Demonstration of negative group delays in a simple electronic circuit

6 Pages·0.15 MB·English

by  T. Nakanishi

Demonstration of negative group delays in a simple electronic circuit
T. Nakanishi, K. Sugiyama, and M. Kitano
Department of Electronic Science and Engineering, Kyoto University
Kyoto 606-8501, Japan
(Dated: February 1, 2008)
Wepresentasimpleelectroniccircuitwhichproducesnegativegroupdelaysforbase-bandpulses.
When a band-limited pulse is applied as the input, a forwarded pulse appears at the output. The
negative group delays in lumped systems share the same mechanism with the superluminal light
propagation, which is recently demonstrated in an absorption-free, anomalous dispersive medium
[Wang et al., Nature 406, 277 (2000)]. In this circuit, the advance time more than twenty percent
ofthepulsewidthcaneasilybeachieved. Thetimeconstants,whichcanbeintheorderofseconds,
2
is slow enough to be observed with thenaked eye bylooking at thelamps driven by thepulses.
0
0
2
I. INTRODUCTION shift is a linear function of the frequency.

n The interferencedoesnotoccuronlyinthe lightprop-
a Brillouin and Sommerfeld investigated the propaga- agation, but also in other wave or oscillation dynamics.
J
tion of light pulse in a dispersive medium described by Signals in electrical circuits also interfere. In lumped
1
the Lorentz model. They pointed out that in the region systems, we cannot define the group velocities, because

1 of anomalous dispersion the group velocity vg could be there exists no finite length scale. Instead, we candefine
v larger than c, the light velocity in a vacuum, or even be the group delay. The group delay is the time difference
1 negative [1]. The anomalous dispersion occurs near the between the input and output signal envelopes. If it is
0 center of absorption lines. In the case of superluminal negative, the output pulse precedes the input as shown
0 group velocities (v > c), the transit time of the light in Fig. 1. The negative delays are closely connected to
g
1 envelope through the medium is shorter than that for a thesuperluminalornegativegroupvelocitiesinspatially
0
vacuum with the same length. In the case of negative extended systems.
2
0 group velocities (vg < 0), the envelope leaves before it In this paper, we propose a simple electronic circuit
/ enters into the medium. which generates negative group delays. While the dis-
h Chu and Wong demonstrated experimentally that the persion relation k(ω) of dispersive media determines the
p
phase shifts in the light propagation, the transfer func-
light in a Ga:N crystal can be propagated at such ex-
-
t traordinary group velocities [2]. But in these kind of ex- tion H(ω) of the circuit determines the phase shifts of
n
a perimentsitisinevitablethattheshapeofthelightpulse the output in reference to the input. Mitchell and Chiao
u is largely distorted due to the absorption. The standard [6] demonstratednegativedelaysin a bandpass filter cir-
q definition of the group velocity, v = dω/dk, which de- cuit to which the carrier signal modulated with a pulse
g
v: scribes the velocity of the light envelope or peak, tends is applied. They showed that if the carrier frequency
i to lose its physical meaning under the strong distortion is located outside of the pass band, the envelope of the
X of the waveform. A new definition of group velocities, output pulse precedes the input. In practical implemen-
r considering the reshape of the spectrum caused by the tation,alargeinductorisrequiredanditisreplacedbya
a
absorption, was proposed recently [3].
Wang et al. realizedanomalousdispersionwithoutab-
negative delay circuit
sorptionusingthe gain-assistedlinearanomalousdisper- peak
sion,anddemonstratedthelightpropagationattheneg-
Η(ω)
ative groupvelocities [4, 5]. For the light pulse propaga-
IN OUT
tioninanabsorption-freemedium, the normaldefinition
of the group velocity describes precisely the propagating
Η(ω)
speed of the waveform, or the envelope.
The pulse propagation with superluminal or negative IN OUT peak
group velocities is very counterintuitive and is liable to
Η(ω)
causemany misunderstandings. However,itis the direct
IN OUT
results of the interference of the wave and is consistent time
with the relativistic causality.
The anomalous dispersion with no absorption induces
FIG. 1: An electronic black box for negative delays. The
phase shifts depending on the frequencies, thereby en- inputand theoutputare monitored by LEDs. When a pulse
hances the front part of the pulse by constructive in- is fed to the input, the LED at the output lights up before
terference and cancels the rear part by the destructive theLED at the input.
interference. The pulse shape is conserved if the phase
2
simulated inductor with operational amplifiers. The use The group delay t is defined as
d
ofthecarriersignalalsoincreasesthecomplicationofthe
circuit. dφ
t = . (4)
d
The circuit that we present here generates negative −dω(cid:12)ω0
(cid:12)
delays for baseband signals or signals with zero carrier (cid:12)
ThentheenvelopeoftheoutputisobtainedfromEqs.(2)
frequency. It is easy to produce negative delays as large
and (3) as
as a few seconds, therefore, we can observe it with the
nakedeye by watching two LEDs (light emitting diodes)
(t)= (t t )eiφ(ω0). (5)
each of which is driven by the input and output pulses. out in d
E E −
It is helpful and illuminating to translate fundamental
This means that, aside from the phase factor, the enve-
physical concepts into a simple circuit which replicates
lope ofthe output is shifted by the groupdelayt , while
the essence of the phenomena [7, 8]. d
maintaining the shape. For t > 0, the input precedes
Inthenextsectionweexplaintherelationbetweenthe d
the output (normal delay) and for t < 0, the output
superluminalornegativegroupvelocityinthelightprop- d
precedes the input (negative delay). We will represent
agationandthe negativegroupdelay inlumped circuits.
a circuit that satisfies the latter condition in the next
Then, in Sec. III, we present a circuit generating nega-
section.
tive group delays and the pulse generator used for the
Inordertotranslatetheabovediscussionintothelight
input. In Sec. IV, we propose a method for larger nega-
propagation through a dispersive medium with length
tivedelaysbycascadingthecircuits. Itisshownthatthe
L, we suppose E and E represent the field of in-
negative delay time can be increased as √n without in- in out
put and that of output of the medium, respectively.
troducingadditionaldistortionofthepulseshape,where
Whenthemonochromaticlightwithfrequencyωisprop-
nisthenumberofthestages. Butforlargen,thecircuit
agatedinthe medium, the phase ofthe field is shifted as
becomes susceptible to the noise.
φ(ω) = k(ω)L, where k(ω) is the wavenumber in the
−
medium. If k(ω) is linear in the bandwidth, with the
II. TRANSFER FUNCTION FOR NEGATIVE help of Eq. (4), we have
GROUP DELAY
dk L
t =L = , (6)
d
In this section we deal with a transfer function H(ω) dω(cid:12)ω0 vg
(cid:12)
in order to study negative group delays. The discus- (cid:12)
where dω/dk is the group velocity v . The envelope
sion can be applied to both electric circuits and light |ω0 g
of the light is delayed by L/v .
propagation in a dispersive medium. A band-limited g
If the difference between the propagation time of the
input signal E (t) can be expressed as a product of
in envelopesinthedispersivemediumandthatinavacuum
the carrier exp(iω t) and the envelope (t); E (t) =
0 Ein in with the same length L is negative, i.e.,
(t)eiω0t+c.c., where c.c. represents the complex con-
in
E
jugate term. Our circuit deals with the baseband signal ∆t=t L/c=L(v−1 c−1)<0, (7)
(ω0 = 0), but we include the carrier for the purpose of d− g −
comparisonwithothercases. Thesignalisexpandedwith
then the light propagation in the medium is called su-
the Fourier component ˜ (u) of the envelope (t) as
Ein Ein perluminal. There are two cases which satisfy this con-
Ω/2 dition; vg >c andvg <0. Inthe formercase,the output
E (t)= du ˜ (u)ei(ω0+u)t+c.c., (1) envelope precedes the output for the vacuum case but
in in
Z−Ω/2 E does not precede the input. In the latter case, the out-
where Ω is the bandwidth and u is the offset frequency putprecedesthe input, orthe wholesystemprovidesthe
from ω0. negative group delay td <0.
The transfer function H(ω) A(ω)eiφ(ω) is defined
≡
for each frequency ω = ω +u and the output E (t)
0 out
can be written as III. CIRCUITS AND EXPERIMENT
Ω/2
E (t)= du ˜ (u)H(ω)ei(ω0+u)t+c.c. A. Negative delay circuit for baseband signals
out in
Z−Ω/2 E
Ω/2 In Fig. 2, we show a negative delay circuit for base-
= du ˜ (u)A(ω)ei(ω0+u)teiφ(ω)+c.c. (2)
Z−Ω/2 Ein band (ω0 = 0) signals. This is basically a non-inverting
(imperfect) differentiator. Its transfer function is easily
Weassumethatwithinthebandwidth(u <Ω/2),the obtained as
| |
amplitude A(ω) is nearly unity and the phase φ(ω) can
be approximated by a linear function, i.e., H(ω)=A(ω)eiφ(ω) =1+iωT, (8)
A(ω) 1, φ(ω) φ(ω ) ut . (3)
0 d where T =RC .
∼ ∼ −
3
R
2.5
C
V 2
out
V
in e
d 1.5
u
plit
m 1
FIG. 2: Negative delay circuit for baseband signal can be a
constructedwithanoperationalamplifier. Thetransferfunc-
0.5
tion is H(ω)=1+iωT.
0
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
We note thatthe correspondingactionin the time do-
ωT
main is 1+T(d/dt). The advancement of a pulse can be
understood qualitatively. For the rising edge, which has
a positive slope, the two terms interfere constructively, 80
while for the falling edge with a negative slope, they in- 60
terfere destructively. Thus the pulse is forwarded.
e 40
In the low-frequency region ( ω 1/T ), H(ω) is e
r
| | ≪ g 20
approximated as e
d
A(ω)=1+O(ω2T2), (9) e / 0
s -20
φ(ω)=ωT +O(ω3T3), (10) ha -40
p
which mean that the amplitude is nearly constant and -60
the phase increases linearly with frequency. Then the -80
group delay becomes negative:
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
dφ ωT
t = = T <0. (11)
d
−dω(cid:12)ω=0 −
(cid:12)
As seen in Fig. 3, the a(cid:12)mplitude A(ω) and the phase FIG. 3: The amplitude of the transfer function A(ω) =
|H(ω)|(upper)and thephase φ(ω)=argH(ω)(lower).
φ(ω) of the transfer function are not linear except in
ω T 1 owing to the higher order terms in Eqs. (9)
| | ≪
and(10). Theseterms induce waveformdistortionofthe
output. Inorderto keepthe distortionas smallaspossi-
TABLE I: Circuit parameters.
ble, the spectrum of the input signal must be restricted
within the frequency region ω 1/T. For this purpose R0 6.8MΩ R 1MΩ
low-pass filters are needed. | |≪ C0 0.22µF C 0.22µF
Trec 1.5s R′ 10kΩ
R1 2.2MΩ C′ 22nF
B. Low-pass filter
C1 0.22µF T 0.22s
R2 10kΩ
In order to prepare a band-limited, base-band pulse
R3 2.2kΩ
weintroducelow-passfilters. Theinitialsourceisarect-
ωc 1.6Hz
angular pulse from a timer IC (integrated circuit). The
rectangular pulse has high-frequency components, and
the negative delay circuit does not work correctly. We
sothattheovershootforrectangularwavesissmall. The
musteliminatethehigh-frequencycomponentswithlow-
cut-off frequency is defined by ω =0.7861/T .
pass filters. We introduce two 2nd-order low-pass filters c LP
shown in Fig. 4. Each transfer function is
α
H (ω)= , (12) C. Experiment
LP 1+iωT (3 α)+(iωT )2
LP LP
−
where T = R C and α = (1+R /R ). By changing We show the overall circuit diagram for the negative
LP 1 1 3 2
α, we can tailor the characteristic of the filter. In our delay experiment in Fig. 4 and the parameters in Table
experiment we choose the value α = 1.268, which corre- I.
sponds to a Bessel filter [9]. The Bessel filter is designed The pulse generator in the upper section of Fig. 4 is
4
Pulse generator
2.5
timer IC
9V 7555 9V Low-pass filters
R0
10k 2 output input
R3 R3
TRGDIS
SW 20k OREUSTTVHC C0 RR21 TL082 RR21 TL082 e / V 1.5
R1 C1 R1 C1 d
C1 C1 u 1
plit
m 0.5
a
0
input Negative delay circuit output
Vin C’ C’ Vout -0.5
-1 0 1 2 3 4 5 6 7
LED
10k R’C R R’C R LED time / sec
100 100
20k
TL082 TL082 FIG. 5: Experimental results. The oscilloscope traces show
the input and output pulses. The output precedes the input
owing to the negative delay.
FIG. 4: Overall circuits. Upper section: pulse generator.
Lower section: negative delay circuit.
0.5
subdivided into the generator of single-shot rectangular
m=2
4 6 8
wave and the low-pass filters. In the first part, when 10
0.4
triggered by the switch, the timer IC generates a single
pulse, whose width is determined by the time constant U.
T = R C = 1.5s. The rectangular pulse is shaped A. 0.3
byrecthe tw0o-s0tagelow-passfilters. The total order of low- e /
d 0.2
pass filter is m = 4. We set the cut-off frequency of u
the low-pass filter as ωc = 0.35/T, so that A(ω) and mplit 0.1
φ(ω)canbeconsideredtobeconstantandlinear,respec-
a
tively, below the cut-off frequency (see Fig. 3). Finally, 0
the band-limited single pulse is sent out for the input of
the negative delay circuit in the lower section of Fig. 4. -0.1
0 1 2 3 4 5 6 7
Two delay circuits shown in Fig. 2 are cascaded for
larger advance time. The input and output terminals time ω t
c
are monitored by LEDs. Their turn-on voltage is about
1.1V. The variable resistor at the input is adjusted so FIG. 6: Responses of the Bessel filter to a rectangular wave.
that the input and the output have the same height. The orders of filters are m=2, 4, 6, 8, and10.
The experimental result is shown in Fig. 5. The input
and output waveforms are recorded with a oscilloscope.
The origin of the time (t = 0) is the moment when the
switch in Fig. 4 is turned on. We see that the output
precedes the input considerably (more than 20% of the
pulse width). The slight distortion of the output wave- waveforms. We could dispense with an oscilloscope.
form is causedby the non-idealfrequency dependence of
A(ω) and φ(ω), as mentioned in Sec. IIIA. The actualnegative delay circuit in Fig. 4 differs from
′ ′
The expected negative delay derived from Eq. (11) is thatshowninFig.2. TheresistorR andthecapacitorC
2T = 0.44s; we have connected two circuits (each time are supplemented for the suppression of high-frequency
constantT =0.22s)inseriesforlargereffect. Thisquan- noises. As shown in Fig. 3, the gain at ω T > 1 is
| |
titatively agrees with the experimental result, where the large. Although the high-frequency components of the
time difference between the output and the input peaks input signal are suppressed by the low-pass filters, the
is about 0.5s. The time scale is chosen so that one can internal and external noises with high frequency are un-
′ ′
directly observe the negative delay with two LEDs con- avoidable. With R and C , the high-frequency gain is
′ ′
nected at the input and the output terminals. We could limited. The parameters are chosen as RC,C R T so
≪
alsousetwovoltmeters(orcircuittesters)tomonitorthe that the phase φ(ω) for ω T <1 is not affected.
| |
5
IV. REALIZATION OF LARGER NEGATIVE
1.2
DELAY
e 1
d
Theobtainednegativedelayintheaboveexperimentis u
about20%ofthepulsewidth. Thisislargerthantheval- plit 0.8
m output input
ues obtained in other superluminal-velocity or negative-
a 0.6
delay experiments. We consider the way to make even d
e
larger negative delays. We assume noise-free environ- z 0.4
ment for a simplicity. ali
m
Cascadingthenegativedelaycircuits,thetimeadvance r 0.2
o
can be increased. One might simply expect that, by in- n
0
creasing the number of stages n, the total time advance
canbeincreasedlinearlywithn. But,unfortunately,this
0 1 2 3 4 5 6 7
is not the case. It is obvious from Eqs.(9) and (10) that
time /sec
the distortion of the waveform of the output is also in-
creased. When n circuits are connected in cascade, the
total transfer function can be written as Hn(ω). Corre- FIG. 7: A simulation result for multi-stage negative delay
circuit (n=10).
spondingly, the amplitude and the phase are given as
n(ωT)2
An(ω) 1+ , (13)
∼ 2 butthe higherthe orderoffilterthe morethe risingpart
nφ(ω) nωT. (14) is delayed. Hence we need a high order filter to attain
∼
large time advance. In other words, the pulse must be
Inordertokeepthewavedistortionbelowacertainlevel, delayed appropriately in advance in order to get a large
we have to limit the excess gain An(ω) 1 within the negative delay.
−
bandwidth by some value γ; Moreover the short-time behavior of the total circuit
including the low-pass filters and the negative delay cir-
n(ωT)2 n(ω T)2
c cuitsisdeterminedbythecompositetransferfunctionat
=γ. (15)
2 ≤ 2 highfrequency(ω ). Theorderofthelow-passfilter
→∞
mshouldnotbe smallerthanthenumberofthe stagesn
Then the advance time per circuit should satisfy
of the negative delay circuits. Otherwise the total trans-
fer function would diverge at ω and the derivative
T = 2γω−1 = 2γT , (16) of the rectangular pulse would a→ppe∞ar at the output.
r n c r n w
Figure 7 represents a result of simulation for the re-
sponse of ten negative delay circuits with the input of
where T is the pulse width. It is determined by the
w
a tenth order Bessel filter. The used parameters are
cut-off frequency of the low-pass filter.
T = 1s and γ = 0.2. The total time advance is esti-
If we want to increase the time advance in conserving w
mated 2s from Eq. (17). This estimation is consistent
the pulse widthandthe distortionofthe signal,wemust
with the result of the simulation. The advance time is
reducethetimeadvanceT percircuitbythefactor1/√n.
comparable to the pulse width. It is so large that the
Therefore, the total time advance T scales as
total
input starts to rise when the output begins to fall.
T =nT = 2nγT , (17)
total w
p
V. DISCUSSION AND CONCLUSION
which is a slowly increasing function of n.
Inaddition,thereisanotherfactortobeconsidered. A
causal transfer function cannot generate negative delays Letusconsiderapositivedelaycircuit. Acircuitcalled
unconditionally. It is impossible to advance the signal all-pass filter has the transfer function [9]:
beyond the time when the switch is turned on in the
1 iωT
rectangular pulse generator in Fig. 4. The reason of the H (ω)= − .
P
1+iωT
advancement is that the slow rising part of the pulse,
whichhasbeensuppressedbythelow-passfilter,isreem- The all-pass filter can be build with an operational am-
phasized by the negative delay circuit. The slowness of plifier, three resistors, and a capacitor. It is very conve-
rising part of the pulse is determined by the order m of nientfor generatingpositive delaysforbase-bandsignals
the low-pass filter. Figure 6 represents responsesof vari- because the amplitude and phase of the transfer func-
ousorderBesselfilterswiththesamecut-offω toarect- tions are given as A(ω)=1, and φ(ω)= 2tan−1ωT
c
angular wave with a unit height and a pulse width ω−1. 2ωT, respectively. The flat amplitude r−esponse allow∼s
c −
Thepulseshapes,especiallythe widthsofthepulses,are ustocascadethencircuitswithoutintroducingpulsedis-
similartoeachotherowingtothesamecut-offfrequency, tortionandresultsinthe delayt =2nT. Unfortunately
d
6
the negative version (T T) of all-pass filter negativedelay(0.44s)couldeasilybeachieved. Itisslow
→−
enoughtobeobservedwiththe LEDsorvoltmeterswith
1+iωT H(ω) the naked eye. The negative delay amounts to 20% of
H (ω)= = ,
N
1 iωT 1 iωT the pulse width. In the light experiment [4], t L/c is
d
− − −
62ns and is only a few percents of the pulse width, 4µs.
will not work because it is not causal. The pole of
Includingthepulsegeneratorfortheinputofthenega-
H ( is), s = 1/T, is located in the right half plain.
N − tivedelaycircuit,theapparatusconsistsofcommonparts
If one makes this circuit, it will be unstable owing to
availableinanylaboratories. Itissosimplethatabegin-
the time response function exp(t/T). This example tells
ner can build it in an hour. The setup can be operated
us aboutthe asymmetry betweenthe negative delay and
stand-aloneandnoexpensiveinstrumentssuchasoscillo-
the positive delay. The former is much more difficult to
scopes and function generators are required. It is useful
achieve than the latter.
for understanding the physics of superluminal propaga-
The group velocity has no direct connection with the
tion as well as the negative group delays.
relativistic causality, therefore, it can exceed the speed
of light c in a vacuum. But the front velocity v (or the
f
wavefrontvelocity)is constrainedby the causalityandis
equal to c, namely, v = L/t = c. In lumped systems
| f| | f| Acknowledgments
(L = 0), the wavefront delay t must vanish. Actually
f
all of the pulses in our system have their wavefronts at
t = 0, the moment when the original rectangular pulse This research was supported by the Ministry of Ed-
rises or the switch is turned on. ucation, Culture, Sports, Science and Technology in
To conclude, we demonstrated the negative group de- Japan under a Grant-in-Aid for Scientific Research
lay in a simple electronic circuit. A considerably large No. 11216203and No. 11650043.
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