Demonstration of Negative Group Delays in a Simple Electronic Circuit

Demonstration of Negative GroupDelays in a Simple Electronic Circuit

Masao [email protected], 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


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!


Experiment

button

LED

pulse

pulse generatorLED

input output

negative delay circuit

1. rectangular pulse generator + low-pass filter

2. negative delay circuit


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


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.


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)


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)


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


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


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


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.


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


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


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


Circuit Board


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


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.


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


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


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


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)


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


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)


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!


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


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.


Demonstration of Negative Group Delays in a Simple Electronic Circuit

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