demonstration of negative group delays in a simple electronic circuit
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Demonstration of Negative GroupDelays in a Simple Electronic Circuit
Masao Kitanokitano@kuee.kyoto-u.ac.jp, Room #1213
Kyoto University
Workshop on Quantum Optics, UCSB – p.1/34
Group Delays in Circuits
Group delays in lumped systems (L = 0 or L c/ω)
Mitchell and Chiao: Am. J. Phys. 66, 14 (1998)Bandpass Amplifier (LC + opamp)Arbitrary Waveform Generator
Nakanishi, Sugiyama, and MK: quant-ph/0201001(to appear in Am. J. Phys.)
Highpass Amplifier (RC + opamp)Baseband pulse (No carriers)Band limited signal from rectangular pulser +lowpass filter
Workshop on Quantum Optics, UCSB – p.2/34
Negative delay circuit
The output LED is lit earlier than the input LED.— Negative delay
Time constants could be order of seconds.— We can see it!
Workshop on Quantum Optics, UCSB – p.3/34
Experiment
button
LED
pulse
pulse generatorLED
input output
negative delay circuit
1. rectangular pulse generator + low-pass filter
2. negative delay circuit
Workshop on Quantum Optics, UCSB – p.4/34
Group delays – ideal case
Group delay for base-band signals (delay time: td)
VOUT(t) = (h∗VIN)(t) = VIN(t − td)
h(t) = δ (t − td)
Fourier Transformed: VOUT(ω) = H(ω)VIN(ω)
H(ω) = (Fh)(ω) =∫
dt h(t)e−iωt = exp(−iωtd)
time
Vin(t) Vout(t)H(ω)td
Workshop on Quantum Optics, UCSB – p.5/34
Positive and Negative delays
td Causality Physical realization> 0 causal distributed system= 0 (locally, mutually) causal lumped system< 0 non causal impossible
Positive delays are easy, if you have an appropriatespace.
“Record and play” is also possible.
No way to make ideal (unconditional) negative delays.
No lumped systems (L = 0) can produce ideal positiveor negative delays.
Workshop on Quantum Optics, UCSB – p.6/34
Approximate delay with lumped systems
Ideal response function H(ω)
A(ω) = |H(ω)| = 1,
φ(ω) = argH(ω) = −tdω
Approximate realization #1 with lumped systems
H(ω) =1+ iωT1− iωT
(1 pole, 1 zero)
A(ω) = 1 (flat response)
φ(ω) = 2tan−1 ωT ∼ 2T ω
Stability condition → T ≤ 0 (only positive delays)
Workshop on Quantum Optics, UCSB – p.7/34
Approximate delay (2)
Approximate realization #2 with lumped systems
H(ω) = 1+ iωT (1 zero)
A(ω) =√
1+(ωT )2 ∼ 1+(ωT )2
2(→ distortion)
φ(ω) = tan−1 ωT ∼ T ω
No sign restriction on T (positive and negative delay)
Workshop on Quantum Optics, UCSB – p.8/34
Asymmetry — positive / negative
Transfer function positive delay T < 0 negative delay T > 0
H(ω) = 1+ iωT
0
zero
Re ω
Im ω
0
zero
Re ω
Im ω
H(ω) =1+ iωT1− iωT
0
pole
zero
Re ω
Im ω
0
pole
zero
Re ω
Im ω
The arrows shows the direction of the phase increase of H(ω).
Stability condition requires that poles can be only in the upper-
half plane, but zero can be anywhere.Workshop on Quantum Optics, UCSB – p.9/34
Negative Group Delay Circuit
Vin Vout
V- R
1/iωC
V− =(iωC)−1
R+(iωC)−1Vout =
11+ iωCR
Vout
Vin ∼V− for large gain of operational amplifier
Vout = (1+ iωCR)Vin
A pole is converted into a zero by the feedback circuit.Workshop on Quantum Optics, UCSB – p.10/34
Finite bandwidth
Transfer Function
H(ω) = 1+ iωT (T = CR > 0)
A(ω) = 1+O(ω2T 2),
φ(ω) = ωT +O(ω3T 3)
Spectral condition for input signals
|ω | < 1T
Workshop on Quantum Optics, UCSB – p.11/34
Bandwidth
Works only for band-limited signals— otherwise outputs are distorted.
-80-60-40-20
020406080
-2 -1.5 -1 -0.5 0 0.5 1 1.5 20
0.5
1
1.5
2
2.5
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
Am
plitu
deA(ω
)
Frequency ωT Frequency ωTPh
ase
φ(ω
)/de
gree
Workshop on Quantum Optics, UCSB – p.12/34
Band-Limit Circuit (Low-pass filter)
Vout
R1 R1
C1C1
(α−1)R2
R2
Vin
Bessel filter (2nd order; m = 2)
HLP(ω) =α
1+ iωTLP(3−α)+(iωTLP)2
TLP = R1C1, α = (1+R3/R2) = 1.268
Cutoff frequency: ωC = 0.7861/TLP
Workshop on Quantum Optics, UCSB – p.13/34
Low-pass filters
A rectangular pulser and a series of lowpass filters (m stages)are used to generate band-limited pulses.
RectangularPulser
LPF 1 LPF 2 LPF m
....
Pushing the button (at t = 0) starts the event.
The band-limited output has a smooth leading edge.
A delay comparable to the pulse width is unavoidable.
Workshop on Quantum Optics, UCSB – p.14/34
Low-pass filters(2)
0
0.1
0.2
0.3
0.4
0.5
0.6
0 1 2 3 4 5 6 7
ampl
itude
/A.U
.
time ωc t
m=24 6 8 10
Workshop on Quantum Optics, UCSB – p.15/34
Circuit Diagram
TRGOUT
VCRESTHDIS
timer IC7555
10k R0
C0
20k
R3
R1 C1
9V 9V
R1R2
SW
C R
C’
R’
20k
10k C R
C’
R’
C1 C1C1
R3
R2
R1
R1
100
TL082 TL082
TL082 TL082
Pulse generator
100
LEDLED
Vin Vout
input output Negative delay circuit
Low-pass filters
Trec
Workshop on Quantum Optics, UCSB – p.16/34
Circuit parameters
Pulse generator Negative delay circuitR0 6.8MΩC0 0.22 µFTrec 1.5sR1 2.2MΩC1 0.22 µFR2 10kΩR3 2.2kΩωc 1.6Hz
R 1MΩC 0.22 µFR′ 10kΩC′ 22nFT (= RC) 0.22s
Workshop on Quantum Optics, UCSB – p.17/34
Circuit Board
Workshop on Quantum Optics, UCSB – p.18/34
Experimental result
m = 4,n = 2
-0.5
0
0.5
1
1.5
2
2.5
-1 0 1 2 3 4 5 6 7
input
ampl
itude
/V
time / sWorkshop on Quantum Optics, UCSB – p.19/34
Experimental result
m = 4,n = 2
-0.5
0
0.5
1
1.5
2
2.5
-1 0 1 2 3 4 5 6 7
inputoutput
ampl
itude
/V
time / sWorkshop on Quantum Optics, UCSB – p.19/34
Interference in time domain
In time domain (cf. H(ω) = 1+ iωT )
vout(t) =(
1+Tddt
)vin(t) = vin(t)+T
dvin
dt(t)
Two terms interfere — constructively at the leading edge anddestructively at the trailing edge.
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
-4 -2 0 2 4
exp(-x**2)-2*x*exp(-x**2)
(1-0.2*x)*exp(-x**2)
constructive destructive
time
vindvin/dt
vout
Workshop on Quantum Optics, UCSB – p.20/34
All-passive circuit
i(t)
v(t) C R
i(t) =1R
v(t)+Cddt
v(t) =1R
(1+RC
ddt
)v(t)
This circuit gives negative group delays, if we consider v(t) as
the input and i(t) as the output.
Workshop on Quantum Optics, UCSB – p.21/34
Unbalanced interferometer
Reflectivity of beam splitters: R = 1/2−ρ , Delay: τ = 2L/c
Edark(t)= (1−R)Ein(t)−REin(t−τ)∼ 2ρ(
1+τ
4ρddt
)Ein(t)
L
dark port
bright port
input
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
-4 -3 -2 -1 0 1 2 3 4
ampl
itude
t/T
inputbrightdark
dark (blow up)
ρ = 0.08, τ = 0.15T , T : gaussian pulse width Workshop on Quantum Optics, UCSB – p.22/34
Cascading — for larger advancement
Transfer function for n stages: Hn(ω) = (1+ iωT )n
An(ω) ∼ 1+n(ωT )2
2= 1+
(√
nωT )2
2, nφ(ω) ∼ nωT
-80
-60
-40
-20
0
20
40
60
80
-0.6 -0.4 -0.2 0 0.2 0.4 0.60.8
0.9
1
1.1
1.2
1.3
1.4
1.5
-0.6 -0.4 -0.2 0 0.2 0.4 0.6
n = 1
n = 1
n = 2
n = 2
n = 3n = 3
Am
plitu
deA(ω
)
Frequency ωT Frequency ωT
Phas
ear
g[φ(
ω)]
Bandwidth is reduced as n increased.Workshop on Quantum Optics, UCSB – p.23/34
Cascading (2)
Usable bandwidth decreases as 1/√
n, if T is given.
For a given input pulse (Tw), T = CR must be reduced asTw/
√n.
Total advancement scales 1/√
n;
−td ∝ n× (Tw/√
n) =√
nTw
Order m of lowpass filters must be increasedaccordingly;
m > n
Workshop on Quantum Optics, UCSB – p.24/34
Cascading, n = 10
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6 7
input
norm
aliz
edam
plitu
de
time / s
Workshop on Quantum Optics, UCSB – p.25/34
Cascading, n = 10
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6 7
inputoutput
norm
aliz
edam
plitu
de
time / s
Workshop on Quantum Optics, UCSB – p.25/34
Problems in cascading
It is possible to increase the advancement as large as thepulse width or more, but
The advancement increases slowly; 1/√
n.
The order m of lowpass filters must be increased.
The system becomes very sensitive to noises. (A hugegain outside of the band)
Workshop on Quantum Optics, UCSB – p.26/34
Pushing the limit
0 5 10 15 20 25 300
2
4
6
8−0.5
0
0.5
1
Total=nT
time t/sec
n
0 5 10 15 20 25 3
0
10
20
30
40
50−0.5
0
0.5
Ttotal=sqrt(n)T
time t/sec
n
Simple-minded cascading 1/√
n cascading
Workshop on Quantum Optics, UCSB – p.27/34
Pushing the limit — two pulse case
0 5 10 15 20 25 30 35−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
time t/sec
input
output(n=49)
output(linear,n=7)
Workshop on Quantum Optics, UCSB – p.28/34
Out-of-band gain
Outside gain — Significant obstacle toward large n.
Realistic transfer function (finite gain at ω = ∞):
H(ω) =1+ iωT
1+ iωT/α, α > 1
φ(ω) = ω(1−α−1)T
A(ω) =
√1+(ωT )2
1+(ωT/α)2∼
1 (ω = 0)α (ω = ∞)
For α = 0.5, n = 50; An(∞) = αn = 250 = 1015!
Workshop on Quantum Optics, UCSB – p.29/34
Role of lowpass filters
Predictability — Your move will be predicted if somerestrictions are imposed.
JERKLIMIT
0.034ACCEL.LIMIT
10 0.11SPEEDLIMIT
35
...
∣∣∣∣dmxdtm
∣∣∣∣
Resemblance of functions — A negative delay circuitreveals the difference.
dist( f ,g) = max0≤ j≤m
∥∥∥∥d j fdt j
− d jgdt j
∥∥∥∥Workshop on Quantum Optics, UCSB – p.30/34
Role of filters (2)
On the condition m > n
0 1 2 3 4 5 6 7 8
0
0.5
n=8n=9n=10n=11n=12n=13
time t/sec 0 0.02 0.04 0.06 0.08 0.1 0.12
−4
−2
0
2
4
6
x 10−3
n=8n=9n=10n=11n=12n=13
time t/sec
Workshop on Quantum Optics, UCSB – p.31/34
Causality
Pulser LPF (m) ND (n)
A B C
O
O
A BC
t
Causal relation in a casual sense
O → A → B?→C
Causal relation in a strict sense
A ⇔ B ⇔C (⇔ A)Workshop on Quantum Optics, UCSB – p.32/34
Conclusion
A stand-alone (battery-operated) demonstration boxBattery operatedNo extra equipment such as an oscilloscope or afunction generator needed
Large negative delays ∼ 0.5s are achievedexperimentally.
25% of pulse width
Larger delay with cascading is possible (in priciple).Advancement can be larger than the pulse width.
Filter stage plays an essential role.The start of event (cause) is located before thefilters.
Workshop on Quantum Optics, UCSB – p.33/34
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