# GUP and Correction to BTH

Sumit Ghosh  Rabin Banerjee, Sumit Ghosh
S. N. Bose National Centre for Basic Sciences,
JD Block, Sector III, Salt Lake, Kolkata-700098, India
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Generalised Uncertainty Principle, Remnant Mass
and
Singularity Problem in Black Hole Thermodynamics

Sumit Ghosh  Rabin Banerjee, Sumit Ghosh
S. N. Bose National Centre for Basic Sciences,
JD Block, Sector III, Salt Lake, Kolkata-700098, India
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We have derived a new Generalised Uncertainty Principle (GUP) based on certain
general assumptions. This GUP is consistent with predictions from string
theory. It is then used to study Schwarzschild black hole thermodynamics.
Corrections to the mass-temperature relation, area law and heat capacity are
obtained. We find that the evaporation process stops at a particular mass,
referred as the remnant mass. This is instrumental in bypassing the well known
singularity problem that occurs in a semiclassical approach.

##  1 Introduction

The introduction of gravity into quantum field theory brings an observer
independent minimum length scale in the picture [1]. A minimal length also
occurs in string theory [2], non commutative geometry [3] or can be obtained
from gedanken experiment [4]. This minimal length is expected to be close or
equal to Planck length ($L_{P}$). The manifestation of the inclusion of a
minimal length in theories has been observed from different perspectives - the
generalised uncertainty principle (GUP), modified dispersion relation (MDR),
deformed special relativity (DSR), to name a few. The modification or
deformation affect the well known semiclassical laws of black hole
thermodynamics [5, 6, 7, 8]. For instance, the black hole entropy is no longer
proportional to the horizon area [9, 10]. Another interesting result is the
existence of a remnant mass of a black hole. The existence of a remnant mass
of a black hole is verified by different approaches - using a generalised
uncertainty principle [11] or analysing the tunnelling probability [12].

In this paper we study the modifications to the laws of Schwarzschild black
hole thermodynamics by starting from a new GUP which is derived from certain
basic assumptions. The consequences of these modifications are investigated in
details. We show the existence of a critical (ans also singular) mass for the
black hole below (at) which the thermodynamic entities become complex (ill
defined). However both the critical and singular mass are less than another
mass - the remnant mass. Our analysis reveals that the black hole evaporation
does not lead to a singularity. This process terminates at a finite mass which
we call the remnant mass. Since, as already stated, both the critical and
singular masses are less than the remnant mass, the problematic situations are
avoided.

Section 2 gives the derivation of the GUP. Corrections induced by this GUP to
the black hole thermodynamics are given in section 3. The connection of
remnant mass with the singularity problem is discussed in section 4. Section 5
contains the discussions.

##  2 A Generalised Uncertainty Principle (GUP)

A particle with energy close to the Planck energy $E_{P}$ will disturb the
space time significantly at least upto a length of the order of the Planck
length. It is very natural to take the metric to be a function of the particle
energy111This is the effect of back reaction [13]. [14] . One can find the
explicit dependence by solving corresponding Einstein equation where the right
hand side is given by the energy momentum tensor of the particle. If we assume
that the a particle field is a linear superposition of plane wave solutions
($\thicksim e^{ik^{\mu}x_{\mu}}$), then one can easily guess that on
quantisation the particle momentum$(p)$ and energy$(E)$ may be non linear in
wave vector$(k)$ and angular frequency$(\omega)$ [14, 15] . In general we may
write

$\displaystyle\it{k^{\mu}=f(p^{\mu})}$ (1)

It is easy to show that both $k^{\mu}$ and $p^{\mu}$ can transform like a
Lorentz vector only for special types of function $f$. The standard form is
the obvious $p^{\mu}=\hbar k^{\mu}$ and a more general form is
$p^{\mu}=\phi\left(k^{\nu}k_{\nu}\right)k^{\mu}$ where
$\phi\left(k^{\nu}k_{\nu}\right)$ is a scalar function of the invariant
$\left(k^{\nu}k_{\nu}\right)$; the more general form is clearly equivalent to
generalising Planck’s constant to a function. For simplicity in this paper we
will forego Lorentz invariance and consider the following relations [16],

$\displaystyle\it{k=f(p)\hskip 28.45274pt\omega=f(E)}$ (2)

The function $f$ satisfies certain properties [14, 17] :

  1. 1.

The function $\left(f\right)$ and its inverse $\left(f^{-1}\right)$ have to be
an odd function to preserve parity.

  2. 2.

For small momentum/energy ($E<<E_{P}$) the function should be chosen to
satisfy the relationship $p=\hbar k$ .

  3. 3.

We assume the existence of a minimum length, identified as the Planck length
$(L_{P})$ [1, 14, 17] that cannot be resolved. So the wave vector $k=f(p)$
should have an upper bound $\dfrac{2\pi}{L_{P}}$. Since the wave vector
$k=f(p)$ shows a saturation with respect to the momentum $p$, the momentum
$p=f^{-1}(k)$ will be a monotonically increasing function of $k$.

We also assume that the commutation relations

$\displaystyle[x,k]=i\;\;\;,\;\;\;[x,p(k)]=i\dfrac{\partial p}{\partial k}$
(3)

hold which lead to a uncertainty relation [18]

$\displaystyle\Delta p\Delta
x\geqslant|\left\langle\dfrac{1}{2}[x,p]\right\rangle|\;=\;\dfrac{1}{2}|\left\langle\dfrac{\partial
p}{\partial k}\right\rangle|$ (4)

Observe that we are not using the field theory commutator between the field
and its conjugate momentum. Rather our analysis is based on the algebra (3)
which is plausible.

The properties of the function $f(p)$, enlisted below (2) cannot be satisfied
by a finite order polynomial. A possible choice is

$k=f(p)=\dfrac{1}{L_{P}}\sum_{i=0}^{\infty}a_{i}(-1)^{i}{\left(\dfrac{L_{P}\;p}{\hbar}\right)}^{2i+1}$
(5)

Only odd powers of $p$ appear in the polynomial ensuring that $f(p)$ is odd in
$p$ (property 1). The coefficients $\\{a_{i}\\}$ are all positive with
$a_{0}=1$ (to satisfy $p=\hbar k$ at small energy(property 2)). The factor
$(-1)^{i}$ ensures saturation (property 3). The third property further implies
that for $p\rightarrow\infty,\hskip
14.22636ptk\rightarrow\dfrac{2\pi}{L_{P}}$, i.e.

$\displaystyle\sum_{i=0}^{\infty}a_{i}(-1)^{i}{\left(\dfrac{L_{P}\;p}{\hbar}\right)}^{2i+1}$
$\displaystyle\longrightarrow\;2\pi$ (6)

From (5) we get

$\dfrac{\partial k}{\partial
p}=\dfrac{1}{\hbar}\sum_{i=0}^{\infty}a_{i}(2i+1)(-1)^{i}\left(\dfrac{pL_{P}}{\hbar}\right)^{2i}$
(7)

Inverting this we obtain

$\dfrac{\partial p}{\partial
k}=\hbar\sum_{i=0}^{\infty}a^{\prime}_{i}\;\left(\dfrac{pL_{P}}{\hbar}\right)^{2i}$
(8)

where the new coefficients of expansions $\\{a^{\prime}_{i}\\}$ are functions
of $\\{a_{i}\\}$. It is very easy to show that the first coefficient of this
inverted series will be inverse of $a_{0}$, i.e. 1.

Hence the GUP following from (4) takes the form,

$\Delta x\Delta
p\geqslant\left\langle\dfrac{\hbar}{2}\sum_{i=0}^{\infty}a^{\prime}_{i}\left(\dfrac{L_{P}p}{\hbar}\right)^{2i}\right\rangle\geqslant\dfrac{\hbar}{2}\sum_{i=0}^{\infty}a^{\prime}_{i}\left(\dfrac{L_{P}}{\hbar}\right)^{2i}\left((\Delta
p)^{2}+\left\langle p\right\rangle^{2}\right)^{i}$ (9)

where we have used $\left\langle p^{2i}\right\rangle\geqslant\left\langle
p^{2}\right\rangle^{i}$. For minimum position uncertainty we put $\left\langle
p\right\rangle=0$ and our GUP becomes

$\Delta x\Delta
p\geqslant\dfrac{\hbar}{2}\sum_{i=0}^{\infty}a^{\prime}_{i}\left(\dfrac{L_{P}\Delta
p}{\hbar}\right)^{2i}$ (10)

Note that all $a^{\prime}_{i}$’s are positive. If only the first two terms in
(10) are considered we reproduce the GUP predicted by string theory [4, 19].

##  3 Thermodynamics of Schwarzschild black hole with corrections

The object of this section is to use the GUP (10) to evaluate different
thermodynamic entities of a Schwarzschild black hole and thereby find
relations among them.

Let us consider a Schwarzschild black hole with mass M. Let a pair (particle-
antiparticle) production occur near the event horizon. For simplicity we
consider the particles to be massless222For massive particle the expression
for temperature (11) will be modified.. The particle with negative energy
falls inside the horizon and that with positive energy escapes outside the
horizon and observed by some observer at infinity. The momentum of the emitted
particle$(p)$, which also characterises its temperature $(T)$ 333For
simplicity we consider the emitted spectrum to be thermal., is of the order of
its momentum uncertainty $(\Delta p)$ [11]. Consequently

$\displaystyle T=\dfrac{\Delta pc}{k_{B}}$ (11)

For thermodynamic equilibrium, the temperature of the particle gets identified
with the temperature of the black hole itself.

It is now possible to relate this temperature with the mass (M) of the black
hole by recasting the GUP (10) in terms of T and M. In that case the GUP has
to be saturated

$\Delta p\Delta
x={\epsilon}_{1}\dfrac{\hbar}{2}\sum_{i=0}^{\infty}a^{\prime}_{i}\;\left(\dfrac{\Delta
pL_{P}}{\hbar}\right)^{2i}$ (12)

where the new dimensionless parameter ${\epsilon}_{1}$ is a scale factor
saturating the uncertainty relation. We can later adjust it by calibrating
with some known result. We add that the product of $\Delta x$ and $\Delta p$
may be arbitrarily large but we assume that the lower limit can be achieved.

Near the horizon of a black hole the position uncertainty of a particle will
be of the order of the Schwarzschild radius of the black hole [6, 11],

$\displaystyle\Delta x\;=\;{\epsilon}_{2}\dfrac{2GM}{c^{2}}$ (13)

The new dimensionless parameter ${\epsilon}_{2}$ is introduced as a scale
factor and will be calibrated soon.

Substituting the values of $\Delta p$ (11) and $\Delta x$ (13) in (10), the
GUP is recast as

$\displaystyle
M\;=\;\epsilon\dfrac{M_{P}}{4}\sum_{i=0}^{\infty}a^{\prime}_{i}\;{\left(\dfrac{k_{B}T}{M_{P}c^{2}}\right)}^{2i-1}$
(14)

(where we have used the relations
$\epsilon=\dfrac{{\epsilon}_{1}}{{\epsilon}_{2}},~{}M_{P}=\dfrac{L_{P}c^{2}}{G}$
and $\dfrac{c\hbar}{L_{P}}=M_{P}c^{2}~{}~{},~{}M_{P}$ being the Planck mass.)

In the absence of correction due to quantum gravity effects only
$a^{\prime}_{0}=1$ will survive and we should reproduce the semi classical
result. In this approximation (14) reduces to

$M=\epsilon\dfrac{M_{P}^{2}c^{2}}{4k_{B}T}$ (15)

This will fix $\epsilon$. Comparison with the standard semi classical Hawking
temperature [9] $\left(T_{H}=\dfrac{M_{P}^{2}c^{2}}{8\pi Mk_{B}}\right)$
yields $\epsilon=\dfrac{1}{2\pi}$.

So the mass temperature relationship is

$\displaystyle
M\;=\;\dfrac{M_{P}}{8\pi}\sum_{i=0}^{\infty}a^{\prime}_{i}\;{\left(\dfrac{k_{B}T}{M_{P}c^{2}}\right)}^{2i-1}$
(16)

The heat capacity of the black hole , by definition, is given by

$\displaystyle C\;=\;c^{2}\dfrac{dM}{dT}$ (17)

Therefore from (16) we find that

$\displaystyle
C\;=\;\dfrac{k_{B}}{8\pi}\sum_{i=0}^{\infty}a^{\prime}_{i}\;(2i-1)\;\left(\dfrac{k_{B}T}{M_{P}\;c^{2}}\right)^{2i-2}$
(18)

The nature of the heat capacity will become more illuminating if we express it
in terms of particle energy $E=k_{B}T$ and Planck energy $E_{P}=M_{P}c^{2}$.
Then

$C\;=\;\dfrac{k_{B}}{8\pi}\sum_{i=0}^{\infty}a^{\prime}_{i}\;(2i-1)\;\left(\dfrac{E}{E_{P}}\right)^{2i-2}$
(19)

For $E<<E_{P}$ the first term will predominate, and since it is with a
negative signature the heat capacity will also be negative in this region. The
heat capacity increases monotonically as $E\rightarrow E_{P}$. There will be a
point at which the heat capacity vanishes. We consider the corresponding
temperature to be the maximum temperature attainable by a black hole during
evaporation. The process stops thereafter.

So a Schwarzschild black hole with a finite mass and temperature, by radiation
process, loses its mass and in turn its temperature increases. This state
corresponds to a negative heat capacity. Then it attains a temperature at
which $\dfrac{dM}{dT}$ becomes zero (zero heat capacity), i.e there will be no
further change of black hole mass with its temperature. The radiation process
ends here with a finite remnant mass with a finite temperature.

One can also determine the black hole entropy(S) in a similar way. According
to the first law of black hole thermodynamics it is given by

$S\;=\;\int\dfrac{c^{2}dM}{T}$ (20)

For technical simplification this definition is expressed in terms of the heat
capacity (17). Then exploiting (18 ) and carrying out the above integration we
finally obtain

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int\dfrac{CdT}{T}$ (21)
$\displaystyle=$
$\displaystyle\dfrac{k_{B}}{16\pi}\left[\left(\dfrac{M_{P}c^{2}}{k_{B}T}\right)^{2}\;+a^{\prime}_{1}\ln\left(\dfrac{k_{B}T}{M_{P}c^{2}}\right)^{2}\;+\sum_{i=2}^{\infty}a^{\prime}_{i}\;\dfrac{(2i-1)}{(i-1)}\;\left(\dfrac{k_{B}T}{M_{P}c^{2}}\right)^{2(i-1)}\right]$

If we want to express the heat capacity and the entropy in terms of the mass
we have to obtain an expression for $T^{2}$ in terms of M. We can do this by
squaring (16)

$\displaystyle\left(\dfrac{8\pi M}{M_{P}}\right)^{2}$ $\displaystyle=$
$\displaystyle\left(\dfrac{M_{P}c^{2}}{k_{B}T}\right)^{2}+2a^{\prime}_{1}+\left({a^{\prime}_{1}}^{2}+2a^{\prime}_{2}\right)\left(\dfrac{k_{B}T}{M_{P}c^{2}}\right)^{2}+2\left(a^{\prime}_{1}a^{\prime}_{2}+a^{\prime}_{3}\right)\left(\dfrac{k_{B}T}{M_{P}c^{2}}\right)^{4}+\ldots$

Then considering a finite number of terms, dictated by the order of the
approximation, we can obtain an expression for
$\left(\dfrac{k_{B}T}{M_{P}c^{2}}\right)^{2}$ in terms of M by inverting
(LABEL:mt ).

###  3.1 First order correction

We will next discuss the effect of first order correction. Consequently we can
neglect the contribution of $\left(\dfrac{k_{B}T}{M_{P}c^{2}}\right)^{2}$ and
higher order terms in (LABEL:mt). Then we get

$\left(\dfrac{k_{B}T}{M_{P}c^{2}}\right)^{2}\;=\;\dfrac{1}{\left(\dfrac{8\pi
M}{M_{P}}\right)^{2}-2a^{\prime}_{1}}$ (23)

The critical mass below which the temperature becomes a complex quantity is
given by

$M_{cr}=\dfrac{\sqrt{2a^{\prime}_{1}}}{8\pi}M_{P}$ (24)

We will soon show that the evaporation process terminates with a mass greater
than this.

Substituting (23 ) in (21) we get

$\displaystyle\dfrac{S}{k_{B}}$ $\displaystyle=$
$\displaystyle\dfrac{S_{BH}}{k_{B}}-\dfrac{2a^{\prime}_{1}}{16\pi}-\dfrac{a^{\prime}_{1}}{16\pi}ln\left(\dfrac{S_{BH}}{k_{B}}-\dfrac{2a^{\prime}_{1}}{16\pi}\right)-\dfrac{a^{\prime}_{1}}{16\pi}ln(16\pi)$
(25)

where $S_{BH}=k_{B}\dfrac{4\pi M^{2}}{M_{P}^{2}}$ is the Bekenstein-Hawking
entropy.

We now obtain the area theorem from equation (25). This theorem will appear
more tractable if we introduce a new variable $A^{\prime}$ (reduced area)
defined as

$\displaystyle
A^{\prime}\;=\;16\pi\dfrac{G^{2}M^{2}}{c^{4}}-\dfrac{2a^{\prime}_{1}}{4\pi}\dfrac{G^{2}M_{P}^{2}}{c^{4}}\;=\;A-\dfrac{2a^{\prime}_{1}}{4\pi}L_{P}^{2}$
(26)

where A is the usual area of the horizon.

In terms of the reduced area the expression for entropy takes a familiar form

$\displaystyle\dfrac{S}{k_{B}}$ $\displaystyle=$
$\displaystyle\dfrac{A^{\prime}}{4L_{P}^{2}}-\dfrac{a^{\prime}_{1}}{16\pi}\ln\left(\dfrac{A^{\prime}}{4L_{P}^{2}}\right)-\dfrac{a^{\prime}_{1}}{16\pi}\ln(16\pi)$
(27)

This is the area theorem in presence of the GUP (10) upto the first order
correction. The usual Bekenstein-Hawking semiclassical area law is reproduced
for $a^{\prime}_{1}=0$.

The important feature of (27) is that entropy is explicitly expressed as a
function of the reduced area and not the actual area [5, 6]. It has a
singularity at zero reduced area which corresponds to a singular mass given by

$M_{sing}=\dfrac{\sqrt{2a^{\prime}_{1}}}{8\pi}M_{P}$ (28)

We will subsequently prove that the reduced area is always positive in
presence of quantum gravity effect and the singularity is thereby avoided
during the evaporation process. Observe that, to this order, the critical mass
(24) and the singular mass (28) are identical.

The variation of temperature and entropy with mass for different values of
$a^{\prime}_{1}$ is shown in $fig~{}1,2$. The interesting fact about these
curves is their termination at some finite mass for non zero $a^{\prime}_{1}$.
We will explain the reason in the next section.

###  3.2 Second order correction

In this subsection the various thermodynamic variables are computed upto
second order. This implies that terms upto
$\left(\dfrac{k_{B}T}{M_{P}c^{2}}\right)^{2}$ in (LABEL:mt) are retained. With
a simple rearrangement one can easily obtain

$\displaystyle\left(\dfrac{k_{B}T}{M_{P}c^{2}}\right)^{2}$ $\displaystyle=$
$\displaystyle\dfrac{\left[\left(\dfrac{8\pi
M}{M_{P}}\right)^{2}-2a^{\prime}_{1})\right]\pm\sqrt{\left[\left(\dfrac{8\pi
M}{M_{P}}\right)^{2}-2a^{\prime}_{1})\right]^{2}-4({a^{\prime}_{1}}^{2}+2a^{\prime}_{2})}}{2\left({a^{\prime}_{1}}^{2}+2a^{\prime}_{2}\right)}$
(29)

Only the (-) sign is acceptable from the $(\pm)$ part, because the (+) sign
will not produce the semi classical result if we put
$a^{\prime}_{1}=a^{\prime}_{2}=0$.

At a first glance, one observes that the first order expression for
temperature (23) cannot be retrieved simply by putting $a^{\prime}_{2}=0$.
Instead, one has to put ${a^{\prime}_{1}}^{2}+2a^{\prime}_{2}=0$, because both
${a^{\prime}_{1}}^{2}$ and $2a^{\prime}_{2}$ bear the signature of the second
order approximation (see (LABEL:mt)). To make this retrieval simpler we will
rearrange (29) using binomial expansion.

$\displaystyle\left(\dfrac{k_{B}T}{M_{P}c^{2}}\right)^{2}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\left(\dfrac{8\pi
M}{M_{P}}\right)^{2}-2a^{\prime}_{1}}\left[1+\dfrac{\left({a^{\prime}_{1}}^{2}+2a^{\prime}_{2}\right)}{\left[\left(\dfrac{8\pi
M}{M_{P}}\right)^{2}-2a^{\prime}_{1}\right]^{2}}+\ldots\right]$

From this expression it is clear that the first order expression for
temperature can be obtained from the second order expression by putting
${a^{\prime}_{1}}^{2}+2a^{\prime}_{2}=0$. The critical mass is found by
equating the discriminant of (29) to zero. It is given by

$\displaystyle\left(\dfrac{8\pi
M_{cr}}{M_{P}}\right)^{2}=2a^{\prime}_{1}+2\sqrt{{a^{\prime}_{1}}^{2}+2a^{\prime}_{2}}$
(31)

which is greater than (24 ) with first order correction (since
$a^{\prime}_{1},a^{\prime}_{2}$ are all positive).

We conclude this section by computing the corrections (upto second order) to
the entropy and the area law. The corrected entropy follows from (21)

$\displaystyle S$ $\displaystyle=$
$\displaystyle\dfrac{k_{B}}{16\pi}\left[\left(\dfrac{M_{P}c^{2}}{k_{B}T}\right)^{2}\;+a^{\prime}_{1}\ln\left(\dfrac{k_{B}T}{M_{P}c^{2}}\right)^{2}\;+3a^{\prime}_{2}\left(\dfrac{k_{B}T}{M_{P}c^{2}}\right)^{2}\right]$
(32)

Substituting (LABEL:mt2e) in (32) we get

$\displaystyle\dfrac{S}{k_{B}}$ $\displaystyle=$
$\displaystyle\left(\dfrac{S_{BH}}{k_{B}}-\dfrac{2a^{\prime}_{1}}{16\pi}\right)-\dfrac{a^{\prime}_{1}}{16\pi}ln\left(\dfrac{S_{BH}}{k_{B}}-\dfrac{2a^{\prime}_{1}}{16\pi}\right)+\sum_{j=0}^{\infty}c_{j}(a^{\prime}_{1},a^{\prime}_{2})\left(\dfrac{S_{BH}}{k_{B}}-\dfrac{2a^{\prime}_{1}}{16\pi}\right)^{-j}-\dfrac{a^{\prime}_{1}}{16\pi}\ln(16\pi)$

where $S_{BH}=k_{B}\dfrac{4\pi M^{2}}{M_{P}^{2}}$ is the Bekenstein-Hawking
entropy and these new coefficients of expansion $c_{j}$ are functions of
$a^{\prime}_{1}$ and $a^{\prime}_{2}$. The coefficients $c_{j}$ will have an
explicit dependence on ${a^{\prime}_{1}}^{2}+2a^{\prime}_{2}$, which follows
from (LABEL:mt2e). Hence we can get our first order result (25) by putting
${a^{\prime}_{1}}^{2}+2a^{\prime}_{2}=0$.

We can now obtain our area theorem in terms of the reduced area (26) from
(LABEL:s2),

$\dfrac{S}{k_{B}}\;=\;\dfrac{A^{\prime}}{4L_{P}^{2}}-\dfrac{a^{\prime}_{1}}{16\pi}\ln\left(\dfrac{A^{\prime}}{4L_{P}^{2}}\right)+\sum_{j=0}^{\infty}c_{j}(a^{\prime}_{1},a_{2})\left(\dfrac{A^{\prime}}{4L_{P}^{2}}\right)^{-j}-\dfrac{a^{\prime}_{1}}{16\pi}\ln(16\pi)$
(34)

This is the area theorem with second order correction. The expression looks
like the standard corrected area theorem [5, 6], with the role of the actual
area ($A$) being played by the reduced area ($A^{\prime}$).

Some comments are in order. The singularity is again at zero reduced area,
corresponding mass being given by (28). As shown in the next section, this
singular mass is also less than the remnant mass with second order correction.

##  4 Remnant mass and singularity problem

As discussed earlier here we show that the black hole evaporation terminates
at a finite mass which is greater than the either the critical mass
$M_{cr}$(24,31) or the singular mass (28). This demonstrates the internal
consistency of our calculation scheme. Consequently the usual singularity
problem whereby the temperature blows up, is avoided.

###  4.1 First order correction

Considering the first two terms in the series expansion for the heat capacity
(18), we obtain

$C\;=\;\dfrac{k_{B}}{8\pi}\left[-\left(\dfrac{M_{P}c^{2}}{k_{B}T}\right)^{2}+a^{\prime}_{1}\right]$
(35)

Substituting the value of $T^{2}$ from (23)

$C\;=\;\dfrac{k_{B}}{8\pi}\left[-\left(\left(\dfrac{8\pi
M}{M_{P}}\right)^{2}-2a^{\prime}_{1}\right)+a^{\prime}_{1}\right]$ (36)

The variation of heat capacity with mass for different values of
$a^{\prime}_{1}$ is shown in $fig~{}3$.

The collapse of the black hole is terminated when the heat capacity becomes
zero. The mass of the black hole now remains unchanged. This mass is called
the remnant mass. Its value is obtained by solving

$\dfrac{k_{B}}{8\pi}\left[-\left(\left(\dfrac{8\pi
M}{M_{P}}\right)^{2}-2a^{\prime}_{1}\right)+a^{\prime}_{1}\right]\;=\;0$ (37)

leading to,

$\displaystyle M_{rem}$ $\displaystyle=$
$\displaystyle\dfrac{\sqrt{3a^{\prime}_{1}}}{8\pi}M_{P}$ (38)

Alternatively the value for remnant mass can also be obtained by minimising
the entropy (28),

$\dfrac{dS}{dM}\;=\;0$ (39)

and looking at the second derivative $\left(\dfrac{d^{2}S}{dM^{2}}>0\right)$.
The result (38) is reproduced.

The most important fact is that the value (38) is greater than the singular
mass (28) and also the critical mass (24). So we can say that the singularity
will be avoided during the evaporation process and the reduced area will be
positive. At the same time we also managed to avoid any possibility of dealing
with complex values for the thermodynamic entities
$\left(since,~{}M_{rem}>M_{cr}\right)$.

The remnant value of area, reduced area, temperature and entropy are now
expressed in terms of the coefficient $a^{\prime}_{1}$.

$\displaystyle A_{rem}$ $\displaystyle=$ $\displaystyle\dfrac{16\pi
G^{2}M_{rem}^{2}}{c^{4}}\;=\;\dfrac{3a^{\prime}_{1}}{64{\pi}^{2}}\dfrac{16\pi
G^{2}M_{P}^{2}}{c^{4}}$ (40) $\displaystyle A^{\prime}_{rem}$ $\displaystyle=$
$\displaystyle\dfrac{a^{\prime}_{1}L_{P}^{2}}{4\pi}\;=\;\dfrac{16\pi
G^{2}M_{rem}^{2}}{3c^{4}}=\dfrac{A_{min}}{3}$ (41) $\displaystyle T_{rem}$
$\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{a^{\prime}_{1}}}\dfrac{M_{P}c^{2}}{k_{B}}$ (42)
$\displaystyle S_{rem}$ $\displaystyle=$
$\displaystyle\dfrac{k_{B}}{16\pi}\left[a^{\prime}_{1}-a^{\prime}_{1}\ln\left(a^{\prime}_{1}\right)\right]$
(43)

The expressions for different thermodynamic entities including the remnant
mass involves only one parameter $a^{\prime}_{1}$ as a measure of quantum
gravity effect. If we put $a^{\prime}_{1}=0$ we get back all our semi
classical results and the remnant mass becomes zero. The final entropy and
heat capacity become zero while the temperature becomes infinite.

###  4.2 Second order correction

The heat capacity with the second order contribution is given by

$\displaystyle C$ $\displaystyle=$
$\displaystyle\dfrac{k_{B}}{8\pi}\left[-\left(\dfrac{M_{P}c^{2}}{k_{B}T}\right)^{2}+a^{\prime}_{1}+3a^{\prime}_{2}\left(\dfrac{k_{B}T}{M_{P}c^{2}}\right)^{2}\right]$
(44)

The expression for remnant mass can be found either from zero heat capacity
condition or by minimising the entropy. Adopting the first approach we will
compute the remnant mass here. Replacing the value of $T^{2}$ from (29) in
(44) and equating the r.h.s. to zero we obtain the remnant mass

$\left(\dfrac{8\pi
M_{rem}}{M_{P}}\right)^{2}=\dfrac{1}{6a^{\prime}_{2}}\left[-a^{\prime}_{1}\left({a^{\prime}_{1}}^{2}-13a^{\prime}_{2}\right)+\left({a^{\prime}_{1}}^{2}+5a^{\prime}_{2}\right)\sqrt{{a^{\prime}_{1}}^{2}+12a^{\prime}_{2}}\;\right]$
(45)

One can easily show that the r.h.s. is greater than $3a^{\prime}_{1}$ which is
the value (38) for $\left(\dfrac{8\pi M_{rem}}{M_{P}}\right)^{2}$ with first
order correction. The remnant mass is also greater than the critical mass (31)
for positive $a^{\prime}_{1}$ and $a^{\prime}_{2}$. The difference between
remnant mass and the critical mass for different values of $a^{\prime}_{1}$
and $a^{\prime}_{2}$ is shown in $fig.~{}4$ .

##  5 Discussion

The laws of black hole thermodynamics are known to be modified by the presence
of a generalised uncertainty principle (GUP) [5, 6, 7, 8].Here we have derived
a new GUP (based on the presence of a minimal length scale $(L_{P})$) which,
at the lowest orders, is also shown to be compatible with string theory
predictions. Using this GUP various aspects of Schwarzscild black hole
thermodynamics were examined.

Our calculations were performed upto two orders (in $L_{P}$) of corrections.
Corrected structures of mass-temperature relation, area theorem and heat
capacity were obtained. The usual semiclassical expressions were easily
derived.

An important consequence was that the black hole evaporation terminated at a
finite mass. This (remnant) mass was found to be greater than either the
critical mass (below which the thermodynamic variables become complex) or the
singular mass (where the thermodynamic variables become infinite).
Consequently the ill defined situations were bypassed. Also, contrary to
standard results [5, 6] using GUP our modified area law (27,34) is more
transparent when expressed in terms of reduced area defined in (26).

To put our results in a proper perspective let us compare with earlier
findings. A remnant mass was also found in [11] using stringy GUP and in [12]
employing notions of tunnelling. In the first case the remnant mass was given
by $M=M_{P}$ (which is consistent with our findings) and successfully avoided
the singularity. However, in contrast to our analysis, the calculations were
confined to the leading order only. Also, the remnant and the critical mass
became identical. Hence it was not possible to distinguish between the
termination of black hole evaporation and complexification of thermodynamic
variables. The calculation of [12], on the other hand, led to the result that
although there was a remnant mass, the singularity problem could not be
avoided.

## Acknowledgement

One of the authors (S.G.) thanks the Council of Scientific and Industrial
Research (C.S.I.R), Government of India, for financial support. We also thank
B. R. Majhi and S. K. Modak for discussions.

Figure 1:  Temperature - mass curve with $a^{\prime}_{1}=1$ (blue),
$a^{\prime}_{1}=.75$ (red), $a^{\prime}_{1}=\dfrac{1}{3}$ (green) and
$a^{\prime}_{1}=0$ (black) [Semiclassical]

Figure 2:  Entropy - mass curve with $a^{\prime}_{1}=1$ (blue),
$a^{\prime}_{1}=.75$ (red), $a^{\prime}_{1}=\dfrac{1}{3}$ (green) and
$a^{\prime}_{1}=0$ (black) [Semiclassical]

Figure 3:  heat capacity - mass curve with $a^{\prime}_{1}=1$ (blue),
$a^{\prime}_{1}=.75$ (red), $a^{\prime}_{1}=\dfrac{1}{3}$ (green) and
$a^{\prime}_{1}=0$ (black) [Semiclassical]
Figure 4:  difference between the remnant mass and the critical mass
$(M_{rem}-M_{cr})$ in unit of $M_{P}$ for different values of $a^{\prime}_{1}$
and $a^{\prime}_{2}$ shown from two different angles. The difference becomes
zero only at $a^{\prime}_{2}=0$. This point is not considered, since for
$a^{\prime}_{2}=0$, second order correction is not meaningful.

## References

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[1]Dylan Heberle

[1]Jeffrey Moses

1]School of Applied and Engineering Physics, Cornell University, 142 Sciences
Drive, Ithaca, 14853, New York, United States

2]INRS-emt, 1650 blvd Lionel-Boulet, Varennes, QC, J3X 1S2, Canada

3]Laboratoire d’Optique Appliquée, ENSTA Paris, CNRS, Ecole polytechnique,
Institut Polytechnique de Paris, 91762 Palaiseau, France

# Self-dispersion-managed adiabatic frequency conversion

<EMAIL_ADDRESS>Noah Flemens  Connor Davis  Philippe Lassonde  Adrien
Leblanc  François Légaré<EMAIL_ADDRESS>[  [  [

###### Abstract

Dispersion management of few-cycle pulses is crucial for ultrafast optics and
photonics. Often, nontrivial dispersion is compensated using complex optical
systems or minimized through careful design of waveguides. Here, we present
dispersion-managed adiabatic frequency conversion enabling efficient
downconversion of an 11.1-fs near-IR pulse to an 11.6-fs mid-IR pulse spanning
an octave of bandwidth from 2–4 $\mu$m. The adiabatic frequency converter is
designed to impart a constant group delay over the entire bandwidth,
eliminating the need for complex dispersion management and opening a new
avenue for dispersion engineering in ultrafast optics and photonics. Notably,
dispersion engineering through the position-dependent conversion position of
an adiabatic conversion device constitutes an additional mechanism for
dispersion control in ultrafast optics and photonics while maintaining high
conversion efficiency for broadband pulses.

##  1 Introduction

Generating octave-spanning coherent bandwidths and compressing them to their
transform-limits are among the most challenging aspects of optical systems
used for ultrafast spectroscopy and control of matter, especially at the
single-cycle limit. Typically, octave-spanning bandwidths are generated using
super-continuum generation, noncollinear or frequency domain optical
parametric amplification [1, 2], coherent beam synthesis [3], nonlinear multi-
mode mixing in gas-filled hollow core fibers [4], or adiabatic frequency
conversion [5, 6]. However, as absolute bandwidths grow large, precise
dispersion management in these systems plays an increasingly important role;
transmission through even a few millimeters of an optical device can become a
limiting technical factor. Standard compressors using bulk material, gratings,
and prisms can offset large amounts of low-order dispersion but struggle to
mitigate higher-order dispersion and can even exacerbate it.

To address the challenge of higher-order dispersion, modern octave-spanning
systems incorporate multiple double-chirped mirror pairs or programmable
dispersive filters [3] in combination with the standard approaches outlined
above. However, increasingly complex dispersion management schemes adversely
impact the cost, stability, flexibility, and construction time of ultrafast
laser architectures. A greatly simplified single-cycle mid-IR source of CEP-
stable transients could significantly extend the boundaries of ultrafast
science for applications including electronics at optical clock rates [7],
attosecond science [8], and field-resolved detection of plasmonics [9].

Additionally, standard experiments in 2DIR spectroscopy require
interferometrically stable identical pulse pairs, and more advanced
experiments require the ability to impress differences on these pairs such as
relative phase, amplitude, and spectral content. An interferometer produces
identical mid-IR pulse pairs but lacks stability and introduces timing jitter.
A spatial light modulator inside a grating pair or an acousto-optic
programmable dispersive filter (AOPDF) would enable arbitrary shaping of post-
generation pulses; however, such devices are usually limited to sub-octave-
spanning bandwidths. Piecewise dispersion compensation is possible through
coherent pulse synthesis schemes but requires substantial engineering [3].

Several researchers have incorporated dispersion management and pulse shaping
into nonlinear frequency conversion via chirped quasi-phase-matched (QPM)
periodically poled gratings. Arbore et al. [10] proposed compressing optical
pulses using second-harmonic generation (SHG) with a chirped QPM grating and
demonstrated pulse compression of 110-fs FWHM frequency-doubled Ti:Sapphire
pulses [11]. Furthermore, Imeshev et al. demonstrated pulse shaping and
arbitrary dispersion management in SHG [12, 13, 14] and extended the platform
to include difference frequency generation (DFG) in the undepleted pump and
unamplified signal regimes [15]. Charbonneau-Lafort et al. [16, 17] used
tandem chirped QPM optical parametric amplification (OPA) devices to engineer
both the gain and group delay.

Figure 1: Self-dispersion managed ADFG concept: a compressed multi-cycle near-
IR wave undergoes bandwidth and phase-conserving adiabatic down-conversion to
a compressed, CEP-stable, single-cycle mid-IR pulse in a chirped QPM grating
with a local poling frequency G(z) designed to impart zero GDD and higher
order dispersion.

While these studies have simplified ultrafast sources, the use cases are
limited by fundamental and signal sources (in the case of SHG and OPA
respectively) and linearity (in the case of DFG). We extend dispersion
management and pulse shaping to adiabatic frequency conversion [18] which
enables nearly complete photon-number transfer between spectral regions while
preserving the absolute bandwidth of the pulse and can be configured for
adiabatic difference frequency generation (ADFG) or adiabatic sum frequency
generation (ASFG) greatly increasing the output spectral range. Here, we
demonstrate a highly efficient adiabatic frequency downconverter that
intrinsically manages its own large material dispersion. The device concept is
outlined in Fig. 1a. The ADFG device converts a multi-cycle, compressed near-
IR pulse at Ti:Sapphire wavelengths (680-820) to a single-cycle, compressed
mid-IR pulse (2–4 µm). This device allows a linear transfer of amplitude and
phase from the near-IR pulse to an octave-spanning bandwidth in the mid-IR
without the need for additional complex dispersion management. Furthermore,
our concept can accommodate arbitrary dispersion functions as shown in Fig. 1b
where a near-IR pulse with high-order dispersion is converted to chirped mid-
IR pulse with dispersion that can be easily compressed with a typical
compressor. The group delays of example designs are plotted in 1c including
constant group delay (our device), linear group delay (constant group delay
dispersion (GDD)), quadratic group delay (constant third-order dispersion
(TOD)), and combined linear and quadratic group delay (constant GDD and TOD).
Possible group delay functions are bounded by the group delay curves
representing complete propagation as a near-IR pulse (blue curve) and as a
mid-IR pulse (orange curve). We note our technique is not restricted to
adiabatic difference frequency generation and can be extended to adiabatic sum
frequency generation and adiabatic four-wave mixing.

Below, we report a 2-cm LiNbO3 device that converts a transform-limited pulse
in the near-IR to an octave-spanning transform-limited pulse in the mid-IR.
The result is a compressed 11.1-fs near-IR pulse converted to an 11.6-fs mid-
IR pulse with high photon conversion efficiency in the presence of a strong,
narrowband pump. This constitutes a nearly complete elimination of GDD and a
dramatic improvement over the $>10^{4}$ fs2 GDD imparted by a standard, quasi-
linear QPM domain poling function, as used in [19].

##  2 ADFG concept

###  2.1 ADFG

Ultrafast ADFG is a conversion technique for efficient down-conversion of
photons over a wide bandwidth [5, 6]. It is achieved using a monotonic, slowly
varying position-dependent poling frequency $G(z)=2\pi/\Lambda(z)$ to create a
chirped QPM device where $\Lambda(z)$ is the domain period of the structure at
the propagation coordinate $z$. Each input angular frequency $\omega$ is
locally converted within the device at a position $z_{c}(\omega)$ when the
phase matching condition $\Delta k(\omega)=\Delta
k_{0}(\omega)-G(z_{c}(\omega))=0$ where $\Delta k_{0}(\omega)$ is the bulk
wavevector mismatch of the DFG process. Here, $\Delta
k_{0}(\omega)=k_{NIR}(\omega)-k_{MIR}(\omega)-k_{p}$ where $k_{NIR}(\omega)$,
$k_{MIR}(\omega)$, and $k_{p}$ are the near-IR, mid-IR, and pump wavevectors.
Note that since the pump is very narrowband, there is a one-to-one
correspondence between near-IR and mid-IR frequencies. When the poling
function varies slowly enough, the adiabatic condition is achieved and photons
at each frequency undergo unidirectional conversion to the difference
frequency field, in this case the mid-IR field. The efficiency of this process
approaches saturation at unity with increasing pump power [20].

ADFG has been used to produce single-cycle, energetic pulses when accompanied
by a complex dispersion management scheme incorporating grism pairs and a
programmable dispersive filter [19]. That work also showed the ability of ADFG
to preserve complex pulse features through the conversion process. However,
the device required a complicated phase mask applied by a programmable
dispersive filter to compress the pulses. This was required because of the
complicated dispersion profile imparted by the conversion device due to the
longitudinally varying conversion positions in addition to the bulk material
dispersion.

Figure 2: Sub-20-fs peak-to-peak residual group delay for the dispersion
managed device compared to >5 ps for a device of identical conversion
bandwidth and length, but with a standard, linearly chirped G(z).

The dispersion that results from a monotonically varying aperiodic grating has
been studied for both adiabatic optical parametric amplification (AOPA) [16]
and ADFG [19] devices. The group delay $\tau(\omega)$ as a function of
conversion position $z_{c}(\omega)$ is given by:

$\tau(\omega)=k_{NIR}^{{}^{\prime}}(\omega)z_{c}(\omega)+k_{MIR}^{{}^{\prime}}(\omega)(L-z_{c}(\omega))$
(1)

where the primes denote differentiation with respect to $\omega$. Intuitively,
this equation states that the total group delay is the sum of that acquired by
the input near-IR frequency up to the conversion position plus the group delay
acquired by mid-IR frequency after the conversion position. A consequence of
Eq. 1 is that the device dispersion is highly dependent on the design. A
linear poling function $G(z)$ is the simplest design to maintain the adiabatic
condition over a wide bandwidth, and yet it can result in a non-trivial group
delay that can not be compensated without the aid of a pulse shaper. Fig. 2
shows that for a linearly chirped 2-cm LiNbO3 device, the resulting group
delay (orange curve) spans several picoseconds with large amounts of high
order dispersion indicated by the line shape.

###  2.2 Dispersion Engineering

The frequency-dependent and localized conversion of ADFG can be used to cancel
the device’s own material dispersion while generating an octave-spanning
bandwidth in the mid-IR. A similar concept was described theoretically for
adiabatic OPA where two devices were used in tandem to engineer a dispersion
profile for the seeded field [16]. Since the desired output of and ADFG device
is the unseeded field, this can be accomplished in a single device. To cancel
the material dispersion of 2 cm of LiNbO3, we design a QPM grating such that
the group delay profile is flat, i.e., $d\tau(\omega)/d\omega=0$, to minimize
group delay dispersion and higher order dispersion imparted by the device
(Fig. 2, blue).

To design the device, we solve Eq. 1 for $z_{c}(\omega)$ given
$\tau(\omega)=C$ where $C$ is a constant. The constant can take any value that
lies between $k_{NIR}^{{}^{\prime}}(\omega)L$ and
$k_{MIR}^{{}^{\prime}}(\omega)L$ for all relevant values of $\omega$ such that
$z_{c}(\omega)$ is monotonic. The grating can then be constructed by choosing
a poling period $\Lambda(z)$ such that $G(z_{c}(\omega))=k_{0}(\omega)$ at
each $z_{c}(\omega)$. Since $z_{c}(\omega)$ is continuous but the grating
structure is made of discrete domains, this can be accomplished by sampling
$z_{c}(\omega)$ according to the current position in the device after placing
each domain.

Figure 3: Propagation simulations of the device under study, showing (a) the
position-dependent conversion of each spectral component of an input 10-fs
near-IR field, (b) temporal evolution dynamics illustrating stretching of the
input near-IR pulse followed by simultaneous conversion to the mid-IR and
recompression to a 10-fs duration. (c) Mid-IR output pulse intensity (solid
red) compared against initial compressed near-IR input pulse (dashed blue).

Fig. 3a shows a simulation of the near-IR and mid-IR spectra as a function of
propagation in our dispersion-managed device. The propagation-dependent
conversion position of each mid-IR wavelength in the device can be seen from
the sharp conversion transition (i.e. blue to red). In the time domain, this
results in a compressed, 10-fs near-IR input converting directly to a
compressed, 10-fs mid-IR output (Fig. 3b,c). Conveniently, the device acts as
its own stretcher/compressor, broadening the input near-IR pulse to several
picoseconds prior to conversion, keeping the peak intensity low. Subsequently,
the generated mid-IR pulse self-compresses back to 10 fs (Fig. 3).
Furthermore, $\tau(\omega)$ can be engineered to suit specific applications
such as compensation of external sources of dispersion so that the input pulse
itself need not be compressed.

##  3 Experimental Results

###  3.1 Setup

Figure 4: Experimental setup. (a) Conversion stage with the dispersion-managed
ADFG device. The near-IR beam is focused into the ADFG crystal using a concave
mirror (CM) and combined with the 1030-nm pump using a dichroic mirror (DM).
The generated mid-IR light is collimated with a second CM. (b) SHG FROG
measurement stage. The near-IR pulse is diverted before the ADFG stage using a
flip mirror (FM) and measured using a dispersion-balanced SHG FROG setup. (c)
FROSt measurement stage used to characterize the mid-IR pulse generated from
the compressed near-IR pulse. The mid-IR pulse is separated from the residual
pump and near-IR beams using a longpass filter (LF). Additional optics include
mirrors (M), beamsplitter (BS), and parabolic mirrors (PM).

The laser source consists of a 1030-nm Yb:YAG laser with two outputs: a 900-fs
low-power (2 µJ) output and a 3-ps high-power (17 mJ) output. The 900-fs low-
power output drives white-light (WL) generation in a 13-mm YAG crystal, from
which a bandwidth of 670–950 nm is selected and amplified to 50 µJ in a series
of noncollinear optical parametric chirped pulse amplifiers pumped with
frequency-doubled light from the high-power output. The amplified near-IR
pulse is split into several paths. The first is sent to the adiabatic
frequency converter to generate the 2–4 µm mid-IR pulse. A 4f fourier-domain
pulse shaper (PhaseTech, Inc.) is used to spectrally filter the near-IR pulse
to 680–820 nm and compress it at the input of the adiabatic frequency
conversion stage. A second near-IR beam, compressed with a chirped mirror
pair, is used as the pump pulse for frequency-resolved optical switching
(FROSt), described below, which is used to characterize the generated mid-IR
pulse.

The ADFG setup is shown in Figure 4a. The compressed 680–820-nm near-IR pulse
is combined with 200 µJ of 1030-nm pump light in the ADFG crystal generating a
compressed 2–4-µm mid-IR pulse. The near-IR beam is focused into the frequency
conversion crystal with a 50-cm concave mirror and combined with the pump
pulse using a shortpass dichroic mirror. A longpass filter (Andover 1650-nm
longpass filter, 1.1-mm Si) is used to separate the mid-IR pulse from the
residual near-IR and pump light. The mid-IR light is collimated using a 50-cm
concave mirror. Through the ADFG device, we achieve a nearly flat 70% photon
conversion efficiency across the input near-IR spectrum.

The near-IR pulse is characterized at the input of the ADFG stage using SHG
FROG as shown in Fig. 4b. The SHG FROG stage is a noncollinear, dispersion-
balanced autocorrelator using BBO for SHG. SHG FROG traces are constructed by
measuring the second harmonic spectrum as a function of pulse delay using an
OceanFX spectrometer. Using these measurements, the near-IR pulse is
compressed by adding a phase with the 4f pulse shaper.

The generated octave-spanning mid-IR pulse is characterized using FROSt. FROSt
is a phase-matching-free characterization technique that was recently used to
characterize two-octave-spanning infrared pulses [21, 22]. In FROSt, a pump
pulse is used to excite free-carriers in a semiconductor sample creating a
sharp absorption edge in time which acts as an optical gate for characterizing
a probe pulse. The probe pulse spectrum is measured as a function of pump-
probe delay, allowing the electric field of the probe pulse to be
reconstructed using a ptychographic algorithm [21]. The FROSt setup is shown
in Fig. 4c. We use a 10-µJ near-IR pulse compressed to 14-fs FWHM to pump the
free-carrier excitation in a 1-mm Si sample. The pump and probe pulses are
combined at the FROSt sample. The probe is focused to a 44-µm 1/$e^{2}$ beam
diameter at the sample. To achieve a strong, uniform free-carrier excitation
across the probe beam, the pump is focused to a 275-µm 1/$e^{2}$ beam
diameter. The mid-IR spectrum is measured using a MCT-array spectrometer as a
function of pump-probe delay.

###  3.2 Results

The measured and retrieved SHG FROG traces for the near-IR pulse are shown in
Fig. 5b with the normalized RMS error, g, described in [23]. Figure 5c shows
the retrieved near-IR spectrum (blue) and group delay (gray). The measured and
retrieved FROSt traces for the mid-IR pulse are shown in Fig. 5d. Figure 5e
shows the retrieved mid-IR spectrum (red) and group delay (gray). A number of
modulations are observed across the mid-IR spectrum and have been identified
with a parasitic OPA process that occur at the end of the device in a region
that is non-critical to the dispersion properties of the device.

Figure 5: (a) Outline of experimental setup indicating components (shaded
region) that must be considered in the calculation of the group delay imparted
by our zero-GDD device using the near-IR and mid-IR pulse measurements. (b)
Measured and retrieved SHG FROG traces for the near-IR pulse. (c) The
retrieved near-IR spectral intensity (blue) and group delay (gray). (d)
Measured and retrieved FROSt traces for the mid-IR pulse. (e) The retrieved
mid-IR spectral intensity (red) and group delay (gray) with the calculated GD
of our zero-GDD device (black). The GD imparted by the device is the
difference between the gray GD curves in (e) and (c) accounting for the
components in the shaded region shown in (a).

The group delay imparted by the device is determined by comparing the near-IR
spectral phase measured with SHG FROG and the output mid-IR spectral phase
measured with FROSt while accounting for all the dispersive components in the
paths to the measurement stages. These components are shown by the shaded
region in 5a and include a 1-mm fused silica beamsplitter used in the SHG FROG
stage, a 1.1-mm silicon (Si) longpass filter used to isolate the mid-IR pulse,
and the air in the beam paths to the measurement devices. Taking these into
account, the spectral phase imparted by our device $\phi_{D}$ and related
group delay $\tau(\omega)=(d\phi_{D}(\omega))\textfractionsolidus d\omega$ is
calculated by $\phi_{D}=\phi_{MIR}-\phi_{NIR}-\phi_{optics}-\Delta\phi_{air}$
where $\phi_{MIR}(\phi_{NIR})$ is the measured spectral phase of the mid-IR
(near-IR) pulse, $\phi_{optics}$ is the measured spectral phase of the
longpass filter minus the analytic spectral phase of the beamsplitter
calculated from the Sellmeier equation [24], and $\Delta\phi_{air}$ is the
difference in phase experienced by the mid-IR and near-IR pulses as they
travel through air to their respective measurement setups. Note, $\phi_{NIR}$
and $\phi_{MIR}$ directly correspond to the gray group delay curves shown in
Fig. 5c and 5d respectively.

Figure 6: (a) Retrieved pulse intensities of the near-IR input (blue) and mid-
IR output (red). The 11.1-fs near-IR pulse is converted to an 11.6-fs mid-IR
pulse. (b) Zoomed-in plots of the near-IR (top) and mid-IR (bottom) pulse
intensities with temporal phases (black).

The group delay imparted by our device is shown as the black curve in Fig. 5e.
Over the target bandwidth (2–4 µm), the group delay remains nearly flat, only
deviating near the edges of the spectrum and at 0.7 and 0.53 PHz where dips
occur in the power spectrum. We believe the dip at 0.7 PHz is caused by water
absorption and the dip at 0.53 PHz is caused by parasitic effects due to the
uniform poling period at the end of the ADFG grating. To demonstrate the
performance of the device, Fig. 6a compares the mid-IR output pulse (red)
using our compressed near-IR pulse (blue). The group delay imparted by the
device results in an 11.6-fs pulse (FWHM duration), which is 1.05x the input
near-IR pulse. Zoomed-in images with the retrieved phase are shown in Fig. 6b.

##  4 Discussion

In summary, we have demonstrated the proof-of-principle of a concept for
generating single-cycle mid-IR pulses efficiently by adiabatic down-conversion
of a few-cycle near-IR source, in a monolithic device imparting near-zero GDD.
Alternatively, the device could be designed to impart a desired dispersion
profile in combination with a small amount of bulk material dispersion. This
would enable high energy frequency conversion by avoiding high peak
intensities in the device.

We believe this work is an important step in advancing the state of the art
for ultrafast lasers and photonics as well as the study of ultrafast science.
By compensating for its own dispersion and/or the dispersion of other
components of the laser architecture, the dispersion managed ADFG concept can
greatly reduce the complexity, cost, and time investment of building ultrafast
laser systems. It also serves to enhance the modularity and compactness of
ultrafast frequency conversion stages by eliminating the need for the
sophisticated dispersion compensation stages. Such stages introduce
instability by increasing extra beam paths and optics and have to be tailored
to the specific conversion device, making them inflexible. Finally,
dispersion-engineered adiabatic frequency conversion provides another
mechanism to tune the dispersion of integrated photonics, which is typically
achieved through geometric dispersion of waveguides, greatly increasing the
tunability and range of ultrafast photonic conversion devices.

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