# Dispersive effects in ultrafast non-linear phenomena

Dusan Lorenc  Institute of Science and Technology Austria, Am Campus 1, 3400
Klosterneuburg, Austria  Zhanybek Alpichshev<EMAIL_ADDRESS>Institute of
Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria

###### Abstract

It is a basic principle that an effect cannot come before the cause.
Dispersive relations that follow from this fundamental fact have proven to be
an indispensable tool in physics and engineering. They are most powerful in
the domain of linear response where they are known as Kramers-Kronig
relations. However when it comes to nonlinear phenomena the implications of
causality are much less explored, apart from several notable exceptions. Here
in this work we demonstrate how to apply the dispersive formalism to analyse
the ultrafast nonlinear response in the context of the paradigmatic nonlinear
Kerr effect. We find that the requirement of causality introduces a noticeable
effect even under assumption that Kerr effect is mediated by quasi-
instantaneous off-resonant electronic hyperpolarizability. We confirm this by
experimentally measuring the time resolved Kerr dynamics in GaAs by means of a
hybrid pump-probe Mach-Zehnder interferometer and demonstrate the presence of
an intrinsic lagging between amplitude and phase responses as predicted by
dispersive analysis. Our results describe a general property of the time-
resolved nonlinear processes thereby highlighting the importance of accounting
for dispersive effects in the nonlinear optical processes involving ultrashort
pulses.

##  I Introduction

The amazing developments in ultrafast laser technology of the past few decades
have now enabled probing materials at the unprecedented timescales down to
those associated with quantum transition times between the stationary atomic
states [1]. All the more important in these extreme regimes is to remain
mindful of fundamental limitations imposed on the material response, both
linear and nonlinear, by the basic physical principles. In the present work we
explore the implications of causality on ultrafast nonlinear optical response
in the context of the paradigmatic Kerr effect (KE), whereby the refractive
index experienced by a probing beam depends on the intensity of light. This
effect is one of the most important nonlinear optical effects, responsible for
a great number of practically important phenomena such as self- and cross-
phase modulation of the beams, Kerr lensing, self-focusing, optical soliton
formation, optical switching and passive mode-locking [2, 3]. Since most of
these become particularly prominent in the context of ultrashort laser pulses,
it can be said that Kerr effect owes most of its relevance to the fact that it
can be often treated as an almost instantaneous interaction between the
probing pulse and the pumping one, responsible for the refractive index
changes.

The mechanism of fast Kerr effect relies on off-resonant electronic
transitions whose characteristic response times can be as short as several
femtoseconds [2, 4]. However, as we argue below, it would be a mistake to
assume that quasi-instantaneous electronic response necessarily implies that
the actual response of the probing beam to the pump intensity variations is
also instantaneous. The reason for this is causality [5] which inevitably
leads to the dispersion of response functions [6]. This relation has been
originally realized in the form of celebrated Kramers-Kronig relations [7, 8]
and has since found applications in a broad array of subjects in physics [9,
10, 11, 12, 13, 14] and engineering [15]. Unfortunately despite general
character of the relationship between causality and frequency dependence of
response, the applications of dispersive analysis in nonlinear phenomena has
been comparatively limited, save for several notable exceptions such as [16,
17, 18, 19].

In the present Letter we consider the Kerr effect mediated by quasi-
instantaneous off-resonant electronic cloud polarization and show how to
consistently apply Kramers-Kronig relations with time-dependent intensity
$I(t)$. We apply this approach to the case of two gaussian pulses and
demonstrate that dispersion introduces a considerable lag between the phase-
and amplitude response of the probe. This lag is readily measurable with
ultrafast pulses, therefore we proceed with an experimental realization in a
time-resolved Mach-Zehnder interferometer in the model system GaAs. We confirm
the presence of the dispersive lag with the magnitude and sign of the effect
being consistent with theoretical expectations.

##  II Dispersive analysis of time-resolved Kerr effect

The essence of Kerr effect is the modification of linear refractive index of
the given material in response to the square of applied electric field. In the
case of optical-frequency fast alternating applied field, the correction
$\delta n$ to the refractive index is proportional to the intensity $I$ of the
optical beam: $\delta n=n_{2}I$, where the coefficient $n_{2}$ is referred to
as either Kerr- or nonlinear refractive index. The realization that the end
result of this nonlinear phenomenon (KE) is a correction to a linear response
function (refractive index) has enabled a quantitative analysis of Kerr effect
in terms of the classical Kramers-Kronig relations [16, 17]. In particular, it
was noted that when the intensity $I$ is constant, the refractive index
acquires a constant correction which is usual Kramers-Kronig relations,
automatically applies to the frequency dependence of the nonlinear refractive
index $n_{2}(\omega)$.

In many applications however, one often deals with time-dependent, pulse-
shaped intensity profiles $I(t)$. In this case the analysis of the Kerr effect
is not as straighforward anymore (e.g. one obviously cannot simply write
$\delta n(t)=n_{2}I(t)$). The goal of this section is to extend the dispersive
analysis of Kerr effect of Sheik-Bahae et al. [16, 17] to the case of
ultrafast pulses. Although in many practical instances KE manifests itself the
self interaction of a single beam, it is convenient to think about KE as an
interaction between two distinct beams: pump and probe. Then KE can be viewed
as the change in effective refractive index, experienced by probe beam
$E_{pr}$, by the presence of the pump intensity $I_{p}$. For the purposes of
this work we will ignore the spatial walk-off between the pump and probe beams
due to group velocity dispersion. This can be justified for the case of thin
sample or sufficiently close wavelength values of the two beams. Effectively
this implies that we can ignore the spatial dispersion of light inside the
sample. As demonstrated in the Supplementary Information, this is a valid
assumption for the experimental settings used in this work.

In the absence of spatial dispersion the general third-order nonlinear
response of a system $\delta E_{NL}$ can be written in terms of the incident
(probe) field $E(t)$ as:

$\delta
E_{NL}(t)=\int\displaylimits_{-\infty}^{t}dt_{1}dt_{2}dt_{3}\,\bar{\chi}^{(3)}(t;t_{1},t_{2},t_{3})E(t_{1})E(t_{2})E(t_{3})$
(1)

To remind, in this work we limit our scope to the case of Kerr effect: a
special case of the general third-order nonlinearity where the phase and
amplitude of probe field $E_{pr}(t)$ are modified in the presence of intensity
of light $I_{p}(t)$ which, generally speaking, is time-dependent. In Kerr
effect, the probe in a certain sense is oblivious to the phase of the pump
beam. This sets KE apart from other third order nonlinear processes such as
third harmonic generation or four-wave mixing. In this case we can simplify
the equation above by introducing Kerr susceptibility $\chi^{K}$ as follows:

$\delta
E_{NL}(t)=\int\displaylimits_{-\infty}^{t}dt^{\prime}E_{pr}(t^{\prime})\int\displaylimits_{-\infty}^{t}dt^{\prime\prime}\chi^{K}(t,t^{\prime},t^{\prime\prime})I_{p}(t^{\prime\prime})$
(2)

Figure 1: Absolute value of phase-amplitude lag $|\delta T|$ in the units of
$h/\Delta_{g}$ as a function of frequency as calculated from eq.(6). Red
(blue) segments correspond to positive (negative) lag with phase response
lagging behind (ahead) of amplitude response. Inset: real (blue) and imaginary
(red) parts of the degenerate nonlinear Kerr index
$\bar{n}_{2}(\omega,\Omega=\omega)$ (see Refs.[16, 17, 20])

This dependence can be further simplified if Kerr response can be treated as
instantaneous. This is approximately the case for the Kerr effect mediated by
off-resonant electronic transitions, where response time of electrons is on
the order of $\tau_{res}\sim h/\Delta_{gap}$, where $\Delta_{gap}$ is the band
gap. The latter typically has a value on the order of 1eV, which makes
$\tau_{res}\sim 4$fs, which is much shorter than the usual width of ultrafast
pulses. If the main nonlinear interaction channel in the medium is the
cooperative effect of pump and probe beams (such as 2PA or Raman effect), the
nonlinear Kerr susceptibility $\chi^{K}$ can be written as:

$\chi^{K}(t,t^{\prime},t^{\prime\prime})=f(t-t^{\prime})\delta(t^{\prime}-t^{\prime\prime})$
(3)

This is because in the absence of real (on-shell) intermediate excitations,
the nonlinear interaction between pump and probe can only be present when both
overlap in time. Now since
$\chi^{K}(t,t^{\prime},t^{\prime\prime})=\chi^{K}(t-t^{\prime},t-t^{\prime\prime})=0$
whenever $t^{\prime}$ or $t^{\prime\prime}$ precede $t$, $f(t)=0$ for $t<0$.
Which means $f(t)$ is a causal function, and as such its Forurier transform
must satisfy Kramers-Kronig relations. The equations above can be used to
describe the Kerr effect with time-dependent pump intensity $I_{p}(t)$ as long
as it doesn’t vary too fast ($1/I_{p}\,(\partial I_{p}/\partial
t)<\Delta_{gap}/h$). This condition is satisfied automatically whenever pump
pulse can be described using slowly varying envelope approximation.

This way we reduce the full nonlinear susceptibility $\chi^{(3)}$ to a
function of single variable $f(t)$. Importantly $f(t)=0$, whenever $t<0$,
therefore it must be subject to Kramers-Kronig relations. In fact, the
function $f(t)$ is directly related to the nonlinear Kerr index
$\bar{n}_{2}(\omega,\Omega)$, where $\omega$ and $\Omega$ are the frequencies
of the probe and pump beams respectively [20]:

$f(\omega)=\int dte^{i\omega t}f(t)\approx i\left({\omega
L}/{c}\right)\bar{n}_{2}(\omega,\Omega)$ (4)

Combined, equations (2), (3) and (4) describe how to use dispersive relations
with time-dependent pump intensity $I_{p}(t)$. As an example we can compute
$\delta E_{NL}$ for the case of gaussian-shaped pump and probe pulses:

$E_{pr}(t)=E_{0}e^{-t^{2}/\tau^{2}}\cos(\Omega
t),\,\,\mathrm{}\,\,I_{p}(t)=I_{0}e^{-2(t-t_{0})^{2}/\tau_{1}^{2}}$

The modification of the (complex) refractive index of the medium implies that
the phase and amplitude of the transmitted beam will acquire a change in phase
$\Delta\phi$ and amplitude $\Delta A$ encoded in $\delta E_{NL}$. One can show
that, as a function of delay $t_{0}$ between the pump and probe pulses, the
amplitude and phase response will read ([20]):

$\displaystyle\Delta A$ $\displaystyle\propto-
I_{0}\mathrm{Im}\\{\bar{n}_{2}\\}\exp\left[-\frac{1}{\tilde{\tau}^{2}}\left(t_{0}-\frac{\nicefrac{{\partial\mathrm{Re}\\{\bar{n}_{2}\\}}}{{\partial\omega}}}{\scriptstyle
2\mathrm{Im}\\{\bar{n}_{2}\\}}\right)^{2}\right]$ $\displaystyle\Delta\phi$
$\displaystyle\propto
I_{0}\mathrm{Re}\\{\bar{n}_{2}\\}\exp\left[-\frac{1}{\tilde{\tau}^{2}}\left(t_{0}+\frac{\nicefrac{{\partial\mathrm{Im}\\{\bar{n}_{2}\\}}}{{\partial\omega}}}{\scriptstyle
2\mathrm{Re}\\{\bar{n}_{2}\\}}\right)^{2}\right]$ (5)

where $\tilde{\tau}^{2}=(\tau^{2}+\tau_{1}^{2})/2$. According to these
expressions, the dispersion of $\bar{n}_{2}$ produces a lag between the phase
and amplitude response:

$\delta
T(\omega,\Omega)=\frac{1}{2}\left[\frac{\nicefrac{{\partial\mathrm{Re}\\{\bar{n}_{2}\\}}}{{\partial\omega}}}{\scriptstyle\mathrm{Im}\\{\bar{n}_{2}\\}}+\frac{\nicefrac{{\partial\mathrm{Im}\\{\bar{n}_{2}\\}}}{{\partial\omega}}}{\scriptstyle\mathrm{Re}\\{\bar{n}_{2}\\}}\right]$
(6)

The important feature of this lag is that it only depends on the dispersion of
Kerr index $\bar{n}_{2}(\omega,\Omega)$ and is independent of pump intensity
magnitude $I_{0}$. To find $\delta T(\omega,\Omega)$ we take the complex
$\bar{n}_{2}(\omega,\Omega)$ as obtained by dispersive analysis assuming
$\mathrm{Im}\\{\bar{n}_{2}\\}$ is mostly due to two-photon absorption
(2PA)[20, 17, 16]. In Fig.1 we plot “degenerate” $\delta T$ taken at
$\omega=\Omega$. For reference we also plot the degenerate
$\bar{n}_{2}(\omega,\Omega=\omega)$ in the inset of Fig.1. As is evident from
this figure, the lag is most prominent near one- and two-photon resonances.
These frequencies correspond to special points $\bar{n}_{2}(\omega,\Omega)$ in
the complex plane $\omega$ [20]. Somewhat less expected is the significant lag
around $\hbar\omega\approx 0.7\Delta_{gap}$ which corresponds to the zero-
crossing of $\mathrm{Re}\\{\bar{n}_{2}\\}$ (inset Fig.1). Overall the analysis
demonstrates that the lag $\delta T$ is quite significant especially near
certain special points, therefore we are turning now to its experimental
observation.

Figure 2: A) Schematic layout of the time-resolved Mach-Zehnder
interferometer. Pump and probe beams are near-degenerate ($\lambda=$ 1060nm
and 1028nm respectively). Transient pump-induced changes in refractive index
are detected as shift in the interference pattern recorded by a CCD camera; B)
sample interferogram as recorded by CCD camera; C) 2D Fourier intensity of the
interferogram in B). The real-space shift of the interference pattern is
inferred from the change of the phase of complex amplitude at the peak. D) A
schematic depiction of pump and probe pulses overlapping in time.

##  III Experiment

In order to experimentally observe the lagging effect, we study GaAs where
Kerr effect is known to be dominated by electronic hyperpolarizability [4]. To
measure the nonlinear refractive index in a time-resolved manner we construct
a pump-probe setup based on a Mach-Zehnder interferometer (MZI) as shown in
Fig. 2A. Here one arm of the interferometer is the probe beam that passes
through the GaAs sample,while the other is used a reference. Both beams are
then combined on a CCD camera that records the interferogram. Pump is a
separate beam with a slightly different wavelength and tunable delay $t_{0}$
with respect to probe. Changes in the real and imaginary parts of the
refractive index induced by the pump, are recorded as respective changes in
the position and amplitude of the interference pattern on the CCD matrix:

$I({\bf r},t)\propto|E_{pr}||E_{ref}|\cos({\bf q}\cdot{\bf r}+\phi_{g}+2\pi
n(t_{0})L/\lambda)$ (7)

here $|E_{pr}|$ and $|E_{ref}|$ are the amplitudes of the electric field in
the probe and reference arms; ${\bf q}={\bf k_{1}}-{\bf k_{2}}$ is the
difference between the corresponding beams’ wave vectors ${\bf k_{1}}$ and
${\bf k_{2}}$; $\phi_{g}$ is the static phase difference determined by
interferometer geometry. The last term is the phase accumulated inside the
sample with a thickness $L$ and refractive index $n$ which is in general a
complex number. To extract the phase and amplitude information, we Fourier-
transform each interferogram and evaluate the complex amplitude of the Fourier
peak corresponding to the main peak in the interference pattern (Fig.2B and C)
$K(t_{0})=|K(t_{0})|e^{i\phi(t_{0})}$ for each pump-probe delay $t_{0}$. The
change in the absolute value and phase provide a direct measure of the pump-
induced change in imaginary and real part of the refractive index:

Figure 3: A) Pump-induced amplitude (blue) and phase (red) change as a
function of pump-probe delay for $xx$ configuration at $I_{0}=4.7$W$/$cm2.
Solid lines: gaussian fits to the peaks of corresponding colors; B) the same
for $xy$ configuration; C) The phase-amplitude lag as a function of intensity
$I_{0}$ is the $xx$\- and; D)$xy$ configurations.

The time dependence of pump-induced changes in phase and amplitude of the
transmitted beam is shown in Fig.3. In Fig.3A the pump and probe are polarized
parallel to the [100] crystalline axis of GaAs (“$xx$” configuration) and in
Fig.3B they are crossed (“$xy$”). The phase shift indicates that nonlinear
Kerr index two distinct components: a fast peak and a slowly growing “tail”.
The quasi-instantaneous character of the peak points towards off-resonant
electronic polarizability as its origin, unlike the tail which will be
discussed later. Using phase shift and sample thickness $L$ we can define an
effective refractive index change as $\overline{\delta
n}=\nicefrac{{\Delta\phi}}{{2\pi}}\cdot\nicefrac{{\lambda}}{{L}}$. Due to
slight wavelength mismatch between pump and probe, this effective index cannot
be directly used to accurately evaluate the Kerr index, however it can still
provides a reasonable estimate (see [20]). With this in mind we plot
$\overline{\delta n}$ against peak pump intensity in Fig.4A to find the values
for the effective nonlinear Kerr index in both $xx$ and $xy$ configurations:
$\overline{n}^{xx}_{2}=(-2.8\pm 0.5)\times 10^{-15}$cm2/W ;
$\overline{n}^{xy}_{2}=(-1.5\pm 0.3)\times 10^{-15}$cm2/W. Numeric simulations
indicate that these numbers underestimate the actual Kerr indices by about a
factor of 2 ([20]). Comparing these estimates against $n_{2}$ values measured
at other wavelengths with similar pulse duration (e.g.
$n_{2}(\lambda=1.7\mu$m$)\approx 3\times 10^{-13}$cm2/W [21]), shows that our
working wavelengths $\lambda\approx 1\mu$m are close to the zero-crossing
value of $\mathrm{Re}\\{\bar{n}_{2}\\}$ which implies that one should expect
to see a significant lagging effect (see eq.(6)).

Amplitude response $\Delta A$ is directly related to the transient
differential absorption $\Delta T/T$ plotted in Fig.3A and Fig.3B in blue.
Similarly to the main peak in phase response, $\Delta T/T$ exhibits quasi-
instantaneous behavior and its sign is negative in agreement with 2PA. The
fact that $n_{2}<0$ and $\lambda_{\mathrm{pump}}>\lambda_{\mathrm{probe}}$
(see Supplemental Material [20] and Fig.2A) rules out coherent power transfer
between probe and pump beams (“two-beam coupling” [2]) as an alternative to
2PA as a mechanism for the probe amplitude variations.

Having established that temporal peaks in both $\Delta A$ and $\Delta\phi$ are
coming from intrinsic electronic response of GaAs we finally turn to the delay
between amplitude $\Delta T/T$ and phase $\phi$ responses that can be clearly
seen in Fig.3A and Fig.3B. As shown in Fig.3C and Fig.3D the magnitude of the
delays are $\delta T_{xx}\approx 100$fs and $\delta T_{xy}\approx 120$fs for
the parallel and crossed configurations respectively. These values correspond
to $\delta T\cdot\Delta_{gap}/h\approx 30$ which is consistent with the
simplified model in Fig.1 (including the sign). Most importantly the lags do
not show any dependence on pump intensity in line with the dispersive
analysis. This observation rules out alternative lagging mechanisms based on
nonlinear phenomena. This observation of a clear intensity-independent lag
between phase and amplitude responses in agreement with theoretical
expectations is the main result of this work.

Figure 4: A) Dependence of the effective instantaneous nonlinear refractive
index $\overline{\delta n}$ on peak pump intensity $I_{0}$(see text). Solid
lines are fits including saturation intensity $I_{sat}=0.8\times
10^{11}$W/cm2,
$\overline{\delta}n(I_{0})=\overline{n}_{2}I_{0}/(1+I_{0}/I_{sat})$; B) Pump
intensity dependence of the long-lived tail in the phase response in Fig.3. In
both panels the blue and red markers/lines correspond to $xx$ and $xy$
configurations respectively.

##  IV Discussion

As mentioned above, in addition to the quasi-instantaneous peak, the phase
response in Fig.3 features a prominent long lived “tail” that is building up
over the course several picoseconds and stays constant for the entire
accessible delay range (see Fig.1 in Supplemental Material [20]). This tail
clearly has a different origin as compared to the to the main peak. This can
already be seen in the different dependence on pump intensity $I_{p}$ between
the peak and tail shown in Fig.4. While the tail grows linearly with $I_{p}$,
the peak exhibits a clear saturation with a characteristic intensity
$I_{sat}=8\times 10^{10}$ W/cm2 (both $xx$ and $yy$). Next, extended lifitime
implies that the tail owes its origin to some real excitations produced by the
pump. The linear pump intensity dependence (Fig. 4B) excludes nonlinear
absorption mechanisms, such as 2PA, as the possible origin of the tail;
thermo-optic effect should also be excluded, since thermo-optic coefficient in
GaAs is positive [22]. With these considerations we believe the most likely
origin of the tail is the photo-excitation of the deep defect states (as
previously reported in [23]) followed by long-lived low-density electron-hole
plasma formation. The low plasma frequency ensures negative contribution to
the refractive index.

The presence of the tail illustrates the usefulness of time resolved methods
for determining the intrinsic electronic Kerr index $n_{2}$. Our value for
nonlinear Kerr index at $\lambda=1028$nm can be compared to a significantly
larger one obtained in an earlier measurement at $1060$nm by Said et. al.
[24]. The origin of the discrepancy likely lies in the fact that in [24] they
were using 40ps laser pulses. At such timescales the tail will be giving an
integrated contribution (see Fig.3A,B) that will overwhelm the intrinsic
instantaneous response. Therefore we disregard this number in the present
study and rely on the values obtained with ultrashort pulses as in [21].

Finally, apart from the amplitude-phase lag the physics described above have
further practically important implications. Indeed, the common case of a self-
Kerr effect of a beam can be thought of as a special case of eq.(5) with
$t_{0}=0$. In this case as these equations indicate, the Kerr changes of both
amplitude and phase will be systematically less than what one would expect
without taking dispersive effects in account. Here we use the full expression
for $\bar{n}_{2}(\omega,\Omega)$ (see [20]) to evaluate the correction factors
$F_{\mathrm{ph}}$ and $F_{\mathrm{amp}}$ for phase and amplitude changes
respectively, which we define as:

$F_{ph}=|\Delta\phi_{\mathrm{max}}/\Delta\phi_{0}-1|,\,\,\,F_{amp}=|\Delta\mathrm{A}_{\mathrm{max}}/\Delta
A_{0}-1|$

where $\Delta\phi_{\mathrm{max}}$ and $\Delta\mathrm{A}_{\mathrm{max}}$ are
the maximum phase and amplitude changes achievable by tuning $t_{0}$;
while$\Delta\phi_{0}$ and $\Delta A_{0}$ are the same quantities evaluated at
$t_{0}=0$. These correction factors show by how much the Kerr response of a
system can exceed the nominal values calculated without regard to the
dispersive effects discussed above. The correction factors calculated as
function of frequency for a generic pulse duration $\tau=\tau_{1}=70$fs is
plotted in Fig.5. As can be seen from the plot, the response renormalization
becomes significant near the $\mathrm{Re}\\{n_{2}\\}$ sign-switching
wavelength and near single- and two-photon resonances.

Figure 5: Dispersion-induced renormalization factors for amplitude (blue) and
phase (purple) response of a Kerr-active medium (see text).

##  V Conclusion

To summarize, we propose a procedure to extend the dispersive analysis of
Sheik-Bahae et al. [16, 17] to the situations where the pump intensity cannot
be approximated as quasi-static and show that the resulting causal Kerr
susceptibility $\chi_{K}(t,t^{\prime},t^{\prime\prime})$ can be directly
related to the dispersive Kerr index $\bar{n}_{2}(\omega,\Omega)$. We find
that dispersive effects can lead to significant modifications in phase and
amplitude dynamics which lead to a significant underestimation of the peak
Kerr response of a nonlinear medium. Our results shed light on the fundamental
effects of causality for the ultrafast nonlinear phenomena and pave way for
their efficient application.

###### Acknowledgements.

Acknowledgments. The work was supported by the Institute of Science and
Technology Austria (ISTA). We thank Prof. John M. Dudley, Dr. Ugur Sezer and
Dr. Artem Volosniev for valuable discussions.

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##  VI Supplementary Information

Experimental details. The GaAs sample used in this study is a 675$\mu$m thick,
(100)-oriented, semi-insulating GaAs wafer ( Freiberger Compound Materials
GmbH). An amplified femtosecond laser system (Light Conversion Pharos) coupled
with an optical parametric amplifier (Light Conversion Orpheus) is used as the
principal laser source. The laser produces a train of pulses centred at 1028nm
with a repetition rate of 3kHz, pulse duration of 290fs and a pulse energy of
2mJ. A small part of the beam (5%) is sent to the Mach-Zehnder interferometer
(MZI) through a variable attenuator consisting of a half-wave plate (HWP) and
a Glan-Taylor polarizer; the remaining part of the beam pumps the optical
parametric amplifier with a tunable output wavelength.

In this work, the fundamental wavelength of $1028$nm is used as a probe while
the 1060nm idler output from the optical parametric amplifier is used as the
pump beam. The pump wavelength was chosen such that it is nearly degenerate
with probe, but is still different enough to be effectively separable from the
latter by means of a spectral filter. The delay between pump and probe is
controlled with a motorized linear stage. Both pump and probe pass through a
set of wave-plates and polarizers to attenuate their respective powers and to
clean the polarizations. The pump beam is then passed through a additional HWP
in order to control its polarization. The pump and probe beams are loosely
focused and spatially overlapped in the sample in a slightly non-collinear
arrangement. Care was taken to make sure that the probe diameter
(D${}_{\mathrm{probe}}\approx 120\mu$m (1/$e^{2}$)) at the sample position is
considerably smaller than that of the pump, the latter being evaluated as
D${}_{\mathrm{pump}}\approx 360\mu$m (1/$e^{2}$) by means of razor-blade
method. This is made to ensure that pump intensity is a well defined quantity.
The probe light leaving the output port of MZI passes through a $1030\pm 5$nm
bandpass filter and the resulting interferograms are recorded by a CCD camera
(PointGrey Research Inc. CM3-U3-50S5M). The pump intensity was tuned in a
broad range up to $\sim 5\times 10^{11}$W/cm2, while the probe intensity was
kept fixed at $I_{\mathrm{probe}}=4\times 10^{6}$ W/cm2.

The pump-induced change in complex refractive index is obtained through
measuring the changes in the phase $\phi(t_{0})$ and amplitude $|K(t_{0})|$ of
the corresponding Fourier peak as function of delay $t_{0}$ between the pump
and probe beams:

$\displaystyle\mathfrak{Re}\\{\Delta n(t_{0})\\}$
$\displaystyle=\frac{\lambda}{2\pi
L}\left\\{\phi(t_{0})-\phi(-\infty)\right\\}$
$\displaystyle\mathfrak{Im}\\{\Delta n(t_{0})\\}$
$\displaystyle\approx-\frac{\lambda}{2\pi
L}\frac{|K(t_{0})|-|K(-\infty)|}{|K(-\infty)|}$

here by “$t=-\infty$” we mean a large enough negative delay when probe beam
precedes the pump. The delay between the pump and probe pulses $t_{0}$ is
controlled by a fast motorized delay stage (Newport LTLMS800). To extract the
time dependence of transient changes to $n_{2}$ the stage is moved
incrementally and an interferogram is recorded for each position of the stage.
In order to avoid the detrimental effects of mechanical vibrations, for every
delay stage position we record two consecutive interferograms: one at the
actual desired position and the other at a fixed reference position pre-dating
pump arrival (“$t=-\infty$”). The changes in both the amplitude and phase are
found from the difference between the two interferograms. Given the high speed
of the delay stage (500mm/s) this method helps minimize the effect of the low
frequency noise coming from mechanical deformations of MZI.

Signal for large pump-probe delay values. Here we study the temporal behavior
of the “tail” whose onset is shown in Fig.3 of the main text. A long scan has
been performed and the fig.6 shows that the signal remains virtually unchanged
for pump-probe delays of multiple tens of picoseconds. On the one hand, the
fact the feature has a long lifetime means that the pump is producing real
excitation, on the other, the pump photon energy of 1.17eV which is less that
GaAs band-gap energy of about $\Delta_{\mathrm{GaAs}}\approx$1.5eV. The most
obvious mechanism for absorption of sub-gap photons is two-photon absorption.
However this is ruled out by the scaling of the “tail” magnitude with pump
intensity, which is linear, as can be seen in Fig.4B of the main text. Here it
might be important to clarify, that this ruled-out real two-photon absorption
is different from the virtual two photon process giving rise to the coherent
feature around t=0 that is the main subject of this work. In the ruled-out two
photon process both photons would be coming from the pump beam, while in the
t=0 virtual process one of the two photons is still provided by the pump while
the other comes from the probe. Therefore we can conclude the observed “tail”
signal must originate from single-photon absorption. While such process is
nominally prohibited for sub-gap frequencies, this is only strictly true for
ideal samples. In realistic materials there always exist deep donor levels
which can be excited with sub-gap photons. Similar processes have been
reported reported previously in e.g. ref.[23].

Figure 6: Pump-induced phase shift in GaAs sample as a function of pump-probe
delay in an extended range.

Propagation of nearly degenerate pulses around $\lambda\approx 1\mu$m in GaAs.
The difference in wavelength between the pump and probe implies that the two
have different group velocities. This gives rise to the walk-off between
pulses as they propagate through the sample, which means that the effective
interaction length is smaller than sample thickness $L$. In the main text the
effective change in refractive index is defined as $\overline{\delta
n}=(\Delta\phi/2\pi)\cdot(\lambda/L)$ from which the effective Kerr index is
inferred as $\overline{\delta n}=\overline{n}_{2}I_{0}$, $I_{0}$ being (peak)
pump pulse intensity. Now, since the interaction length is in fact less than
$L$, $\overline{n}_{2}$ always underestimates the actual Kerr index value.
Here we perform a simulation of pump-probe pulse interaction in order to find
out the extent to which $\overline{n}_{2}$ falls behind $n_{2}$ in magnitude
when walk-off effects are taken into account.

For the present purpose we ignore absorption effects and go ahead with the
standard equations for cross-phase modulation [Agrawal]:

$\frac{\partial A_{j}}{\partial z}+\beta_{1j}\frac{\partial^{2}A_{j}}{\partial
t^{2}}=in_{2}(\omega_{j}/c)\left\\{f_{jj}|A_{j}|^{2}+2f_{jk}|A_{k}|^{2}\right\\}A_{j}$
(8)

here the index $j=1,2$ corresponds to the probe and pump pulses respectively;
$A_{j}(z,t)$ is the amplitude of $j$-th pulse, $z$ and $t$ being the distance
in the propagation direction and time respectively; $\beta_{1j}$ and
$\beta_{2j}$ are the group velocity and GVD coefficients of $j$-th pulse
respectively; while $f_{jk}$ is the overlap coefficient (having the meaning of
the inverse of the effective overlap area) between the $j$-th and $k$-th
pulses with modal distributions $F_{j}(x,y)$:

$f_{jk}=\frac{\int dxdy|F_{j}(x,y)|^{2}|F_{k}(x,y)|^{2}}{\left(\int
dxdy|F_{j}(x,y)|^{2}\right)\left(\int dxdy|F_{k}(x,y)|^{2}\right)}$

To compare with experiment the quantity of interest is $A_{1}(L,t)$, that is
the amplitude of probe field at the exit from the sample. To mimic the
experimental setup in which we measure the shifts of interference patterns, we
first calculate the amplitude $A^{0}_{1}(L,t)$ after the sample in the absence
of pump pulse. We will use it as a reference pulse playing the role of the
pulse in the reference arm of the Mach-Zehnder interferometer if Fig.2 of the
Main Text. Next, we add the pump pulse $A_{2}$. The temporal position of the
pump in relation with probe (pump-probe delay) is kept as a free parameter
$\tau_{\mathrm{delay}}$ and for each value, we calculate the field amplitude
of the probe pulse that has interacted with the pump at the exit of the sample
$A_{1}(L,t,\tau_{\mathrm{delay}})$.

In order to simulate the experimental interference pattern we calculate the
“interference” between sample field $A_{1}(L,t,\tau_{\mathrm{delay}})$ the
reference field $A^{0}_{1}(L,t,\phi)\equiv A^{0}_{1}(L,t)\exp(i\phi)$ retarded
by some variable phase $\phi$. The meaning of this phase is to simulate the
actual interference pattern where e.g. depending on the point on the screen
the probe and reference beams have different mutual phase dictated by the
wavelength of the beams and their intersection angle. It is this space-
dependent phase that gives rise to the interference pattern. With this in mind
we calculate the cumulative intensity $G(\phi)$ as:

$G(\phi,\tau_{\mathrm{delay}})=\int
dt\left(\left(A_{1}(L,t,\tau_{\mathrm{delay}})\right)^{*}A^{0}_{1}(L,t)\exp(i\phi)+{A_{1}(L,t,\tau_{\mathrm{delay}})}\left(A^{0}_{1}(L,t)\right)^{*}\exp(-i\phi)\right)$
(9)

Figure 7: “Interference patterns” as a function of delay between pump and
probe with (right) and without (left) group index mismatch between pump and
probe pulses taken into account. Simulation was carried out at peak pump
intensity $I_{0}=9.8\times 10^{14}$ W/m2 and $n_{2}=1.2\times 10^{-19}$m2/W.

In Fig.7 we plot $G(\phi,\tau_{\mathrm{delay}})$ with and without group-index-
mismatch effects taken into account. Here we use the following parameters:

  * Wavelength: $\lambda_{\mathrm{probe}}=1030$nm (probe); $\lambda_{\mathrm{pump}}=1060$nm (pump);

  * Pulse duration (pump and probe): $\tau_{\mathrm{pulse}}=270$fs

  * Spot area (gaussian profile): Aeff1 = $\pi(45\times 10^{-6})^{2}$ m2 (probe); Aeff1 = $\pi(180\times 10^{-6})^{2}$ m2 (pump);

  * Sample thickness: $L=700\mu$m; Kerr nonlinear refractive index: $n_{2}=1.2\times 10^{-19}$ m2/W

  * Group refractive index: $n_{g1}=3.9899$ (probe); $n_{g2}=3.9254$ (pump)(ref. [Skauli2003])

  * GVD coefficient: $\beta_{11}=4.5532$ (probe); $\beta_{12}=3.8172$ (pump) (ref. [Skauli2003]);

As can be seen, introducing the walk-off does not give give rise to an orders-
of magnitude change in peak phase shift. To see this more clearly, in fig.8 we
plot $G(\phi,\tau_{\mathrm{delay}})$ with $\tau_{\mathrm{delay}}$ fixed at
maximum phase shift position. The peak phase shifts with and without walk-off
compare as 0.65 to 1. It is in place to mention that the peak phase
$\Delta\phi\approx 1.2$rad shift in this simulation for $I_{0}\approx
10^{11}$W/cm2 is seemingly smaller than the experimentally measured
$\Delta\phi_{\mathrm{exp}}\approx 0.5$rad, however this discrepancy is
actually due to the fact that our simulation does not take saturation effects
into account (see Fig.4 in the main text).

When walk-off effects are taken into account, the phase shift as can be seen
in Fig.8 is reduced by about a factor of 2. This means that the ignoring the
walk-off underestimates Kerr index only by the same factor of 2. This
justifies the usage of the effective Kerr index in the main text for the
purpose of the order-of-magnitude estimate of its actual value.

Figure 8: Comparison of the “interference patterns” between the probe and
reference beams (see main text of SM) in the absence of pump (blue); pump

###  VI.1 Dispersive relations for quasi-instantaneous off-resonant electron-
mediated Kerr effect

Relationship between Kerr susceptibility $\chi^{K}$ and nonlinear refractive
index. At this point it is important to establish the relation between the
function $f(t)$ and nonlinear Kerr index $n_{2}$. With the assumptions made
above one can express the field amplitude after the sample $E_{out}$ in terms
of incoming field $E_{in}(t)=E_{0}\exp(-i\omega t)$ as follows:

$E_{out}(t)=E_{0}\exp(-i\omega t)\times\exp(i\,2\pi
n_{0}(\omega)\nicefrac{{L}}{{\lambda}})$ (10)

where $L$ is the sample thickness; $\lambda=(c\,\omega)/2\pi$ is the
wavelength of the probing beam (in vacuum); and $n_{0}(\omega)$ is the linear
refractive phase index of the medium which is in general a complex number. As
was noted by Sheik-Bahae et al. (ref.[16]), when pump intensity $I_{p}$ is
independent of time, the net result of nonlinear Kerr effect is a modification
of the (linear) refractive index $n(\omega)$:

$E_{out}(t)=E_{0}(t)\exp\\{i\,2\pi\left[n_{0}(\omega)+n_{2}(\omega,\Omega)I_{p}\right]\nicefrac{{L}}{{\lambda}}\bigr{\\}}$
(11)

where $n_{2}(\Omega,\omega)$ is the nonlinear Kerr index of the sample
material which in general depends on both pump ($\Omega$) and probe ($\omega$)
frequencies. If we define $\delta E_{NL}=E_{out}(I_{p})-E_{out}(I_{p}=0)$ as
the nonlinear response of the medium, we can write:

$\delta E_{NL}(t)\approx E_{0}(t)\times\exp(i\,2\pi
n_{0}(\omega)\nicefrac{{L}}{{\lambda}})\cdot\left(i\left(\raise
2.15277pt\hbox{$\omega L$}\kern-1.00006pt/\kern-1.49994pt\lower
1.07639pt\hbox{$c$}\right)\bar{n}_{2}(\omega,\Omega)I_{p}\right)$ (12)

Noting that in practice it is usually the case that $n_{0}(\omega)$ varies by
about $\sim 15$% in the relevant frequency range of $\hbar\omega<\Delta_{gap}$
we follow Sheik-Bahae et al in refs.[16, 17], and ignore the frequency
dependence of linear refractive index putting $n_{0}(\omega)=n_{0}$. Then the
contribution of the linear term in the above expression is just a constant
phase shift. In the experiment described in this work this term results in a
constant phase offset of the beam in the probing arm of the interferometer
independent of pump, which is subtracted away according to the measurement
protocol. Alternatively one can imagine putting the same amount of sample
material in the reference arm of the interferometer with no pump beam. Then
the linear phase shift is equal in both arms and cancels out on the
interferogram. Either way, the equilibrium linear phase shift is
inconsequential for our analysis and will be dropped from now on. With this in
mind we can re-write eq.(12) as:

$\delta E_{NL}(\omega)=i\left(\raise 2.15277pt\hbox{$\omega
L$}\kern-1.00006pt/\kern-1.49994pt\lower
1.07639pt\hbox{$c$}\right)\bar{n}_{2}(\omega,\Omega)I_{p}E_{pr}(\omega)$ (13)

Now take eq.(2) and eq.(3) and substitute $I_{p}(t)=I_{0}$. Integrating over
delta-function we get:

$\delta
E_{NL}(t)=\int\displaylimits_{0}^{t}dt^{\prime}E_{pr}(t^{\prime})f(t-t^{\prime})I_{0}$

and

$\delta E_{NL}(\omega)=\int dt\exp(i\omega t)\delta
E_{NL}(t)=f(\omega)E_{pr}(\omega)I_{0}$ (14)

where $f(\omega)=\int dt\exp(i\omega t)f(t)$. Comparing this with eq.(13) we
see that

$f(\omega)=i\left(\raise 2.15277pt\hbox{$\omega
L$}\kern-1.00006pt/\kern-1.49994pt\lower
1.07639pt\hbox{$c$}\right)\bar{n}_{2}(\omega,\Omega)$ (15)

which is the relationship between $f(t)$ and nonlinear refractive (Kerr) index
$\bar{n}_{2}(\Omega,\omega)$, where $\omega$ and $\Omega$ are the central
frequencies of the probe and pump pulses respectively.

Application to the case of two gaussian pulses. To see implications of the
formalism described in the previous paragraph, let’s consider the case when
both pump and probe have a gaussian profile shifted relative to each by
$t_{0}$:

$\displaystyle I_{p}(t)=$
$\displaystyle\,I_{0}\exp(-2(t-t_{0})^{2}/\tau_{1}^{2})$ (16) $\displaystyle
E_{pr}(t)=$ $\displaystyle\,E_{0}\exp(-t^{2}/\tau^{2})\cos(\omega_{0}t)$ (17)

then

$\delta E_{NL}(t,t_{0})=\int
dt_{1}f(t-t_{1})I_{p}(t_{1})E_{pr}(t_{1})\propto\\\ \propto
I_{0}E_{0}\int\frac{d\omega}{2\pi}f(\omega)\left\\{\exp\left[\frac{\left(2t_{0}/\tau_{1}^{2}+i(\omega+\omega_{0})/2\right)^{2}-\frac{2t_{0}^{2}}{\tau_{1}^{2}}-i\omega
t}{2/\tau_{1}^{2}+1/\tau^{2}}\right]+\exp\left[\frac{\left(2t_{0}/\tau_{1}^{2}+i(\omega-\omega_{0})/2\right)^{2}-\frac{2t_{0}^{2}}{\tau_{1}^{2}}-i\omega
t}{2/\tau_{1}^{2}+1/\tau^{2}}\right]\right\\}$ (18)

In practice, in order to characterize a rapidly oscillating pulse $E(t)$ one
can interfere it with some reference pulse $E_{\mathrm{ref}}(t,\theta)$ with
same central frequency as the characterized pulse, and a controllable phase
$\theta$ relative to it. The measured signal is the total energy of the mixed
pulse $\delta\mathcal{E}(\theta)\propto\int
dt(E(t)+E_{\mathrm{ref}}(t,\theta))^{2}$. If the phase or amplitude of $E(t)$
changes for some reason, this will in general affect $\delta\mathcal{E}$. More
specifically, the phase and amplitude shifts of $E(t)$ are correspondingly
proportional to:

$\displaystyle\Delta\phi\propto\,\delta\mathcal{E}(\theta=0)$
$\displaystyle-\delta\mathcal{E}(\theta=\pi)$ $\displaystyle\Delta
A\propto\,\delta\mathcal{E}(\theta=\pi/2)$
$\displaystyle-\delta\mathcal{E}(\theta=-\pi/2)$ (19)

Indeed, as an illustration one can imagine two laser beams being interfered at
a screen while intersecting at some finite angle. Then if the amplitude of one
of the beams has changed, this will affect the contrast ratio of the
interference pattern, i.e. difference in intensity between the maximum (when
mutual phase difference $\theta=0$) and minimum intensity positions
(destructive interference, phase $\theta=\pi$). Likewise, the change in phase
in one of the beams results in the shift of the interference pattern as a
whole. The magnitude of the latter can be found as the difference in intensity
changes at the positions of the highest spatial gradient of intensity on the
pattern, i.e. when $\theta=\pm\pi/2$.

The relevant part of $\delta\mathcal{E}(\theta)$ is the one that comes from
the mixing of $E(t)$ and $E_{\mathrm{ref}}(t)$:

$\delta\mathcal{E}_{\mathrm{mix}}(\theta)=2\int dtE(t)E_{\mathrm{ref}}(t)$
(20)

note that since $\theta$-dependence of $\delta\mathcal{E}(\theta)$ is entirely
due to $\delta\mathcal{E}_{\mathrm{mix}}(\theta)$ one can substitute the
former for the latter in eq.(19).

In our experiment $E(t)=E_{pr}+\delta E_{NL}(t)$ with $E_{pr}$ and $\delta
E_{NL}(t)$ defined in eq.(17) and eq.(18) respectively;
$E_{\mathrm{ref}}(t,\theta)=E_{pr}(t+\theta/\omega_{0})$. Now putting these
into $\delta\mathcal{E}_{\mathrm{mix}}(\theta)$ we write:

$\displaystyle\begin{multlined}\delta\mathcal{E}_{\mathrm{mix}}(t_{0},\theta)=\mathrm{const}\cdot
E_{0}^{2}I_{0}\int\displaylimits_{-\infty}^{+\infty}dt\int\displaylimits_{-\infty}^{+\infty}\frac{d\omega}{2\pi}f(\omega)\left[e^{-\nicefrac{{t^{2}}}{{\tau^{2}}}+i\omega_{0}t+i\theta}+e^{-\nicefrac{{t^{2}}}{{\tau^{2}}}-i\omega_{0}t-i\theta}\right]\cdot\\\
\cdot\left[\exp\left\\{\frac{\left[\nicefrac{{2t_{0}}}{{\tau_{1}^{2}}}+\nicefrac{{i(\omega+\omega_{0})}}{{2}}\right]^{2}}{\nicefrac{{2}}{{\tau_{1}^{2}}}+\nicefrac{{1}}{{\tau^{2}}}}-\frac{2t_{0}^{2}}{\tau_{1}^{2}}-i\omega
t\right\\}+\exp\left\\{\frac{\left[\nicefrac{{2t_{0}}}{{\tau_{1}^{2}}}+\nicefrac{{i(\omega-\omega_{0})}}{{2}}\right]^{2}}{\nicefrac{{2}}{{\tau_{1}^{2}}}+\nicefrac{{1}}{{\tau^{2}}}}-\frac{2t_{0}^{2}}{\tau_{1}^{2}}-i\omega
t\right\\}\right]\end{multlined}\delta\mathcal{E}_{\mathrm{mix}}(t_{0},\theta)=\mathrm{const}\cdot
E_{0}^{2}I_{0}\int\displaylimits_{-\infty}^{+\infty}dt\int\displaylimits_{-\infty}^{+\infty}\frac{d\omega}{2\pi}f(\omega)\left[e^{-\nicefrac{{t^{2}}}{{\tau^{2}}}+i\omega_{0}t+i\theta}+e^{-\nicefrac{{t^{2}}}{{\tau^{2}}}-i\omega_{0}t-i\theta}\right]\cdot\\\
\cdot\left[\exp\left\\{\frac{\left[\nicefrac{{2t_{0}}}{{\tau_{1}^{2}}}+\nicefrac{{i(\omega+\omega_{0})}}{{2}}\right]^{2}}{\nicefrac{{2}}{{\tau_{1}^{2}}}+\nicefrac{{1}}{{\tau^{2}}}}-\frac{2t_{0}^{2}}{\tau_{1}^{2}}-i\omega
t\right\\}+\exp\left\\{\frac{\left[\nicefrac{{2t_{0}}}{{\tau_{1}^{2}}}+\nicefrac{{i(\omega-\omega_{0})}}{{2}}\right]^{2}}{\nicefrac{{2}}{{\tau_{1}^{2}}}+\nicefrac{{1}}{{\tau^{2}}}}-\frac{2t_{0}^{2}}{\tau_{1}^{2}}-i\omega
t\right\\}\right]$ (23)

dropping the oscillating terms (rotating wave approximation) and integrating
over $t$ gives:

$\delta\mathcal{E}_{\mathrm{mix}}(t_{0},\theta)\approx\mathrm{const}\cdot
E_{0}^{2}I_{0}\int\frac{d\omega}{2\pi}f(\omega)\cdot\\\
\cdot\left[\exp\left\\{-\frac{2t_{0}^{2}}{\tau_{1}^{2}}-\left(\frac{\omega-\omega_{0}}{2}\right)^{2}\tau^{2}+i\theta+\frac{\left[2t_{0}/\tau_{1}^{2}+i(\omega-\omega_{0})/2\right]^{2}}{2/\tau_{1}^{2}+1/\tau^{2}}\right\\}+(\omega_{0}\rightarrow-\omega_{0},\theta\rightarrow-\theta)\right]$
(24)

This integral can be evaluated by using steepest descent method. However,
noticing that as long as $\tau\gg h/\Delta_{gap}$, $f(\omega)$ is a slowly
varying function as compared to the gaussian in the rectangular brackets
above. Therefore the latter can be approximated as delta-function and the
integral can then be evaluated trivially to give:

$\delta\mathcal{E}_{\mathrm{mix}}(t_{0},\theta)\approx\mathrm{const}\cdot
E_{0}^{2}I_{0}\cdot
e^{-\left(\nicefrac{{t_{0}}}{{\bar{\tau}}}\right)^{2}}\cdot\left[f(\omega_{0}+i\eta)e^{i\theta}+f(-\omega_{0}+i\eta)e^{-i\theta}\right]$
(25)

here we introduced $\bar{\tau}^{2}=(\tau^{2}+\tau_{1}^{2})/2$ and
$\eta=t_{0}/\bar{\tau}^{2}$. Now substituting this into eq.(19) and using
eq.(15)we can write down the amplitude $\Delta A$ and phase response
$\Delta\phi$ as a function of pump-probe delay $t_{0}$ in terms of nonlinear
refractive index $n_{2}(\omega)$ as:

$\displaystyle\Delta\phi(t_{0})\propto$
$\displaystyle\,I_{0}\cdot\left[-n_{2}(\omega_{0}+i\eta)+n_{2}(-\omega_{0}+i\eta)\right]$
$\displaystyle\Delta A(t_{0})\propto$ $\displaystyle\,I_{0}\cdot
i\left[n_{2}(\omega_{0}+i\eta)+n_{2}(-\omega_{0}+i\eta)\right]$ (26)

For typical pulses $\eta\ll\omega_{0}$, therefore the expressions above can be
evaluated to a good precision by means of Taylor expansion near
$\pm\omega_{0}$. In general $n_{2}(\omega)$ is a complex-valued function of
$\omega$ on a complex plane, however given that away from the branching points
at $\omega_{0}=0.5\Delta_{gap}$ and $\omega_{0}=\Delta_{gap}$ $n_{2}(\omega)$
analytical and thus satisfies Cauchy-Riemann conditions. Then it is sufficient
to know how $n_{2}(\omega)$ depends on $\omega$ on real axis alone to find
$n_{2}$ values in its immediate vicinity:

$\displaystyle\Delta A(t_{0})\propto$
$\displaystyle\,I_{0}\,\mathrm{Im}\\{n_{2}(\omega_{0})\\}\cdot\exp\left[-\frac{1}{\bar{\tau}^{2}}\left(t_{0}-\frac{\partial(\mathrm{Re}\\{n_{2}\\})/\partial\omega}{2\mathrm{Im}\\{n_{2}\\}}\right)^{2}\right]$
$\displaystyle\Delta\phi(t_{0})\propto$
$\displaystyle\,I_{0}\,\mathrm{Re}\\{n_{2}(\omega_{0})\\}\cdot\exp\left[-\frac{1}{\bar{\tau}^{2}}\left(t_{0}+\frac{\partial(\mathrm{Im}\\{n_{2}\\})/\partial\omega}{2\mathrm{Re}\\{n_{2}\\}}\right)^{2}\right]$
(27)

the derivatives here are evaluated at $\omega=\omega_{0}$. In order to proceed
further we need to know the functional dependence of the nonlinear refractive
index $n_{2}(\omega)$ on frequency.

Frequency dependence of nonlinear refractive index. In this paragraph we write
down for the record the expression for the nonlinear refractive index as
calculated by means of the dispersive approach of Sheik-Bahae et al in
refs.[16, 17]. Generally speaking nonlinear refractive (Kerr) index is a
function of both pump- and probe frequencies, $\Omega$ and $\omega$
respectively. In the cited papers the authors notice that when pump intensity
is constant $I_{p}(t)=I_{0}$, the total Kerr-modified refractive index of the
medium can still be treated as linear index when when used to describe the
propagation of the probe beam. This means that, among other things, the
correction to the refractive index $\delta
n(\omega)=n_{2}(\omega,\Omega)I_{0}$ must satisfy Kramers-Kronig relations:

$\mathrm{Re}\\{n_{2}(\omega,\Omega)\\}=\frac{1}{\pi}\int\displaylimits_{\infty}^{+\infty}\frac{\mathrm{Im}\\{n_{2}(\omega^{\prime},\Omega)\\}}{\omega^{\prime}-\omega}d\omega^{\prime}$
(28)

Imaginary part of the Kerr index corresponds to pump-induced absorption of the
probe beam. In the simplest case this can happen due to two-photon absorption
(2PA). The magnitude for 2PA was calculated for a direct-gap semiconductor in
refs.[SheikBahae1991, 17]. Being only interested in the frequency dependence
of $n_{2}$ we write down the result of that work as:

$\mathrm{Im}\\{n_{2}(\omega,\Omega)\\}=\mathrm{const}\cdot\frac{\left(\omega+\Omega-\Delta\right)^{3/2}}{\omega^{2}\Omega^{2}}\left(\frac{1}{\omega}+\frac{1}{\Omega}\right)^{2}\cdot\Theta(\omega+\Omega-\Delta)$
(29)

where $\Theta(x)$ is the Heaviside step-function. Since in eq.(28) we
integrate along the entire real axis, we need to fix what $n_{2}$ is for
negative $\omega$. The reality of $n_{2}(t-t^{\prime})$ requires that
$n_{2}(\omega)=n_{2}^{*}(-\omega)$. Therefore, the 2PA part of $n_{2}$ should
read:

$\begin{multlined}\mathrm{Im}\\{n^{\mathrm{2PA}}_{2}(\omega,\Omega)\\}=\frac{\mathrm{const}}{\omega^{4}\Omega^{4}}\cdot\left\\{{\left(\omega+\Omega-\Delta\right)^{3/2}}{}\left({\omega}+{\Omega}\right)^{2}\cdot\Theta(\omega+\Omega-\Delta)+\right.\\\
\left.{+\left(\Omega-\omega-\Delta\right)^{3/2}}{}\left({\Omega}-{\omega}\right)^{2}\cdot\Theta(\Omega-\omega-\Delta)\right\\}\end{multlined}\mathrm{Im}\\{n^{\mathrm{2PA}}_{2}(\omega,\Omega)\\}=\frac{\mathrm{const}}{\omega^{4}\Omega^{4}}\cdot\left\\{{\left(\omega+\Omega-\Delta\right)^{3/2}}{}\left({\omega}+{\Omega}\right)^{2}\cdot\Theta(\omega+\Omega-\Delta)+\right.\\\
\left.{+\left(\Omega-\omega-\Delta\right)^{3/2}}{}\left({\Omega}-{\omega}\right)^{2}\cdot\Theta(\Omega-\omega-\Delta)\right\\}$
(30)

Further, as will be clear below, we need to also symmetrize the above
expression with respect to $\Omega$. These new terms achieved by substituting
$\Omega\rightarrow-\Omega$ can be identified with Raman contribution to
$n_{2}$:

$\displaystyle\mathrm{Im}\\{n^{\mathrm{2PA+Raman}}_{2}(\omega,\Omega)\\}=\frac{\mathrm{const}}{\omega^{4}\Omega^{4}}\cdot\left\\{{\left(\omega+\Omega-\Delta\right)^{3/2}}{}\left({\omega}+{\Omega}\right)^{2}\cdot\Theta(\omega+\Omega-\Delta)+\right.$
$\displaystyle\left.{+\left(\Omega-\omega-\Delta\right)^{3/2}}{}\left({\Omega}-{\omega}\right)^{2}\cdot\Theta(\Omega-\omega-\Delta)+\right.$
$\displaystyle\left.+{\left(\omega-\Omega-\Delta\right)^{3/2}}{}\left({\omega}-{\Omega}\right)^{2}\cdot\Theta(\omega-\Omega-\Delta)+\right.$
$\displaystyle\left.{+\left(-\Omega-\omega-\Delta\right)^{3/2}}{}\left(-{\Omega}-{\omega}\right)^{2}\cdot\Theta(-\Omega-\omega-\Delta)\right\\}$
(31)

Now we can perform the integral in eq.(28), however in addition to the
physical branching points at $\omega=\pm\left(\Delta\pm\Omega\right)$ that
correspond to certain two-photon absorption thresholds, there is an unphysical
singularity at $\omega=0$. As discussed in ref.[17] this is an artifact of the
derivation of two-photon absorption probability by means of dipole
(A$\cdot{\bf p}$) approximation. At low frequencies this singularity is
cancelled by taking into account the omitted diamagnetic term A2. Knowing that
singularity at $\omega=0$ is fictitious, we can simply amend the expression
above with a proper counter-term that will remove the divergence at
$\omega=0$.

Figure 9: Real(left) and imaginary (right) parts of $n_{2}(\omega,\Omega)$ as
a function of complex $z=\omega/\Delta$ based of eq.(32). Here we took
$\Omega=0.8\Delta$

An inspection of the expression for non-degenerate two-photon absorption above
reveals that it is symmetric in $\omega$ and $\Omega$. This is natural given
how it is derived. What it means however, is that to recover proper $\omega$
dependence of $n_{2}$, one should also include the counter terms to remove the
singularity at $\Omega=0$. Then one finally would need to make sure that
counter terms in $\omega$ don’t diverge at $\omega=0$ and vice versa. Adding
everything up, and performing the Kramers-Kronig integration in eq.(28) we can
write down the full expression for nondegenerate $n_{2}$:

$\displaystyle
n^{\mathrm{2PA+Raman}}_{2}(\omega,\Omega)\propto\frac{\Delta^{9/2}}{\omega^{4}\Omega^{4}}\cdot\left\\{{\left(\Delta-\omega-\Omega\right)^{3/2}}\left({\omega}+{\Omega}\right)^{2}+\right.\left.\left(\Delta+\omega-\Omega\right)^{3/2}\left({\Omega}-{\omega}\right)^{2}+\right.$
$\displaystyle\left.+\left(\Delta-\omega+\Omega\right)^{3/2}\left({\omega}-{\Omega}\right)^{2}+\right.\left.\left(\Delta+\Omega+\omega\right)^{3/2}\left(-{\Omega}-{\omega}\right)^{2}-\right.$
$\displaystyle\left.-2\Omega^{2}(\Delta-\Omega)^{3/2}\left[1+\frac{\omega^{2}}{\Omega^{2}}-\frac{3\omega^{2}}{\Omega(\Delta-\Omega)}+\frac{3}{8}\frac{\omega^{2}}{(\Delta-\Omega)^{2}}\right]-\right.$
$\displaystyle\left.-2\Omega^{2}(\Delta+\Omega)^{3/2}\left[1+\frac{\omega^{2}}{\Omega^{2}}+\frac{3\omega^{2}}{\Omega(\Delta+\Omega)}+\frac{3}{8}\frac{\omega^{2}}{(\Delta+\Omega)^{2}}\right]-\right.$
$\displaystyle\left.-2\omega^{2}(\Delta-\omega)^{3/2}\left[1+\frac{\Omega^{2}}{\omega^{2}}-\frac{3\Omega^{2}}{\omega(\Delta-\omega)}+\frac{3}{8}\frac{\Omega^{2}}{(\Delta-\omega)^{2}}\right]-\right.$
$\displaystyle\left.-2\omega^{2}(\Delta+\omega)^{3/2}\left[1+\frac{\Omega^{2}}{\omega^{2}}+\frac{3\Omega^{2}}{\omega(\Delta+\omega)}+\frac{3}{8}\frac{\Omega^{2}}{(\Delta+\omega)^{2}}\right]+\right.$
$\displaystyle\left.+4\Delta^{3/2}\left(\omega^{2}+\Omega^{2}\right)+9\frac{\omega^{2}\Omega^{2}}{\Delta^{1/2}}\right\\}$
(32)

here both $\omega$ and $\Omega$ are understood as complex variables. In the
degenerate limit $\omega=\Omega$ the above expression can be checked to
coincide with the functional dependence of $n_{2}(\omega)$ given in [17].


# On Modular Gray Map

Manish K Gupta
Dhirubhai Ambani Institute of Information and Communication Technology,
Post Bag Number 4 , Near Indroda Circle
Gandhinagar, 382007, India
<EMAIL_ADDRESS>A part of this work was presented at the Special
Session on Algebraic Coding Theory, Central Section Meeting #995, American
Mathematical Society, Athens,OH, 26-27 March 2004.

###### Abstract

This paper introduces an isometry between the modular rings ${\mbox{\msbm
Z}}_{2^{s}}$ and ${\mbox{\msbm Z}}_{2^{s-1}}$ with respect to the homogeneous
weights. Certain product of these maps gives Carlet’s generalised Gray map and
also Vega’s Gray map. For $s=2$ this reduces to popular Gray map. Several
interesting properties of these maps are studied. Towards the end we list
several interesting problems to work on.

Keywords: Codes over rings, Gray isometry, Gray map.

##  1 Introduction

In $19^{th}$ century a cyclic-permuted binary code was invented by a french
engineer Emile Baudot (now often called Gray code as it was patented by Frank
Gray on $17.03.1953$ [13]). This has represented a major advancement in
telegraphy. After that it has witnessed various practical applications other
than solving puzzles such as Tower of Hanoi and the Brain [29]. It is very
recent when it is used as an isometry between ${\mbox{\msbm Z}}_{4}$ and
${\mbox{\msbm Z}}_{2}^{2}$ with respect to the Lee and Hamming distances to
solve a $30$ year old coding theory puzzle [17]. This isometry can be extended
component-wise between ${\mbox{\msbm Z}}_{4}^{n}$ and ${\mbox{\msbm
Z}}_{2}^{2n}$ and is usually known as the Gray map. Due to an isometry the
Gray map has been used extensively to construct good quality binary linear and
nonlinear codes [28, 15, 17]. Also the ${\mbox{\msbm Z}}_{4}$\- linearity of
various binary codes of even length has been inevstigated. In [22], Ana
$S\breve{a}l\breve{a}gean$\- Mandache has shown that except for the well-known
case $p=s=2,$ it is not possible to construct a weight function on
$\;{\mbox{\msbm Z}}_{p^{s}}$ for which ${\mbox{\msbm Z}}_{p^{s}}$ is isometric
to ${\mbox{\msbm Z}}_{p}^{s}$ with the Hamming metric. A similar result for
the Lee metric was established in [10] by S. R. Costa et al. However there
exist an isometry between ${\mbox{\msbm Z}}_{2^{s}}\rightarrow{\mbox{\msbm
Z}}_{2}^{2^{s-1}}$ was discovered by Carlet in [8]. Using this generalized
Gray map he extendes the concept of ${\mbox{\msbm Z}}_{4}$-linear codes to
${\mbox{\msbm Z}}_{2^{s}}$-linear codes. The generalization of these
isometries up to finite chain rings was considered by Greferath and Schmidt,
Honold and Landjev, San Ling and Blackford and Nechaev in [14, 18, 19, 20]. In
[5] Asch and Tilborg has studied the isometry between ${\mbox{\msbm
Z}}_{p^{2}}$ and ${\mbox{\msbm Z}}_{p}^{p}$. More recently an isometry between
codes over ${\mbox{\msbm Z}}_{2^{s}}$ and codes over ${\mbox{\msbm Z}}_{4}$
has been introduced by Vega and Tapia-Recillas in [25, 26]. Further they have
studied the condition under which a code over ${\mbox{\msbm Z}}_{4}$ is
${\mbox{\msbm Z}}_{2^{s}}$ linear. In this paper we will introduce a Gray
isometry between ${\mbox{\msbm Z}}_{2^{s}}$ and ${\mbox{\msbm Z}}_{2^{s-1}}$.
It is shown that the generalized Gray map introduced by Carlet is equivalent
to the product of $s-1$ such maps. It also covers the various generalizations
introduced earliear. Some basic properties of these maps are studied.

##  2 Preliminaries And Notations

A code ${\cal C},$ of length $n$, over $\;{\mbox{\msbm Z}}_{p^{s}}$ is simply
a subset of ${\mbox{\msbm Z}}_{p^{s}}^{n}$. If in addition it is an additive
subgroup of ${\mbox{\msbm Z}}_{p^{s}}^{n}$ we call ${\cal C}$ a linear code of
length $n$. An element of ${\cal C}$ is called a codeword of ${\cal C}$ and a
generator matrix of a linear code ${\cal C}$ is a matrix whose rows generate
${\cal C}$. The Hamming weight $w_{H}(x)$ of a vector $x$ in $\mbox{\msbm
Z}_{p^{s}}^{n}$ is the number of non-zero components. The Lee weight
$w_{L}(x)$ of a vector $x=(x_{1},x_{2},\ldots,x_{n})$ is
$\sum_{i=1}^{n}\min\\{|x_{i}|,|p^{s}-x_{i}|\\}$. The Euclidean weight
$w_{E}(x)$ of a vector $x$ is
$\sum_{i=1}^{n}\min\\{x_{i}^{2},(p^{s}-x_{i})^{2}\\}$. The Euclidean weight is
useful in connection with lattice constructions. The Chinese Euclidean weight
$w_{CE}(x)$ of a vector $x\in{\mbox{\msbm Z}}_{m}^{n}$ is
$\sum_{i=1}^{n}\left\\{2-2\;\mbox{cos}\;\left(\frac{2\pi
x_{i}}{m}\right)\right\\}$. This is useful for $\\!m\\!-\\!PSK$ coding. In
[9], Constantinescu and Werner introduced the homogeneous weight on the
residue class rings of integers which was later extended to finite Frobenius
rings. The homogeneous weight on ${\mbox{\msbm Z}}_{p^{s}}$ is defined as
follows. For $\;u\neq 0,u\in{\mbox{\msbm Z}}_{p^{s}}$

$w_{HW}(u)=\left\\{\begin{array}[]{cc}&p^{s-2}(p-1),u\notin
p^{s-1}{\mbox{\msbm Z}}_{p^{s}}\\\ &p^{s-1},u\in p^{s-1}{\mbox{\msbm
Z}}_{p^{s}}\end{array}\right.\;\mbox{and}\;w_{HW}(0)=0.$

The Hamming, Lee, Euclidean and Homogeneous distances $d_{H}(x,y)$,
$d_{L}(x,y)$, $d_{E}(x,y)$ and $d_{HW}(x,y)$ between two vectors $x$ and $y$
are $w_{H}(x-y)$, $w_{L}(x-y)$, $w_{E}(x-y)$ and $w_{HW}(x-y)$ respectively.
The minimum Hamming, Lee, Euclidean and Homogeneous weights, $d_{H},d_{L}$,
$d_{E}$ and $d_{HW}$ of ${\cal C}$ are the smallest Hamming, Lee, Euclidean
and Homogeneous weights among all non-zero codewords of ${\cal C},$
respectively.

Carlet’s defenition of generalized Gray map can be extended to ${\mbox{\msbm
Z}}_{p^{s}}$ naturally. The Generalized Gray map $\gamma:\;{\mbox{\msbm
Z}}_{p^{s}}^{n}\rightarrow\;{\mbox{\msbm Z}}_{p}^{p^{s-1}n}$ is the
coordinate-wise extension of the function $\gamma_{g}$ from $\;{\mbox{\msbm
Z}}_{p^{s}}$ to $\;{\mbox{\msbm Z}}_{p}^{p^{s-1}}$ defined as follows. Let
$u=u_{1}+pu_{2}+\cdot+p^{s-1}u_{s}\in{\mbox{\msbm Z}}_{p^{s}}$ be the finite
$p$-adic expansion of $u$, where $u_{i}\in{\mbox{\msbm Z}}_{p}$ also let
$y_{i}\in{\mbox{\msbm Z}}_{p}$. The map
$\gamma_{g}:(y_{1},y_{2},\ldots,y_{s-1})\mapsto
u_{s}+\sum_{i=1}^{s-1}u_{i}y_{i}\in{\mbox{\msbm Z}}_{p}$ defined as a
generalized Boolean function evaluated on $GF(p^{s-1})$. It is easy to see
that this map is equivalent to the generalized Gray map introduced by
Greferath and Schmidt [14] using tensor product. In particular for $s=2$ this
map is also equivalent to a generalized Gray map from ${\mbox{\msbm
Z}}_{p^{2}}$ to ${\mbox{\msbm Z}}_{p}^{p}$ introduced recently by Asch and
Tilborg [5]. The image $\gamma({\cal C})$, of a linear code ${\cal C}$ over
$\;{\mbox{\msbm Z}}_{p^{s}}$ of length $n$ by the generalized Gray map, is a
$p$-ary code of length $p^{s-1}n$. The dual code ${\cal C}^{\perp}$ of ${\cal
C}$ is defined as $\\{x\in{\mbox{\msbm Z}}_{p^{s}}^{n}\mid x\cdot
y=0\;\mbox{for all}\;y\in{\cal C}\\}$ where $x\cdot y$ is the standard inner
product of $x$ and $y$. ${\cal C}$ is self-orthogonal if ${\cal
C}\subseteq{\cal C}^{\perp}$ and ${\cal C}$ is self-dual if ${\cal C}={\cal
C}^{\perp}$. Two codes are said to be equivalent if one can be obtained from
the other by permuting the coordinates and (if necessary) changing the signs
of certain coordinates. Codes differing by only a permutation of coordinates
are called permutation-equivalent.

Any linear code ${\cal C}$ over $\mbox{\msbm Z}_{p^{s}}$ is permutation-
equivalent to a code with generator matrix $G$ (the rows of $G$ generate
${\cal C}$) of the form

(1)
$G=\left[\begin{array}[]{cccccc}I_{k_{0}}&A_{01}&A_{02}&\cdots&A_{0s-1}&A_{0s}\\\
{\bf 0}&pI_{k_{1}}&pA_{12}&\cdots&pA_{1s-1}&pA_{1s}\\\ {\bf 0}&{\bf
0}&p^{2}I_{k_{2}}&\cdots&p^{2}A_{2s-1}&p^{2}A_{2s}\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\ {\bf 0}&{\bf 0}&{\bf
0}&\cdots&p^{s-1}I_{k_{s-1}}&p^{s-1}A_{s-1s}\end{array}\right],$

where $A_{ij}$ are matrices over $\;{\mbox{\msbm Z}}_{p^{s}}$ and the columns
are grouped into blocks of sizes $k_{0},\;k_{1},\;\cdots,\;k_{s-1},\;k_{s},$
respectively. Let $k=\sum_{i=0}^{s-1}(s-i)k_{i}$. Then $|{\cal C}|=p^{k}$ is
the number of codewords in the code ${\cal C}.$ Note that ${\cal C}$ is a free
code if and only if $k_{i}=0\;\;\mbox{for all}\;\;\;i=1,2,\cdots,s-1.$ It is
easy to see that a generator matrix for ${\cal C}^{\perp}$ is of the form

$H=\left[\begin{array}[]{ccccc}B_{0s}&B_{0s-1}&\cdots&B_{01}&I_{k_{s}}\\\
pB_{1s}&pB_{1s-1}&\cdots&pI_{k_{s-1}}&{\bf 0}\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\
p^{s-1}B_{s-1s}&p^{s-1}I_{k_{1}}&\cdots&{\bf 0}&{\bf 0}\end{array}\right],$

where $B_{ij}$ are matrices over $\;{\mbox{\msbm Z}}_{p^{s}}$ and the columns
are grouped into blocks of size $k_{0},k_{1},\cdots,k_{s}.$ Therefore $|{\cal
C}^{\perp}|=p^{sn-k}.$ Note that $sn-k=\sum_{i=1}^{s}ik_{i}.$

A vector ${\bf v}$ is a $p$-linear combination of the vectors ${\bf
v}_{1},{\bf v}_{2},\ldots,{\bf v}_{k}$ if ${\bf v}=\l_{1}{\bf
v}_{1}+\ldots+\l_{k}{\bf v}_{k}$ with $\l_{i}\in{\mbox{\msbm Z}}_{p}$ for
$1\leq i\leq k.$ A subset $S=\\{{\bf v}_{1},{\bf v}_{2},...,{\bf v}_{k}\\}$ of
${\cal C}$ is called a $p$-basis for ${\cal C}$ if for each
$i=1,2,...,k-1,\;p{\bf v}_{i}$ is a $p-$linear combination of ${\bf
v}_{i+1},...,{\bf v}_{k}$, $2{\bf v}_{k}=0,\;{\cal C}$ is the $p$-linear span
of $S$ and $S$ is $p$-linearly independent [15]. The number of elements in a
$p$-basis for ${\cal C}$ is called the $p$-dimension of ${\cal C}.$ It is easy
to verify that the rows of the matrix

(2) ${\cal
B}=\left[\begin{array}[]{cccccc}I_{k_{0}}&A_{01}&A_{02}&\cdots&A_{0s-1}&A_{0s}\\\
pI_{k_{0}}&pA_{01}&pA_{02}&\cdots&pA_{0s-1}&pA_{0s}\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
p^{s-1}I_{k_{0}}&p^{s-1}A_{01}&p^{s-1}A_{02}&\cdots&p^{s-1}A_{0s-1}&p^{s-1}A_{0s}\\\
\hline\cr{\bf 0}&pI_{k_{1}}&pA_{12}&\cdots&pA_{1s-1}&pA_{1s}\\\ {\bf
0}&p^{2}I_{k_{1}}&p^{2}A_{12}&\cdots&p^{2}A_{1s-1}&p^{2}A_{1s}\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\ {\bf
0}&p^{s-1}I_{k_{1}}&p^{s-1}A_{12}&\cdots&p^{s-1}A_{1s-1}&p^{s-1}A_{1s}\\\
\hline\cr\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\ \hline\cr{\bf 0}&{\bf
0}&{\bf 0}&\cdots&p^{s-1}I_{k_{s-1}}&p^{s-1}A_{s-1s}\end{array}\right].$

form a $p$-basis for the code ${\cal C}$ generated by $G$ given in (1).

A linear code ${\cal C}$ over $\;{\mbox{\msbm Z}}_{p^{s}}$ ( over
$\;{\mbox{\msbm Z}}_{2}$) of length $n$, $p$-dimension $k$, minimum distance
$d_{H},d_{L},d_{E},d_{CE}$ and $d_{HW}$ is called an
$\left[n,k,d_{H},d_{L},d_{E},d_{CE},d_{HW}\right]$ $\left([n,k,d_{H}]\right)$
or simply an $\left[n,k\right]$ code.

##  3 Modular Gray Map

Now we define a natural Gray map as follows.

Let

$\begin{array}[]{ccc}A_{1}&=&\\{0,1,2,3,\ldots,2^{s-2}-1\\},\\\
A_{2}&=&\\{2^{s-2},2^{s-2}+1,\ldots,2^{s-1}-1\\},\\\
A_{3}&=&\\{2^{s-1},2^{s-1}+1,\ldots,2^{s-1}+2^{s-2}-1\\}\;\mbox{and}\\\
A_{4}&=&\\{2^{s-1}+2^{s-2},2^{s-1}+2^{s-2}+1,\ldots,2^{s-1}+2^{s-1}-1\\}\end{array}$

be the four consecutive disjoint partitions of the integers of ${\mbox{\msbm
Z}}_{2^{s}}$. Define the modular generalized Gray map $\eta^{s}_{g}$ on
${\mbox{\msbm Z}}_{2^{s}}$ to ${\mbox{\msbm Z}}_{2^{s-1}}$ as

$\eta^{s}_{g}(i)=\left\\{\begin{array}[]{cc}(i,i)&i\in A_{1}\\\
(i-2^{s-2},i)&i\in A_{2}\\\ (i-2^{s-2},i-2^{s-2})&i\in A_{3}\\\
(i-2^{s-1},i-3\cdot 2^{s-2})&i\in A_{4}\end{array}\right.$

###### Remark 1

Each entry of ${\mbox{\msbm Z}}_{2^{s}}$ is mapped onto the first order Reed
Muller code of length $2$ over ${\mbox{\msbm Z}}_{2^{s}}$ [15, 16] generated
by the following generator matrix

$G=\left[\begin{array}[]{cc}1&1\\\ 0&2^{s-2}\end{array}\right]_{2\times 2}.$

###### Remark 2

For $s=2$ $\eta^{s}_{g}$ reduces to ordinary Gray map.

###### Remark 3

Applying $\eta^{s}_{g}$ to a code ${\cal C}$ over ${\mbox{\msbm Z}}_{2^{s}}$
gives a code $\eta^{s}_{g}(C)$ over ${\mbox{\msbm Z}}_{2^{s-1}}$ of length
twice the length of ${\cal C}$

###### Example 1

For $s=2,3,4$ these maps are given below.

$\begin{array}[]{ccc}{\mbox{\msbm Z}}_{4}&\mapsto&{\mbox{\msbm Z}}_{2}\\\
0&\mapsto&00\\\ 1&\mapsto&01\\\ 2&\mapsto&11\\\ 3&\mapsto&10\end{array}$

$\begin{array}[]{ccc}{\mbox{\msbm Z}}_{8}&\mapsto&{\mbox{\msbm Z}}_{4}\\\
0&\mapsto&00\\\ 1&\mapsto&11\\\ 2&\mapsto&02\\\ 3&\mapsto&13\\\
4&\mapsto&22\\\ 5&\mapsto&33\\\ 6&\mapsto&20\\\ 7&\mapsto&31\end{array}$

$\begin{array}[]{ccc}{\mbox{\msbm Z}}_{16}&\mapsto&{\mbox{\msbm Z}}_{8}\\\
0&\mapsto&00\\\ 1&\mapsto&11\\\ 2&\mapsto&22\\\ 3&\mapsto&33\\\
4&\mapsto&04\\\ 5&\mapsto&15\\\ 6&\mapsto&26\\\ 7&\mapsto&37\\\
8&\mapsto&44\\\ 9&\mapsto&55\\\ 10&\mapsto&66\\\ 11&\mapsto&77\\\
12&\mapsto&40\\\ 13&\mapsto&51\\\ 14&\mapsto&62\\\ 15&\mapsto&73\end{array}$

Some interesting facts about modular Gray maps are collected in the following.

###### Theorem 1

The generalized Gray map of Carlet $\Phi_{G}:{\mbox{\msbm
Z}}_{2^{s}}\mapsto{\mbox{\msbm Z}}_{2}$ can written as a product of $s-1$
modular Gray maps i.e,
$\Phi_{G}=\eta^{s}_{g}\eta^{s-1}_{g}\eta^{s-2}_{g}\ldots\eta^{2}_{g}$.

Proof. By remark 1, if we apply the modular gray map $\eta^{s}_{g}$ to the
$2$-basis generator matrix of the first order Reed Muller code over
${\mbox{\msbm Z}}_{2^{s}}$ we get the $2$-basis generator matrix of the first
order Reed Muller code over ${\mbox{\msbm Z}}_{2^{s-1}}$ which is the image
set of $\eta^{s-1}_{g}$ and so on. Thus successive applications yields
Carlet’s generalized Gray map [15, 16].

$\Box$

###### Remark 4

It is interesting to note that now we can get modular Gray map between all
intermediate rings i.e, between

${\mbox{\msbm Z}}_{2^{s}}\mapsto{\mbox{\msbm Z}}_{2^{s-1}}\mapsto{\mbox{\msbm
Z}}_{2^{s-2}}\mapsto\cdots\mapsto{\mbox{\msbm Z}}_{4}\mapsto{\mbox{\msbm
Z}}_{2}.$

For example, the Gray map of Tapia-Recillas and Vega [25, 26] between
${\mbox{\msbm Z}}_{2^{s}}$ and ${\mbox{\msbm Z}}_{4}$ is equivalent to a map
by taking product of $s-2$ such maps
$\eta^{s}_{g}\eta^{s-1}_{g}\ldots\eta^{3}_{g}$. Clearly our map is much more
powerful.

###### Lemma 1

The images of the rows of the matrix ${\cal B}$ given by (2) under the modular
Gray map $\eta^{s}_{g}$ are $2$-linearly independent over $\;{\mbox{\msbm
Z}}_{2^{s-1}}.$

Proof. After taking appropriate block permutations row-wise to the matrix
${\cal B}$ and applying the modular Gray map we see that in the diagonals we
have 2-linearly independent vectors over ${\mbox{\msbm Z}}_{2^{s-1}}$. This
proves our claim.

$\Box$

###### Proposition 1

The modular Gray map $\eta^{s}_{g}$ is an isometry between ${\mbox{\msbm
Z}}_{2^{s}}\mapsto({\mbox{\msbm Z}}_{2^{s-1}})^{2}$ with respect to the
homogeneous weights.

Proof. Let $u(\neq 0)\in{\mbox{\msbm Z}}_{2^{s}}$. Then the homogeneous weight
will be given as

$w_{HW}(u)=\left\\{\begin{array}[]{cc}2^{s-2},&u\neq 2^{s-1}\\\
2^{s-1},&u=2^{s-1}.\end{array}\right.$

Thus homogeneous weight of each entry in the sets
$A_{1},A_{2},A_{3}\;(\mbox{except first entry which is}\;2^{s-1})$ and $A_{4}$
will be $2^{s-2}$. Thus direct computation of homogeneous weight as a rational
sum of the entries in the definition yields the desired result. The case of
$u=0$ is trivial. $\Box$

###### Remark 5

Note that any product of modular Gray maps $\eta^{s}_{g}$ is also an isometry.

###### Remark 6

The modular Gray map $\eta^{s}_{g}$ can be extended coordinate-wise to a map
from $({\mbox{\msbm Z}}_{2^{s}})^{n}$ to $({\mbox{\msbm Z}}_{2^{s-1}})^{2n}$.
With abuse of notation we denote this map also by $\eta^{s}_{g}$.

###  3.1 Permuted Modular Gray Map

We can permute the modular Gray map to get some very interesting results like
perverseness of Octocode. The modified modular Gray map $\xi^{s}_{g}$ is
define as ….

….

Let us modify our examples to get the modify modular Gray map for $s=2,3,4$.

$\begin{array}[]{ccc}{\mbox{\msbm Z}}_{4}&\mapsto&{\mbox{\msbm Z}}_{2}\\\
0&\mapsto&00\\\ 1&\mapsto&10\\\ 2&\mapsto&11\\\ 3&\mapsto&01\end{array}$

$\begin{array}[]{ccc}{\mbox{\msbm Z}}_{8}&\mapsto&{\mbox{\msbm Z}}_{4}\\\
0&\mapsto&00\\\ 1&\mapsto&11\\\ 2&\mapsto&20\\\ 3&\mapsto&13\\\
4&\mapsto&22\\\ 5&\mapsto&31\\\ 6&\mapsto&02\\\ 7&\mapsto&33\end{array}$

$\begin{array}[]{ccc}{\mbox{\msbm Z}}_{16}&\mapsto&{\mbox{\msbm Z}}_{8}\\\
0&\mapsto&00\\\ 1&\mapsto&11\\\ 2&\mapsto&22\\\ 3&\mapsto&13\\\
4&\mapsto&40\\\ 5&\mapsto&15\\\ 6&\mapsto&26\\\ 7&\mapsto&17\\\
8&\mapsto&44\\\ 9&\mapsto&71\\\ 10&\mapsto&62\\\ 11&\mapsto&73\\\
12&\mapsto&04\\\ 13&\mapsto&75\\\ 14&\mapsto&66\\\ 15&\mapsto&77\end{array}$

###### Example 2

Consider the generator matrix of the Octocode over ${\mbox{\msbm Z}}_{8}$.

$G_{8}=\left[\begin{array}[]{cccccccc}5&7&5&6&1&0&0&0\\\ 5&0&7&5&6&1&0&0\\\
5&0&0&7&5&6&1&0\\\ 5&0&0&0&7&5&6&1\end{array}\right].$

Apply the modular Gray map $\eta^{3}_{g}$ on the rows of $G_{8}$. We get the
following generator matrix

$\eta^{3}_{g}(G_{8})=\left[\begin{array}[]{cccccccc|cccccccc}3&3&3&0&1&0&0&0&1&3&1&2&1&0&0&0\\\
3&0&3&3&0&1&0&0&1&0&3&1&2&1&0&0\\\ 3&0&0&3&3&0&1&0&1&0&0&3&1&2&1&0\\\
3&0&0&0&3&3&0&1&1&0&0&0&3&1&2&1\end{array}\right].$

##  4 Properties

In this section we list several interesting properties of modular Gray map
that one can study.

  * •

General results of Theorems $4,5,6,$ page $305$ of [17] these should also
generalize the results of Vega et al in [25] and Carlet’s result of [8].

  * •

Studying the characterizations of cyclic, quasi-cyclic, consta-cyclic,
negacyclic, quasi-twisted codes with respect to modular Gray map in view of
the following papers [31, 30, 19, 26, 27].

  * •

Does modular Gray map preserves self orthogonality of codes ? Under what
conditions ?

  * •

Applications of both modular and permuted modular Gray map for the
constructions of new codes over various modular rings and binary fields. For
example to Hamming codes, Simplex codes, Golay codes and other standard codes.

  * •

Is there any connection for the modular Gray map with the Nenchev’s Gray map ?

  * •

Other non-trivial applications like by taking the generator matrix of Kerdock
codes, Preparata codes, Goethals codes or like codes and applying the modular
Gray map.

  * •

Construction of record breaking codes !

  * •

Effect on Torsion and reduction codes !

  * •

New code construction with respect to 2-basis modular Gray map applications !

  * •

Connection with Hensal-uplift !

  * •

Studying the ${\mbox{\msbm Z}}_{2^{s}}$-linearity of standard uplifted codes
over ${\mbox{\msbm Z}}_{2^{s-1}}$.

  * •

Finding other non-trivial properties !

  * •

…

Note This note is an old contribution and has not been updated with some new
results on Gray map. We refer the reader to look into some new literature in
the area.

## References

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