2 I(ρ,z) = exp(− ) πW 2( z ) W 2( z ) ! The beam radius is taken as the (1/e2) width: spot € size 13 ! For a given pulse duration, it is convenient to use fluence 2.2 - TEM00 mode (Jcm-2) -instead of intensity -as a number of laser Nd:YAG laser, 6 ns, 120 mJ, 266 nm processes are characterised by their fluence (eg. threshold): * $ 2 '- 2ET r Fz (r) = 2 exp,− 2& 2 )/ πWz +, % Wz (./

ET = Total energy in laser beam 1 W = spot size on target (at z) z e2

ET 2 ≡ Average fluence π Wz 14

€ 2.3 Beam Quality: M2 factor ! Gaussian beam is an idealisation ! Deviation of optical beam (waist diameter

2Wm, divergence 2θm) from Gaussian form

(W0, θ0) measures optical quality: quantitative measure is M2 -factor: 2W 2θ 2W 2θ M2 = m m = m m 2W0 2θ0 4λ / π θ M2 = m If the two beams have the same beam waist θ0 M 2 ≥ 1 M 2 ≤ 1.1 (single mode HeNe laser), M 2 ≥ 3,4 (high power multimode) 15

€ 2.4 Transmission of Gaussian beams through thin lenses

! Gaussian beam remains Gaussian if paraxial nature of the wave is maintained. ! Beam is reshaped: waist and curvature are altered. ! Beam shaping, beam focusing can be achieved (optical design)

16 2.4 Transmission of Gaussian beams through thin lenses

! Complex amplitude multiplied by phase factor as it passes through ! Wavefront is altered: new curvature, new phase (beam width W = W’). 17 2.4 Transmission of Gaussian beams through thin lenses ! Parameters of the emerging beam: 1 1 1 " = − Waist radius : W0 = MW0 R " R f Waist location : (z" - f) = M 2(z- f) " W W0 = (Waist radius) 2 2 2 Depth of focus : 2z " = M 2z 1+(πW /λR ") 0 0 2θ R " Divergence angle : 2θ " = 0 -z " = (Beam centre location) 0 2 2 M 1+(λR " /πW ) M Magnification : M = r 2 f = focal length of thin lens 1+ r z 0 f € r = and M r = z − f z − f 18

€ 2.4 Transmission of Gaussian beams through thin lenses

! Beam shaping: use lens or series of lenses to reshape the Gaussian beam ! Lens at beam waist: do z = 0 in W previous equations W " = (Waist radius) 0 2 1+(z 0 / f ) R " z" = 2 (Beam centre location) 1+( f /z 0) 19

€ 2.4 Transmission of Gaussian beams through thin lenses ! Depth of focus z0 >> f ! Small spot size is really important in laser scanning, , CD burning,... ! Need short f, thick beam and short € f ! λ ! W0 ≈ W0 = f = θ0 f (Waist radius) If D (diameter of lens) z0 πW0 = 2W0 z! ≈ f (Beam centre location) Focused spot size: 4 f 4 2W ! ≈ λ = λF # 0 π D π 20 F # → F-number of lens