Wide-bandwidth, tunable, multiple-pulse-width optical delays using slow
  light in cesium vapor
Wide-bandwidth, tunable, multiple-pulse-width optical delays using slow light in
cesium vapor
Ryan M. Camacho, Michael V. Pack, John C. Howell
Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA
Aaron Schweinsberg, Robert W. Boyd
Institute of Optics, University of Rochester, Rochester, NY 14627, USA
We demonstrate an all-optical delay line in hot cesium vapor that tunably delays 275 ps input
pulses up to 6.8 ns and 740 input ps pulses up to 59 ns (group index of approximately 200) with
little pulse distortion. The delay is made tunable with a fast reconfiguration time (100’s of ns) by
optically pumping out of the atomic ground states.
There is considerable practical interest in developing
all-optical delay lines that can tunably delay short pulses
by much longer than the pulse duration. Slow light (i.e.
the passage of light pulses through media with a small
group velocity) has long been considered a possible mech-
anism for constructing such a delay line. Most commonly,
the steep linear dispersion associated with a single gain or
transparency resonance provides the group delay. Most
early work used the dispersion associated with electro-
magnetically induced transparency [1, 2, 3, 4, 5, 6, 7],
but recently other resonances have been explored, includ-
ing coherent population oscillations [8, 9, 10], stimulated
Brillouin scattering [11, 12, 13, 14], stimulated Raman
scattering [15, 16], and spectral hole-burning [17].
In addition to single-resonance systems, double gain
resonances have been used for pulse advancement [18,
19, 20, 21, 22, 23] and delay [24]. Widely spaced gain
peaks create a region of anomalous dispersion, result-
ing in pulse advancement. When the spacing between
the gain peaks is small, a region of normal dispersion
is created, resulting in pulse delay. Pulse advancement
is also possible by the proper spacing of two absorbing
resonances [22]. The possibility of pulse delay between
two absorbing resonances has also received some atten-
tion [25, 26, 27, 28, 29, 30].
Ideally, an optical delay line would delay high band-
width pulses by many pulse lengths in a short propaga-
tion distance without introducing appreciable pulse dis-
tortion and be able to tune the delay continuously with
a fast reconfiguration rate. Minimal pulse absorption
is also desirable, but not necessary because absorption
can be compensated through amplification. Relatively
few experiments [1, 4, 7, 11, 17, 28] have directly mea-
sured pulse delays longer than the incident pulse dura-
tion, and of these, none has used pulses shorter than 2 ns
or reported reconfiguration rates approaching the inverse
pulse delay time.
In this Letter, we demonstrate the tunable delay of a
1.6-GHz-bandwidth pulse by up to 25 pulse widths and
the tunable delay of a 600-MHz-bandwidth pulse by up
to 80 pulse widths by making use of a double absorption
resonance in cesium. Furthermore, we show that the de-
lay can be tuned with a reconfiguration time of 100’s of
nanoseconds.
In a medium with two Lorentzian absorption reso-
nances, as illustrated in Figure 1, the complex index of
refraction can be approximated as
n(δ) = 1− A
δ +∆+ + iγ
δ −∆− + iγ
where 2γ is the homogeneous linewidth (full width at half
maximum, FWHM), g1 and g2 account for the possibility
of different strengths for the two resonances, δ = ω−ω0−
∆ is the detuning from peak transmission, ω0 = (ω1 +
ω2)/2, ω1 (ω2) is the resonance frequency for transition
1 (transition 2), ∆± = ω21 ±∆, ω21 = (ω2 − ω1)/2, and
1 − g
1 + g
ω21. (2)
For example, alkali atoms have two hyperfine levels as-
sociated with their electronic ground state, leading to
two closely spaced absorption resonances. We note that
any other system with two similar absorbing resonances
may also be used (e.g. quantum dots, microresonators,
photonic crystals, etc). For a vapor of alkali atoms, the
detunings satisfy ∆+ ≈ ∆− ≫ γ, and the strength of the
resonance is given in SI units byA = N |µ|2/[ǫ0h̄(g1+g2)],
where µ is the effective far-detuned dipole moment [31],
and g1 and g2 are proportional to the degeneracies of the
hyperfine levels.
Eq. (1) is also applicable for inhomogeneously broad-
ened lines, such as Doppler broadened atomic vapors, if
the detunings ∆− and ∆+ are greater than the inhomo-
geneous linewidth by an order of magnitude or more.
This result holds because the homogenous Lorentzian
lineshape has long wings while the inhomogeneous line-
shape decreases exponentially.
By expanding Eq. (1) about the point δ = 0, we find
that the real part n′ and imaginary part n′′ of the index
of refraction are given by
n′(δ) ≈ 1 +K0 +K1
δ +K3
δ3 (3a)
http://arxiv.org/abs/0704.1279v1
-5 0 5
Signal Detuning (GHz)
FIG. 1: (a) CW signal transmission (asterisks–measured,
solid–fit ) overlayed with the spectrum (dashed) of a 275 ps
pulse and (b) index of refraction (solid) and group velocity
(dashed), all versus signal detuning for cesium at approxi-
mately 114 ◦C. All theory curves taken from Eq. (1) with
A = 4 × 105 rad/s, g1 = 7/16 and g2 = 9/16. High-fidelity
optical delay is observed for light pulses passing through the
nearly transparent window between the two resonances.
n′′(δ) ≈ K1
+ 3K3
δ2, (3b)
where
1 + g
(2−i)/3
1 + (−1)i+1g
(2−i)/3
and where we have assumed that n − 1 ≪ 1 in keeping
only the first few terms in the expansion. Note that for
the special case in which the two resonances are of equal
strength (i.e. g1 = g2 = g), the coefficients are given by
Ki = 2g for i odd and Ki = 0 for i even. For cesium,
which has g1 = 7/16 and g2 = 9/16, the error introduced
by assuming g1 = g2 is approximately 0.5%. For this
reason, we make the simplifying assumption g1 = g2 =
1/2 throughout the remainder of the paper.
Pulse propagation can be described in terms of various
orders of dispersion, which can be determined through
use of Eq. (3a) as
djωn′(ω)
ω=ω0+∆
, (5)
Thus the group velocity is given by vg = 1/β1, and
the group-velocity dispersion (GVD) and third-order-
dispersion are given respectively by β2 and β3. The ab-
sence of second-order (first-order) frequency dependence
in Eq. (3a) (Eq. (3b)) means that near δ = 0 the GVD
(absorption) is minimized regardless of possible differ-
ences between g1 and g2. Thus, between two absorp-
tion resonances, which can be described by Eq. (1), the
maximum transparency is accompanied by a minimum
in GVD.
We next develop a simple model to provide an un-
derstanding of the role of dispersion and absorption on
pulse broadening. We provisionally define the pulse
width as the square root of the variance of the tempo-
ral pulse shape. For an unchirped Gaussian pulse, i.e.
E(0, t) = E0 exp
−t2/2T 20
, the pulse width defined in
this way is simply T0. The pulse width after propagating
through a distance L of dispersive medium is then given
to third-order in δ by [32]
T 2d = T
2T 20
where T0 is the initial pulse width. In the case of cesium,
where ω0 ≈ 2π×3.5×1014 rad/s and ω21 ≈ π×9.2×109
rad/s, β2 can be neglected and Eq. (6) simplifies to
T 2d ≈ T 20 +
ω221T
, (7)
where β3 has been calculated using Eqs. (5) and (3)
and where τd ≈ α0L/2γ is the pulse delay and α0L =
′′L/c is the optical depth at the pulse carrier fre-
quency ω0. We further note that the change in pulse
width due to absorption only can be approximated as
[28, 33]
T 2a = T
, (8)
so long as (Ta/T0 − 1) < 1.
The fractional broadening due to dispersion, defined
as Td/T0 − 1, scales as 1/T 30 , while the broadening due
to absorption scales as 1/T0. In the present study, τd ≈
10−8 s, ω21 ≈ 1011 rad/s, T0 ≈ 10−10 s, and γ ≈ 107
rad/s, indicating that dispersion is the dominant form of
broadening by about three orders of magnitude, and the
absorptive contribution to broadening can be ignored.
Experimentally it is much easier to quantify pulse
widths in terms of their FWHM rather than in terms
of their variance as we have done in Eqs. (6) - (8). In
the remainder of this Letter, we will quote pulse widths
in terms of their FWHM.
Our experimental setup is shown in Fig. 2. The signal
laser is a CW diode laser with a wavelength of 852 nm.
The signal frequency is tuned to obtain maximum trans-
mission between the two Cs D2 hyperfine resonances and
is pulsed at a pulse repetition frequency of 100 kHz using
a fast electro-optic modulator (EOM). The signal beam is
collimated to a diameter of 3 mm, and two different pulse
widths are used, 275 ps or 740 ps FWHM, with a peak
intensity of less than 10 mW/cm2. The pulses then pass
through a heated 10-cm-long glass cell containing atomic
cesium vapor. The 275 ps pulses are measured using a 7.5
GHz silicon photodiode, and the 740 ps pulses are mea-
sured with a 1 GHz avalanche photodiode. All electrical
signals are recorded with a 30 GHz sampling oscilloscope
FIG. 2: Experimental schematic. A signal pulse passes
through a heated cesium vapor cell. Two pump beams com-
bine on a beamsplitter and counter-propagate relative to the
signal beam through the vapor, to provide tunable delay of
the signal pulse.
triggered by the pulse generator. The pump beams are
turned off except for the experiments reported in Figs. 5
and 6.
Figure 1(a) shows the transmission of a CW optical
beam as a function of frequency near the two cesium hy-
perfine resonances, overlayed with the spectrum of a 275
ps Gaussian pulse. The data points are measured values
and the solid line fits these points to the imaginary part
of Eq. (1). The entire pulse spectrum lies well within
the relatively flat transmission window between the res-
onances, resulting in little pulse distortion from absorp-
tion. Figure 1(b) shows the index of refraction (real part
of Eq. (1)) and frequency-dependent group velocity asso-
ciated with the absorption shown in Fig. 1(a). We note
that, in the region of the pulse spectrum, the curvature
of the frequency-dependent group velocity is greater than
that of the absorption, suggesting that dispersion is the
dominant form of pulse distortion. This is not the case
for single-Lorentzian systems, where the spectral varia-
tion of absorption is the dominant form of distortion [34].
While most slow light experiments have worked by mak-
ing highly dispersive regions transparent, we have worked
where a highly transparent region is dispersive.
As shown above, the delay of a pulse is proportional to
the optical depth of the vapor. Figure 3 shows that we
can control the delay by changing the temperature (and
thus optical depth) of the Cs cell. Using a 10 cm cell,
and varying the temperature between approximately 90
◦C and 120 ◦C, we were able to tune the delay of a 275
ps pulse between 1.8 ns and 6.8 ns. The theory curves in
Fig. 3 were obtained using I(x, t) = n′(0)cǫ0|E(z, t)|2/2
where the electric field is given by
E(z, t) =
E0T0 exp [−i(ω0 +∆)t]√
dδexp
ωn(δ)
z − δt
2T 20
, (9)
FIG. 3: Pulse shapes of 275 ps input pulses transmitted
through a cesium vapor cell. Delays a large as 25 pulse widths
are observed. The temperature range from 90 ◦C to 120 ◦C
and where we have used Eq. (1) for the index of refrac-
tion. The atomic density N has been chosen separately
to fit each measured pulse. We note that a pulse may
be delayed by many pulse widths relative to free-space
propagation with little broadening.
Longer pulses lead to delay with reduced pulse broad-
ening because pulse broadening is approximately propor-
tional to 1/T 30 (see Eq. (7)). To study the larger frac-
tional delays enabled by this effect, we used longer 740
ps input pulses for which the dispersive broadening is sig-
nificantly reduced. Figures 4(a) and 4(b) show the delay
and broadening of a 740 ps pulse after passing through
a sequence of three 10 cm cesium vapor cells. The plots
correspond to a temperature range of approximately 110
◦C to 160 ◦C. Even though the pulse experiences strong
absorption at large delays, the fractional broadening of
the pulse FWHM remains relatively low.
FIG. 4: (a) Output pulse shapes and (b) fractional broaden-
ing as functions of fractional delay for a 740 ps input pulse.
Fractional delay is defined as (τd/T0) and fractional broaden-
ing is defined as (T − T0)/T0.
FIG. 5: Pulse output waveforms with auxiliary pump beams
on (dotted) and off (solid). Two 275 ps input pulses separated
by 1 ns are delayed by approximately 5.3 ns without pumping,
but only 4.3 ns with pumping (a change of one bit slot) with
little change in pulse shape.
In addition to temperature tuning, the optical depth
can be changed much more rapidly by optically pumping
the atoms into the excited state using two pump lasers.
As shown in Fig. 2 each pump laser is resonant with
one of the D2 transitions in order to saturate the atoms
without optical pumping from one hyperfine level to the
other. The power of each pump beam is approximately
30 mW, and both pump beams are focused at the cell
center. The signal beam overlaps the pump beams and
is also focused to a 100 µm beam diameter. The pump
beams are turned on and off using an 80 MHz AOM
with a 100 ns rise/fall time. Being on resonance with
the D2 transitions, the pump fields experience significant
absorption (αL ∼ 300), and are entirely absorbed despite
having intensities well above the saturation intensity.
With the pump beams on, the decreases in effective
ground-state atomic density leads to smaller delay. Fig-
ure 5 shows a delayed pulse waveform consisting of two
275 ps input pulses separated by 1 ns, with the pump on
and off. We note that pump fields create no noticeable
change in the waveform shape or amplitude. Also, we
measured that the change in delay is essentially propor-
tional to the pump power.
In Fig. 6 the measured signal delay is shown as a func-
tion of the difference between arrival time ts of the signal
at the cell and the turn-on time tp of the pump. The rise
and fall times lie in the range 300-600 ns and vary slightly
depending on the relative detunings of the pumps.
In summary, we have observed large tunable fractional
time delays of high-bandwidth pulses with fast recon-
figurations rates and low distortion by tuning the laser
frequency between the two ground-state hyperfine reso-
nances of a hot atomic cesium vapor cell. We have shown
that in such a medium dispersion is the dominant form of
broadening, and we have characterized the delay, broad-
ening, and reconfiguration rates of the delayed pulses.
This work was supported by the DARPA/DSO Slow
Light program, the National Science Foundation, andthe
Research Corporation.
FIG. 6: Pulse delay versus time following pump turn-on and
turn-off, showing the reconfiguration time for optically tuning
the pulse delay. The two pump beams are tuned to separate
cesium hyperfine resonances and are switched on at the time
origin and switched off 24 µs later.
[1] A. Kasapi et al., Phys. Rev. Lett. 74, 2447 (1995).
[2] M. M. Kash et al., Phys. Rev. Lett. 82, 5229 (1999).
[3] D. Budker et al., Phys. Rev. Lett. 83, 1767 (1999).
[4] L. V. Hau et al., Nature 397, 594 (1999).
[5] D. F. Phillips et al., Phys. Rev. Lett. 86, 783 (2001).
[6] C. Liu et al., Nature 409, 490 (2001).
[7] A. Turukhin et al., Phys. Rev. Lett. 88, 023602 (2002).
[8] M. S. Bigelow et al., Science 301, 200 (2003).
[9] X. Zhao et al., Optics Express 93, 7899 (2005).
[10] P. Palinginis et al., Optics Express 13, 9909 (2005).
[11] K. Y. Song et al., Opt. Lett. 30, 1782 (2005).
[12] Y. Okawachi et al., Phys. Rev. Lett. 94, 153902 (2005).
[13] M. Gonzalez-Herraez et al., Appl. Phys. Lett. 87, 081113
(2005).
[14] M. Gonzalez-Herraez et al., Optics Express 14, 1400
(2006).
[15] J. E. Sharping et al., Optics Express 13, 6092 (2005).
[16] D. Dahan et al., Optics Express 13, 6234 (2005).
[17] R. M. Camacho et al., Phys. Rev. A 74, 033801 (2006).
[18] R. Y. Chiao, Phys. Rev. A 48, R34 (1993).
[19] L. J. Wang et al., Nature 406, 277 (2000).
[20] A. Dogariu et al., Phys. Rev. A 63, 053806 (2001).
[21] M. D. Stenner et al., Nature 425, 695 (2003).
[22] B. Macke et al., Eur. Phys. J. D 23, 125 (2003).
[23] G. S. Agarwal et al., Phys. Rev. A 70, 023802 (2004).
[24] M. D. Stenner et al., Optics Express 13, 9995 (2005).
[25] D. Grischkowsky, Phys. Rev. A 7, 2096 (1973).
[26] H. Tanaka et al., Phys. Rev. A 68, 053801 (2003).
[27] B. Macke et al., Phys. Rev. A 73, 043802 (2006).
[28] R. M. Camacho et al., Phys. Rev. A 73, 063812 (2006).
[29] Z. Zhu et al., Opt. Exp. 14, 7238 (2006).
[30] R. M. Camacho et al., Physical Review Letters 98,
043902 (2007).
[31] D. A. Steck. Cesium D line data, Technical Report
No. LA-UR-03-7943, Los Alamos National Laboratory
(2003), http://steck.us/alkalidata/.
[32] G. P. Agrawal, Nonlinear Fiber Optics (Academic Press,
1995), p. 79.
[33] R. W. Boyd et al., Phys. Rev. A 71, 023801 (2005).
[34] R. W. Boyd et al., Progress in Optics (Elsevier, 2002),
p. 497.
