Simulation of 2D Maxwell-Bloch Equations - NUSOD · 1> Population Bloch Sphere Tm :YAG3+ Simulation...
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NUSOD 2007 Simulation of 2D Maxwell-Bloch Equations NUSOD – Sep. 24, 2007 Jingyi Xiong, Max Colice, Friso Schlottau, Kelvin Wagner Optoelectronic Computing Systems Center Department of Electrical and Computer Engineering Bengt Fornberg Department of Applied Mathematics University of Colorado at Boulder Material property Simulation schemes and results Experimental results Simulation of 2D Maxwell-Bloch Equations – p.1/20
Transcript of Simulation of 2D Maxwell-Bloch Equations - NUSOD · 1> Population Bloch Sphere Tm :YAG3+ Simulation...
NUSOD 2007
Simulation of 2D Maxwell-Bloch EquationsNUSOD – Sep. 24, 2007
Jingyi Xiong, Max Colice, Friso Schlottau, Kelvin WagnerOptoelectronic Computing Systems Center
Department of Electrical and Computer Engineering
Bengt FornbergDepartment of Applied MathematicsUniversity of Colorado at Boulder
Material propertySimulation schemes and resultsExperimental results
Simulation of 2D Maxwell-Bloch Equations – p.1/20
Inhomogeneously broadened material
x
y
z
Inhomogeneousband
νΙ
νΗ
n k
Grid size: mxnxk
z(mm)
x(µm)
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
−100−80−60−40−20
0
20
40
60
80100
Rare earthdoped crystal
m
Time domain parametersLife time T1 = 10msec
Decoherence time T2 = 5 µsecFrequency domain parameters
Inhomogeneous BW νI = 20GHz
Homogeneous BW νH = 25KHz
Number of bins N = νI/νH ≈ 106
ApplicationsData storage: 1016 bits/cm3
Signal processing: 20+GHz BW
Sun, et. al, J. of Luminescence, Vol.98, p281, 2002Simulation of 2D Maxwell-Bloch Equations – p.2/20
Two-level atoms
Inhomogeneous Bandwidth
(1/T = 20 GHz)
CW Excitation(793.3 nm)
*2
1>
0>
T1
0> Population
1> Population Bloch Sphere
Tm :YAG3+Simulation of 2D Maxwell-Bloch Equations – p.3/20
Bloch equations: Excitation
u: inphase component of Pv: in-quadrature component of Pw: population inversion
E : E-field∆: detuning frequencyT2: decoherence timeT1: population lifetime
Excitation du
dt= −
u
T2
− ∆ · v − κ={E}w
dv
dt= ∆ · u −
v
T2
+ κ<{E}w
dw
dt= κ={E}u − κ<{E}v +
w − 1
T1
Allen and Eberly, Optical resonance and two-level atoms, 1975Simulation of 2D Maxwell-Bloch Equations – p.4/20
Bloch equations: Detuning
u: inphase component of Pv: in-quadrature component of Pw: population inversion
E : E-field∆: detuning frequencyT2: decoherence timeT1: population lifetime
du
dt= −
u
T2
− ∆ · v − κ={E}w
dv
dt= ∆ · u −
v
T2
+ κ<{E}w
dw
dt= κ={E}u − κ<{E}v +
w − 1
T1
Free precession
Simulation of 2D Maxwell-Bloch Equations – p.5/20
Bloch equations: Decoherence
u: inphase component of Pv: in-quadrature component of Pw: population inversion
E : E-field∆: detuning frequencyT2: decoherence timeT1: population lifetime
du
dt= −
u
T2
− ∆ · v − κ={E}w
dv
dt= ∆ · u −
v
T2
+ κ<{E}w
dw
dt= κ={E}u − κ<{E}v +
w − 1
T1
Decoherence
Simulation of 2D Maxwell-Bloch Equations – p.6/20
Bloch equations: Population relaxation
u: inphase component of Pv: in-quadrature component of Pw: population inversion
E : E-field∆: detuning frequencyT2: decoherence timeT1: population lifetime
du
dt= −
u
T2
− ∆ · v − κ={E}w
dv
dt= ∆ · u −
v
T2
+ κ<{E}w
dw
dt= κ={E}u − κ<{E}v +
w − 1
T1
Population relaxation
Simulation of 2D Maxwell-Bloch Equations – p.7/20
Inhomogeneous band
Inhomogeneouslineshape g(∆)
∆
Inhomogeneous averaging
P(t) = α
∫ +∞
−∞
g(∆)[u(t, ∆) + iv(t, ∆)]d∆
Simulation of 2D Maxwell-Bloch Equations – p.8/20
2D Maxwell-Bloch equations
The 2D SEVA wave equation is driven by polarization,∂E(x, z, t)
∂z+
i
2k
∂2E(x, z, t)
∂x2= −i
µ0ω20
2kP(x, z, t)
Bloch equations under RWA driven by the E-field,
∂
∂t
u
v
w
=
− 1
T2−∆ −κ={E}
∆ − 1
T2κ<{E}
κ={E} −κ<{E} 1
T1
u
v
w
+
0
0
− 1
T1
(1)
Polarization given by summation over the inhomogeneous band
P(x, z, t) = α
∫
+∞
−∞
g(∆)[u(x, z, t, ∆) + iv(x, z, t, ∆)]d∆
Simulation of 2D Maxwell-Bloch Equations – p.9/20
Previous numerical solutions
Vector form: FDTD1,2
Slow: ∆z = λ100
∼ λ50
corresponding to < 0.1fsec
Scalar form (wave equation)Cornish3 simulated 1-DChang4 simulated two symmetric angled plane waves butdidn’t give specifics of numerical technique
2D spatial grid: 200µm(x) × 5mm(z) 108 spatial stepsTime span: 2µsec ⇒ 2 · 1010 time stepsSpectral lines: 250 250 lines
Using 2.25GHz computer, it would take about 3 × 104 years.
1. Ziolkowski, et. al, Phys. Rev., Vol 52, p3082, 19952. Schlottau, et. al, Optics Express, Vol. 13, p182, 20053. Cornish, Ph.D. thesis, University of Washington, 20004. Chang, et. al, J. of Luminescence, Vol. 107, p138, 2004
Simulation of 2D Maxwell-Bloch Equations – p.10/20
FFT-BPM IThe 2D SEVA wave equation is driven by polarization,
∂E(x, z, t)
∂z+
i
2k
∂2E(x, z, t)
∂x2= −i
µ0ω20
2kP(x, z, t)
Expand E(x, z, t) and P(x, z, t) in Fourier domain,
E(x, z) =
∫ +∞
−∞
E(kx, z) exp(−ikxx)dkx,
P(x, z) =
∫ +∞
−∞
P(kx, z) exp(−ikxx)dkx
−k2xE(kx, z) − 2ik
dE(kx, z)
dz= −µ0ω
20P(kx, z)
Feit, M. D.et. al, Appl. Opt. Vol. 17 p3990, 1978Simulation of 2D Maxwell-Bloch Equations – p.11/20
FFT-BPM II
SEVA wave equation we want to solve is∂E
∂z=
i
2k(k2
xE − µ0ω20P)
Use trapezoidal rule to evolve wave quation
- Since delayed source terms inconsistant with SS-BPM frame work
For an ODE y′ = f(x, y), the trapezoidal rule is
yn+1 = yn +1
2h [f(xn, yn) + f(xn + h, yn+1)]
Apply trapezoidal rule and solve for En+1,
En+1 =1
1 − i
4kk2
xdz
[(
1 +i
4kk2
xdz
)
En −i
4kdz · 2µ0ω
20Pn
]
Simulation of 2D Maxwell-Bloch Equations – p.12/20
Numerical scheme for Bloch equationsBloch equations under RWA driven by the E-field,
d
dt
u
v
w
=
− 1
T2
−∆ −κ={E}
∆ − 1
T2
κ<{E}
κ={E} −κ<{E} 1
T1
u
v
w
+
0
0
− 1
T1
(2)
Write in matrix-vector form, it becomes y′ = Ay + b
Applying the 4th order Runge-Kutta method,f1 = Ay
l,m,n,k+ b,
f2 = A(yl,m,n,k
+ f1
2) + b
f3 = A(yl,m,n,k
+ f2
2) + b,
f4 = A(yl,m,n,k
+ f3) + bz(mm)
x(µm
)
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
−100
−80
−60
−40
−20
0
20
40
60
80
100
Grid size: mxnxk
k
z n
mx
:y
l+1,m,n,k= y
l,m,n,k+ ∆t(f1 + 2f2 + 2f3 + f4)/6
l: time step, m: x sampling step, n: z step, k: spectral lines
Gross and Manassah, Opt. Lett., Vol.17, p340, 1992Simulation of 2D Maxwell-Bloch Equations – p.13/20
Bloch simulation results
∆
w
2π/τ
τ τ time
3-pulse Echo
> T2
2-pulse Echo
τP P
Start animation
Simulation of 2D Maxwell-Bloch Equations – p.14/20
Advancing the Maxwell-Bloch equations
Initial Value Condition
0 ∆z z
0
∆t
E0,0
E0,1
2∆t
......
l∆t
E0,2
E0,l
......
2∆z ...... n∆z
E1,0 E2,0 En,0......
t
n∆z (n+1)∆zz
t
l∆t
(l+1)∆t
El,n
Pl,n+1
El,n+1 El+1,n
Calculate Maxwell’s equation
Propagate E-field from boundaryinto crystal
Simulation of 2D Maxwell-Bloch Equations – p.15/20
Advancing the Maxwell-Bloch equations
∆t later
0 ∆z z
0
∆t
E0,0
E0,1
2∆t
......
l∆t
E0,2
E0,l
......
2∆z ...... n∆z
P0,1
E1,0 E2,0 En,0
P1,1 P2,1 Pn,1
t
n∆z (n+1)∆zz
t
l∆t
(l+1)∆t
El,n
Pl,n+1
El,n+1 El+1,n
Calculate Bloch equation
E-field drives Bloch vectors to getpolarization
Simulation of 2D Maxwell-Bloch Equations – p.15/20
Advancing the Maxwell-Bloch equations
∆t later
0 ∆z z
0
∆t
E0,0
E0,1
2∆t
......
l∆t
E0,2
E0,l
......
2∆z ...... n∆z
P0,1
E1,0 E2,0 En,0
P1,1 P2,1 Pn,1
E1,1 E2,1
......
...... En,1
t
n∆z (n+1)∆zz
t
l∆t
(l+1)∆t
El,n
Pl,n+1
El,n+1 El+1,n
Calculate Maxwell’s equation
Contribute polarization to propa-gate E-field
Simulation of 2D Maxwell-Bloch Equations – p.15/20
Advancing the Maxwell-Bloch equations
2∆t later
0 ∆z z
0
∆t
E0,0
E0,1
2∆t
......
l∆t
E0,2
E0,l
......
2∆z ...... n∆z
P0,1
E1,0 E2,0 En,0
P1,1 P2,1 Pn,1
E1,1 E2,1
......
...... En,1
P0,2 P1,2 P2,2 Pn,2
t
n∆z (n+1)∆zz
t
l∆t
(l+1)∆t
El,n
Pl,n+1
El,n+1 El+1,n
Calculate Bloch equation
E-field drives Bloch vectors to getpolarization
Simulation of 2D Maxwell-Bloch Equations – p.15/20
Advancing the Maxwell-Bloch equations
2∆t later
0 ∆z z
0
∆t
E0,0
E0,1
2∆t
......
l∆t
E0,1
E0,l
......
2∆z ...... n∆z
P0,1
E1,0 E2,0 En,0
P1,1 P2,1 Pn,1
E1,1 E2,1
......
...... En,1
P0,2 P1,2 P2,2 Pn,2
E1,2 E2,2 ...... En,2
t
n∆z (n+1)∆zz
t
l∆t
(l+1)∆t
El,n
Pl,n+1
El,n+1 El+1,n
Calculate Maxwell’s equation
Contribute polarization to propa-gate E-field
Simulation of 2D Maxwell-Bloch Equations – p.15/20
Advancing the Maxwell-Bloch equations
m∆t later
0 ∆z z
0
∆t
E0,0
E0,1
2∆t
......
l∆t
E0,2
E0,l
......
2∆z ...... n∆z
P0,1
t
E1,0 E2,0 En,0
P1,1 P2,1 Pn,1
E1,1 E2,1
......
...... En,1
P0,2 P1,2 P2,2 Pn,2
E1,2 E2,2 ...... En,2
......
......
......
......
P0,l P1,l P2,l Pn,l
E1,l E2,l ...... En,l
n∆z (n+1)∆zz
t
l∆t
(l+1)∆t
El,n
Pl,n+1
El,n+1 El+1,n
Calculate Maxwell’s equation
Contribute polarization to propa-gate E-field
Simulation of 2D Maxwell-Bloch Equations – p.15/20
Maxwell-Bloch simulation setup200µm×5mm, 2.4µsec, 250 spectral lines, 100MHz of BW
z(mm)
x(µm
)1 2 3 4 5
−100
−50
0
50
100
1o
t-0.8
o
t
1.8µs
x(µm
)300ns
Firstpulse
Pulse 1 comes in at 1◦
Simulation of 2D Maxwell-Bloch Equations – p.16/20
Maxwell-Bloch simulation setup200µm×5mm, 2.4µsec, 250 spectral lines, 100MHz of BW
z(mm)
x(µm
)1 2 3 4 5
−100
−50
0
50
100
1o
t-0.8
o
t
x(µm
)300ns
Secondpulse
∆
w
∆
wPulse 1 and 2 write populationgrating at overlap region
Simulation of 2D Maxwell-Bloch Equations – p.16/20
Maxwell-Bloch simulation setup200µm×5mm, 2.4µsec, 250 spectral lines, 100MHz of BW
z(mm)
x(µm
)1 2 3 4 5
−100
−50
0
50
100
1.8µs300ns
1o
t-0.8
o
t
x(µm
)
Thirdpulse
Expect echo!
Pulse 3 comes in at pulse 1 direction
Population gating diffracts pulse 3 both in space and time
Echo is expected 300ns later in direction of pulse 2Simulation of 2D Maxwell-Bloch Equations – p.16/20
Angled beam photon echo
x(µm)
Sp
ectr
um
(MH
z)
−50 0 50
−20
−10
0
10
20
Volume population grating
Time(µsec)
Ang
le(d
eg)
0.3µs
0.3µs
0.3µs
2−pulse echo 3−pulse echo
0 0.5 1 1.5 2
−4
−2
0
2
4
Time-angle waterfall display
Simulation of 2D Maxwell-Bloch Equations – p.17/20
Chirped angled beam photon echoes
0 2x10−5 4x10−5 6x10−5
Time (sec)
−0.2
0
0.2
0.4
0.6
0.8
1
Inte
nsity
Am
plitu
de (
a.u.
)
Three Pulse Photon Echo − Chirp Correlation, BW = 25 MHz
Intensity
Frequency
Inte
nsity
(a.u
.)
Time (sec)
Chirp photon echo experimental tracesTime domain chirp setupk1
k2
t
t
τ1 τ2
τ1−τ2
τ1
Echo
0.5 1 1.5 2 2.5 30
1
2
3
4
x 107
Time(µsec)
Inte
nsi
ty(a
.u.)
Echo10×
Read Chirp
Echo Efficiency≈2%
CH1CH2
Time(µsec)
An
gle
(deg
)
Echo
0.5 1 1.5 2 2.5 3
−4
−2
0
2
4
Chirp 1
Chirp 2
Chirp 3
Time domain chirp result
Time-angle waterfall display
0 0.5 1 1.5 2 2.50.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Angle(deg)
Ech
o In
tens
ity (
a.u.
)
w0=60µm, αL=1.0
delay= 0.1µsdelay= 0.112µsdelay= 0.16µsdelay=0.2µs
Interaction angle vs. echo efficiency
Simulation of 2D Maxwell-Bloch Equations – p.18/20
Experimental results
0 1 2 3 4 5 6 7−50
0
50
100
150
200
250
300
350
400
450
angle(deg)
Inte
grat
ed e
cho
stre
ngth 20µsec
40µsec
60µsec
80µsec
100µsec120µsec
140µsec
Increasing time delays
0 1 2 3 4 5 6−50
0
50
100
150
200
250
300
350
400
450
500
angle(deg)
Inte
grat
ed e
cho
stre
ngth
20µsec
40µsec
60µsec
80µsec
100µsec
120µsec
140µsec
Increasing time delays
0 0.5 1 1.5 2 2.50.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Angle(deg)
Ech
o In
tens
ity (
a.u.
)
w0=60µm, αL=1.0
delay= 0.1µsdelay= 0.112µsdelay= 0.16µsdelay=0.2µs
Experimental Setup
Simulation ResultsExperimental ResultsSpot size: 40µm x 80µm
CH1
CH2
CH3
Arb ChannelsλGateAOM
APD
AOM
AOM
CH1
CH2 CH3
Cryo
SSHTo Scope
f
Simulation of 2D Maxwell-Bloch Equations – p.19/20
Conclusion
Spatial-spectral Maxwell-Bloch equations solved usingFFT-trapezoidal rule and RK-4
We are now working towardsIncrease the number of transverse samples for larger anglesSimulate many GHz of bandwidthSimulate realistic T2 (T1 effect for accumulation)
z(mm)
x(µm
)
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
−100
−80
−60
−40
−20
0
20
40
60
80
100
Grid size: 128x128x250
250
5mm128
128
200µ
m
:
:
100MH
z:
AcknowledgementThe authors would like to acknowledge the support from S. Papport underthe DARPA AOSP program. B. Fornberg acknowledges support by NSF DMS-0309803. We wish to thank Tiejun Chang for valuable discussions.
Simulation of 2D Maxwell-Bloch Equations – p.20/20
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