Sample Problems > Applications > Lasers > self

{ SELF_FOCUS.PDE

This problem considers the self-focussing of a laser beam of Gaussian profile.

-- Submitted by John Trenholme, LLNL

}

title "2D GAUSSIAN BEAM PROFILE"

select

elevationgrid = 300

variables

realf (threshold=0.1) { real (in-phase) part of field envelope }

imagf (threshold=0.1) { imaginary (quadrature) part of field envelope }

definitions

radX = 2 { X "radius" of beam }

radY = 2 { Y "radius" of beam }

bMax = 2.25 { maximum B integral (= Time)}

zm = 5 { zoom-in factor for plots }

xHi = 7.17 * SQRT( radX * radY) { size of calculation domain... }

yHi = 7.17 * SQRT( radX * radY) { set for field = 0.001 at edge }

x45 = xHi * 0.7071 { point on boundary at 45 degrees }

y45 = yHi * 0.7071

tn =1e-30 { tiny number to force zero on plot scales }

power = PI * radX * radY * 2.73 ^ 2 / 8 { analytic integral }

inten = realf * realf + imagf * imagf { definition for later use }

time { "time" is really B integral }

0 to bMax by 0.03 * bMax

initial values

realf = EXP( - ( x / ( radX * 2.73)) ^ 2 - ( y / ( radY * 2.73)) ^ 2)

imagf = 0

equations { normalized, low-secular-phase nonlinear propagation }

realf: DEL2( imagf) + imagf * ( inten - 1) = -DT( realf)

imagf: DEL2( realf) + realf * ( inten - 1) = DT( imagf)

boundaries

region 1

start ( 0, 0) { bump is at center; only do one quadrant }

natural( realf) = 0 { set slope to zero on boundary }

natural( imagf) = 0 { if boundary value too big, move boundary out }

line to ( xHi, 0)

arc ( center = 0, 0) to ( 0, yHi)

line to ( 0, 0)

to close

monitors

for cycle = 1 { do this every cycle }

elevation( inten) from ( 0, 0) to ( xHi, 0) as "INTENSITY"

range( 0, tn)

contour( inten) as "INTENSITY" zoom ( 0, 0, xHi / zm, yHi / zm)

plots

for t = starttime { at the beginning only }

grid( x, y)

surface( inten) as "INTENSITY" range( 0, tn) viewpoint( 1000, 200, 40)

elevation( LOG10( inten)) from ( 0, 0) to ( xHi, 0) as "LOG10 INTENSITY"

for t = endtime { at the end only }

grid( x, y)

grid( x, y) zoom ( 0, 0, xHi / zm, yHi / zm)

surface( inten) as "INTENSITY" range( 0, tn) viewpoint( 1000, 200, 40)

zoom ( 0, 0, xHi / zm, yHi / zm)

elevation( LOG10( inten)) from ( 0, 0) to ( xHi, 0) as "LOG10 INTENSITY"

for t = starttime by ( endtime - starttime) / 5 to endtime { snapshots }

elevation( ARCTAN( imagf / realf) * 180 / PI) from ( 0, 0)

to ( xHi / zm, 0) as "PHASE (DEGREES)"

elevation( inten) from ( 0, 0) to ( xHi / zm, 0) as "INTENSITY"

range( 0, tn)

histories

history( inten) at ( 0, 0) ( 0.01 * xHi, 0) ( 0.03 * xHi, 0) ( 0.1 * xHi, 0)

( 0.3 * xHi, 0) ( xHi, 0) ( x45, y45) as "INTENSITY" export

history( realf) at ( 0, 0) ( 0.01 * xHi, 0) ( 0.03 * xHi, 0) ( 0.1 * xHi, 0)

( 0.3 * xHi, 0) as "IN-PHASE FIELD"

history( imagf) at ( 0, 0) ( 0.01 * xHi, 0) ( 0.03 * xHi, 0) ( 0.1 * xHi, 0)

( 0.3 * xHi, 0) as "QUADRATURE FIELD"

history( ARCTAN( imagf / realf) * 180 / PI) at ( 0, 0) ( 0.01 * xHi, 0)

( 0.03 * xHi, 0) ( 0.1 * xHi, 0) ( 0.3 * xHi, 0) as "PHASE (DEGREES)"

history( MIN( ( ABS( inten - 0.33)) ^ ( -0.75), 1)) at ( 0, 0)

range ( 0.045, 1) as "( INTENSITY - 0.33) ^ -0.75" { goes linearly to 0}

history( ABS( INTEGRAL( inten) / power - 1)) as "POWER CHANGE (EXACT = 0)"

end

self_focus

self_focus