Temporal Analog of Reflection and Refraction
at a Temporal Boundary
z (m)
80
0
0
–10
–10
60
–20
40
–30
20
0
–10
–5
0
5
10
–40
T (ps)
B. W. Plansinis, G. P. Agrawal,
and W. R. Donaldson
University of Rochester
Laboratory for Laser Energetics
Intensity (dB)
100
Spectral evolution
–20
–30
–6 –4 –2
0
2
Spectral density (dB)
Temporal evolution
–40
o – o0 (THz)
Frontiers in Optics 2015
San Jose, CA
18–22 October 2015
1
Summary
A temporal analog of reflection and refraction occurs
when a pulse encounters a moving temporal boundary
• A moving temporal boundary breaks translational symmetry in time
and space, allowing both photon momentum and energy to change
• We performed numerical simulations to demonstrate that such
a moving boundary produces temporal reflection and refraction
• We derive temporal analogs for Snell’s laws of reflection and refraction
based on the conservation of photon momentum in the moving frame
• Using these equations, a temporal analog of total internal reflection
was found and demonstrated using numerical simulations
• We are developing an experiment to verify these results using
a traveling-wave phase modulator to produce the moving refractive
index boundary
E24516
2
From a physics perspective, a refractive index boundary
breaks translational symmetry
• A spatial boundary breaks translational symmetry in space,
which requires the photon energy (~) to be conserved while
its momentum (i) changes
• A temporal boundary breaks translational symmetry in time,
which requires the photon momentum (b) to be conserved
while its energy (~) changes
• From this comparison, we see that the frequency of the optical
pulse is the temporal analog of the angle in space
• If we allow for a moving temporal boundary, we expect both
energy and momentum to change
E24517
3
A moving temporal boundary is a refractive index
change that propagates through a dispersive medium
mitt
ed
n + Dn
nt
n
Inc
ide
ct
ed
Re
fle
n + Dn
T = t – TB – zYVB
z
z
In
ns
a
r
T
ted
ci
ted
t
i
m
ec
t
n
de
fl
Re
n
Moving frame
Tran
s
Laboratory frame
0 TB
–TB
Lab time (t)
0
Time (T)
E24518
4
A change in refractive index in time shifts the dispersion
relation of the material
• An index change of Dn will shift the dispersion relation by bB = k0 Dn
• We can Taylor expand the dispersion relation in the moving frame
to give
b
b ^~h = b 0 + Db 1 ^~ – ~ 0h + 2 ^~ – ~ 0h2 + b B H ^Th
2
• The final term vanishes for T < 0 and has value bB for T > 0
• Using this dispersion relation together with Maxwell’s equations
leads to the time-domain equation for the pulse envelope A ^z, Th
b 22 A
2A
2A
+ Db 1
+i 2
= ib B H ^Th A
2z
2 2T 2
2T
E24519
5
The temporal evolution is strikingly similar to an optical
beam hitting a spatial boundary
z (m)
80
0
0
–10
–10
60
–20
40
–30
20
0
–10
–5
0
5
10
–40
T (ps)
b B = 0.5 m –1
Db 1 = 0.1 ps/m
Intensity (dB)
100
Spectral evolution
–20
–30
–6 –4 –2
0
2
Spectral density (dB)
Temporal evolution
–40
o – o0 (THz)
b 2 = 0.005 ps 2 /m
Dn = 8 # 10 –8
E24520
6
T<0
(2)
(1)
0.0
–0.5
(3)
–1.0
–6
–4 –2
0
o – o0 (THz)
2
–10
60
–20
40
–30
20
–5
0
T (ps)
5
10
–40
0
Intensity (dB)
0
80
z (m)
0.5
T>0
100
0
–10
1.0
–10
–20
–30
–6
–4 –2
0
o – o0 (THz)
2
–40
Spectral density (dB)
bl – b0 (m–1)
The dispersion relation in the moving frame can be used
to explain the observed frequency shifts
E24521
7
• Setting b = b 0 we find the reflected
and transmitted frequencies
Db 1
~r = ~0 – 2
,
b2
Db 1
2b B b 2
~t = ~0 +
–1 + 1 –
>
2 H
b2
D
b
^ 1h
bl – b0(m–1)
Solving for point (1) and point (3), we can find equations
analogous to Snell’s laws
1.0
0.5
(2)
(1)
0.0
–0.5
(3)
–1.0
–6
–4 –2
0
o – o0 (THz)
2
• By shifting the reference frequency to ~ c = ~ 0 – _Db 1 b 2i
and using the notation D~ = ^~ – ~ ch
we find the temporal analogs to Snell’s laws
D~ r = –D~ 0
D~ t = D~ 0
2b B b 2
1–
2
^Db 1h
E24522
8
When bB is large, the transmitted frequency
can no longer propagate
1–
100
z (m)
80
Spectral evolution
0
0
–10
–10
60
–20
40
–30
20
0
–10
–5
0
5
10
–40
T (ps)
b B = 1.5 m –1
Db 1 = 0.1 ps/m
–20
–30
–6 –4 –2
0
2
Spectral density (dB)
Temporal evolution
2b B b 2
2 becomes complex
^Db 1h
Intensity (dB)
D~ t = D~ 0
–40
o – o0 (THz)
b 2 = 0.005 ps 2 /m
Dn = 2.4 # 10 –7
E24523
9
Looking closer at the reflecting pulse, a temporal
analog of the evanescent wave can be seen
0
z (m)
80
–10
60
–20
40
–30
20
0
2
3
4
5
6
Intensity (dB)
100
–40
T (ps)
E24524
10
We have imposed momentum conservation, even
though a moving boundary can change both photon
momentum and energy
• Momentum was only conserved in the moving frame
where the dispersion relation is given by
b
b ^~h = b 0 + Db 1 ^~ – ~ 0h + 2 ^~ – ~ 0h2
2
where Db 1 = b 1 – 1 V B is a measure of the difference in speed
• In the lab frame, the dispersion relation is given by
bl ^~h = b 0 + b 1 ^~ – ~ 0h +
b2
~ – ~ 0h2
^
2
• Comparing these equations we find that bl = b +
~ – ~0
VB
• Even though b is constant, bl changes because the frequency changes
E24525
11
Summary/Conclusions
A temporal analog of reflection and refraction occurs
when a pulse encounters a moving temporal boundary
• A moving temporal boundary breaks translational symmetry in time
and space, allowing both photon momentum and energy to change
• We performed numerical simulations to demonstrate that such
a moving boundary produces temporal reflection and refraction
• We derive temporal analogs for Snell’s laws of reflection and refraction
based on the conservation of photon momentum in the moving frame
• Using these equations, a temporal analog of total internal reflection
was found and demonstrated using numerical simulations
• We are developing an experiment to verify these results using
a traveling-wave phase modulator to produce the moving refractive
index boundary
E24516
12