Robust and broadband frequency conversion in composite crystals with tailored segment widths and χ nonlinearities of alternating signs

Abstract

We propose an efficient, robust, and broadband nonlinear optical frequency conversion technique, which uses segmented crystals constructed in analogy with the composite pulses in nuclear magnetic resonance and quantum optics. The composite crystals are made of several macroscopic segments of nonlinear susceptibilities of opposite signs and specific thicknesses, which are determined from the condition to maximize the conversion efficiency with respect to variations in the experimental parameters. These crystals deliver broadband operation for significantly lower pump intensities than single bulk crystals. We demonstrate this technique by numerical simulation of sum-frequency generation in MgO : LiNbO 3 crystal.

Full text

Robust and broadband frequency conversion in
composite crystals with tailored segment
widths and χ(2) nonlinearities of alternating signs
A. A. Rangelov,1,* N. V. Vitanov,1and G. Montemezzani2,3
1Department of Physics, Sofia University, James Bourchier 5 blvd., 1164 Sofia, Bulgaria
2Université de Lorraine, LMOPS, EA 4423, F-57070 Metz, France
3Supélec, LMOPS, EA 4423, F-57070 Metz, France
*Corresponding author: rangelov@phys.uni‑sofia.bg
Received March 3, 2014; revised April 11, 2014; accepted April 14, 2014;
posted April 14, 2014 (Doc. ID 207588); published May 12, 2014
We propose an efficient, robust, and broadband nonlinear optical frequency conversion technique, which uses seg-
mented crystals constructed in analogy with the composite pulses in nuclear magnetic resonance and quantum
optics. The composite crystals are made of several macroscopic segments of nonlinear susceptibilities of opposite
signs and specific thicknesses, which are determined from the condition to maximize the conversion efficiency with
respect to variations in the experimental parameters. These crystals deliver broadband operation for significantly
lower pump intensities than single bulk crystals. We demonstrate this technique by numerical simulation of
sum-frequency generation in MgO:LiNbO3crystal. © 2014 Optical Society of America
OCIS codes: (190.0190) Nonlinear optics; (190.4223) Nonlinear wave mixing; (190.2620) Harmonic generation and
mixing.
http://dx.doi.org/10.1364/OL.39.002959
A common notion in nonlinear optics is that efficient
frequency conversion can be achieved only if the
phase-matching condition is satisfied [1,2]. The most
common approaches to achieve phase matching exploit
either the birefringence of the nonlinear material
(birefringence phase matching) or a periodic switch of
the sign of the nonlinearity to compensate for the phase
mismatch (quasi-phase matching, QPM) [2,3]. In both
cases the efficiency of the frequency conversion process
decreases rapidly away from perfect phase matching, for
instance, due to a detuning of the pump wavelength or a
variation in the crystal temperature. However, in some
cases it is necessary to broaden the frequency spectrum
or the temperature tolerance of the nonlinear interaction,
especially if tunable sources or (spectrally broad) ultra-
short-pulse sources are used. For QPM in materials
where domain structuring with sufficiently short periods
is possible, the spectral bandwidth can be widened by
designing chirped [4–6] or other carefully designed aperi-
odic domain structures [7]. However, for single crystals
under birefringence phase matching, the only way to
broaden the spectral response is a reduction of the length
of the nonlinear sample, which obviously requires a
much higher pump power.
In this Letter, we introduce an alternative approach for
robust, efficient, and broadband frequency conversion,
which combines the elements, and the advantages, of
birefringence phase matching and QPM. We suppose that
the nonlinear interaction is phase-matchable by birefrin-
gence phase matching and the sign of the nonlinear
susceptibility is inverted just a few times. In contrast
to the standard QPM, where the domain lengths are typ-
ically of the order of tens or hundreds of wavelengths,
here the domain length is of the order of the interaction
length (i.e., millimeters). Therefore, the approach pre-
sented here can be implemented even in materials
for which ferroelectric poling or other microscopic struc-
turing techniques are not possible, just by stacking a
number of thin crystals of specific thicknesses, mutually
inverted by 180 deg.
The proposed technique is an analog of the composite
pulses in nuclear magnetic resonance (NMR) [8,9] and
quantum optics [10,11]. Unlike most composite pulses,
which use specific relative pulse phases, here we use
the composite sequences of Shaka and Pines [12,13],
which use only sign change of the coupling (i.e., the non-
linear susceptibility here). This is dictated by the neces-
sity to maintain the polarization of each wave after each
interface and by the strict limitation of two directions of
the domains in the case of domain poling.
We consider the symmetrized coupled-wave equations
for collinear three-wave mixing in the slowly varying
amplitude approximation [1]:
i∂zA1~
ΩA
2A3exp−iΔkz;(1a)
i∂zA2~
ΩA
1A3exp−iΔkz;(1b)
i∂zA3~
ΩA1A2expiΔkz;(1c)
where ~
Ω−χ2∕c
ω1ω2ω3∕n1n2n3
pis the coupling co-
efficient, zis the position along the propagation axis, ωj,
kj, and njare the frequencies, the wavenumbers, and the
refractive indices of the electric fields, cis the speed of
light in vacuum, and χ2is the effective second-order sus-
ceptibility. Here, j1, 2, 3 refer to the pump, signal, and
idler fields, respectively. The amplitudes Aj≡
nj∕ωj
pEj
are proportional to the amplitudes Ejof the electric
fields; jAjj2is proportional to the number of photons
associated with the jth wave. The phase mismatch
parameter is Δkk1k2−k3.
May 15, 2014 / Vol. 39, No. 10 / OPTICS LETTERS 2959
0146-9592/14/102959-04$15.00/0 © 2014 Optical Society of America

The three nonlinear coupled Eqs. (1) can be simplified
assuming that the incoming pump wave is much stronger
than the signal and therefore its amplitude remains
constant (undepleted pump approximation) during the
evolution (A1const). This leads to the system of
equations
i∂zBzHBz;(2)
where BzB2z;B
3zT,B2zA2zeiΔkz∕2,
B3zA3ze−iΔkz∕2and
H1
2−ΔkΩ
ΩΔk(3)
with Ω2~
ΩA1. If initially B3zi0, then η
jB3zfj2∕jB2zij2is the conversion efficiency. In the
phase-matched case (Δk0), which corresponds to res-
onant excitation in quantum physics, the frequency con-
version efficiency is ηsin2S∕2, with SRzf
ziΩdz.
Hence, complete energy transfer between the signal
and the idler occurs for Sπor odd multiples of π[1].
The value of ηis sensitive to both variations in Sand
the phase mismatch. This sensitivity is greatly reduced
by replacing the bulk crystal by a composite crystal—
i.e., a stack of crystals with different thicknesses and
alternating signs of the nonlinear susceptibility.
To this end, we note that by mapping the coordinate
onto time dependence, z→t, Eq. (2) becomes the con-
ventional time-dependent Schrödinger equation for a
two-state atom in the rotating-wave approximation
[14], with Hbeing the Hamiltonian. Then the two field
amplitudes B2and B3are the probability amplitudes
for the ground and excited states, the coupling Ωin
Eq. (3) is the Rabi frequency, while Δkis the atom-laser
detuning [14]. This analogy has been used earlier in the
design of aperiodically poled nonlinear crystals [15–17]
in order to perform adiabatic frequency conversion in
a manner similar to the process of rapid adiabatic pas-
sage through a level crossing in quantum physics.
We propose to construct composite crystals in analogy
with the technique of composite pulses in quantum phys-
ics [8–11]. Composite pulses use less energy than adia-
batic techniques while they deliver higher efficiency
and similar robustness to parameter variations; similar
advantages are expected in frequency conversion too.
In optical frequency conversion, instead of a pulse
sequence, we propose to use a sequence of Ncrystal do-
mains with suitably chosen thicknesses l1;l
2;…;l
Nand
alternating positive and negative susceptibilities. In the
undepleted-pump regime the problem is linear and the
total evolution matrix is a product of the evolution
matrices for each segment (acting from right to left),
UulN…ul2ul1;(4)
where uljdescribes the evolution matrix for the jth seg-
ment. Most composite sequences are made of rectangular
pulses and hence the Hamiltonian matrix Hof Eq. (3)is
constant. This is the case for our composite crystals too,
where this condition translates into constant coupling Ω
(i.e., constant nonlinear susceptibility χ2) and constant
frequency mismatch Δkjover the jth segment. For
constant Hjthe evolution matrix reads ulj
exp−iHjljand the total evolution matrix (4) is readily
computed. The “Hamiltonians”Hjj1;2;…;Ndiffer
only by the sign of Ω, which changes from segment to
segment. The segment thicknesses l1;l
2;…;l
Nare free
control parameters, which are chosen from the condition
to maximize the conversion efficiency ηjU21 j2with
respect to variations of Δk(around Δk0) and the cou-
pling Ω(around a selected value Ω0). Such a composite
crystal will tolerate phase mismatch in a certain range of
wavelengths, i.e., it will act as a broadband device. This
optimization is done by making a double Taylor series
expansion of the total evolution matrix around selected
values of the coupling and phase mismatch (Ω0,Δk0):
UΩ;ΔkUΩ0;0U0
ΩΩ0;0Ω−Ω0
U0
ΔkΩ0;0Δk1
2U00
Ω;ΩΩ0;0Ω−Ω02
U00
Ω;ΔkΩ0;0Ω−Ω0Δk
1
2U00
Δk;ΔkΩ0;0Δk2⋅⋅⋅ (5)
then setting jUΩ0;021j21and seeking the lengths
l1;2…;N that nullify as many derivatives as possible.
The physical mechanism of the composite crystals is the
destructive interference of phase-mismatch errors due to
tailored multiple light scattering.
Nearly all composite pulses use the relative phases
between the constituent pulses as control parameters
in order to design a desired interaction profile. Because
of the limitations of our composite crystals, in which we
are only allowed to flip the sign of the coupling Ωfrom
segment to segment (which is done by changing the sign
of χ2), we use the composite pulses of Shaka and Pines
[12,13], which use only sign flips from pulse to pulse (i.e.,
phases 0 and π) and the control parameters are the pulse
durations. These composite sequences, adapted to
composite crystals, are listed in Table 1.
Figure 1illustrates the performance of the composite
crystals with N4and N6segments from Table 1
compared to a single bulk crystal. The contours are
calculated numerically from Eqs. (1) in the cases of
undepleted pump (but depleted signal, left frames) and
depleted pump (right frames). Figure 1shows clearly that
Table 1. Domain Lengths (in Units of Total Crystal
Length L) for Composite Crystals Composed of
NDomains with Alternating Sign of χ2, which
Optimize the Bandwidth and the Robustness
of the Frequency Conversion Processa
NDomain Lengths l1;l2;…;lNin Units L
4 0.053; 0.191; 0.307; 0.449
6 0.161; 0.174; 0.348; 0.148; 0.083; 0.087
11 0.048; 0.091; 0.087; 0.177; 0.059; 0.038; 0.142;
0.074; 0.175; 0.07; 0.039
15 0.039; 0.067; 0.077; 0.043; 0.031; 0.116; 0.052;
0.077; 0.134; 0.06; 0.049; 0.026; 0.12; 0.064; 0.044
aThe values are taken from [12,13].
2960 OPTICS LETTERS / Vol. 39, No. 10 / May 15, 2014

the region of high conversion efficiency expands strongly
for longer composite crystals. In other words, the
composite crystals exhibit much broader acceptance
bandwidths compared to a single crystal. On the other
hand, since the conversion is partially compensated in
the opposite domains, the pump power needed to reach
full signal-to-idler conversion is higher in the composite
case than for a single crystal of equal total length.
To illustrate the matter further, we choose a real
crystal: 5 mol. % magnesium oxide doped lithium niobate
(MgO:LiNbO3). This ferroelectric nonlinear crystal at-
tracts much interest due to its much higher damage
threshold compared to pure LiNbO3, high nonlinear op-
tical coefficient, broad transparency range, and suitabil-
ity for domain poling [18]. We examine sum-frequency
generation (SFG) in both the undepleted and depleted
pump regimes. We assume to be near temperature-tuned
noncritical phase-matching configurations of the type
oo →e(two ordinarily polarized waves generate an
extraordinarily polarized wave) for near-infrared wave-
lengths of the S-band telecommunication window and
its harmonics,
750 nmo1500 nmo→500 nme:(6)
The temperature-dependent Sellmeier equations from
[18] indicate that this process can be noncritically phase
matched at temperature 363 K.
The contour plots in Fig. 2compare the SFG efficiency
ηfor a bulk MgO:LiNbO3crystal and several composite
crystals made of segments of MgO:LiNbO3, versus the in-
put pump intensity (fixed wavelength at 750 nm) and the
signal wavelength near 1.5 μm. The total crystal lengths
in the different cases have been chosen such that the
maximum conversion efficiency is obtained for similar
pump intensities. The figure shows the greatly enhanced
robustness and frequency bandwidth of the SFG process
by the composite crystals compared to the bulk crystal.
We find it remarkable that, although the theoretical argu-
mentation has been derived in the undepleted-pump limit
(left frames), the advantages of the composite crystals
persist for a depleted pump (right frames) too, even if
to a slightly lesser extent. This is seen by comparing
the 50% efficiency bandwidth in Figs. 2(d) and 2(f) with
the one for a bulk crystal [Fig. 2(b)].
Furthermore, in order to test the sensitivity of the
composite crystals to the prescribed domain lengths,
we have conducted the numerical simulations in Fig. 2
by artificially adding a random 10% error in the domain
lengths listed in Table 1. We have verified that this
error, which is far worse than what can be achieved ex-
perimentally, does not change the conversion efficiency
dramatically.
One may argue that a large bandwidth can be obtained
also by using sufficiently thin single-domain crystals.
Such an approach has the drawback of requiring a much
higher pump intensity than for composite crystal struc-
tures proposed here. Table 2gives the necessary total
crystal length Land pump intensity Ipfor a given
0 2 4 6 8 10
(f)
(d)
0
0.75
0.25
0.50
1.00
(b)
depleted
-20
-10
0
10
20 (c)
Phase Mismatch (units of 1/L)
-20
-10
0
10
20
0 2 4 6 8 10
(e)
Coupling (units of 1/L)
-20
-10
0
10
20 (a)
undepleted
Fig. 1. Numerically simulated SFG efficiency from Eqs. (1)
versus the coupling ~
Ωand the phase mismatch Δk. Frames
(a) and (b) are for a single crystal N1, while frames (c)
and (d) are for a composite crystal of N4domains
and frames (e) and (f) are for a composite crystal of N6
domains, with the thicknesses listed in Table 1. The left col-
umns (a), (c), and (e) are for the undepleted-pump regime
jA1zij  10jA2zij, while the right columns (b), (d), and
(f) are for the depleted-pump regime jA1zij  jA2zij.
The inner curves mark the 90% efficiency level, and the outer
curves are for the 50% level. The reciprocal of the total crystal
length Lis used as a unit for ~
Ωand Δk.
1470
1490
1510
1530 (a)
undepleted
(b)
depleted
1470
1490
1510
1530 (c)
0
0.75
0.25
0.50
1.00
Signal Wavelength [nm]
1470
1490
1510
1530
0 1 2 3
(e)
Pump Intensity [GW/cm2]
0 1 2 3
(f)
(d)
Fig. 2. Efficiency of SFG versus the pump intensity and the
signal wavelength for different composite crystals from Table 1.
The contour curves show the 50% efficiency level. Frames (a)
and (b), a bulk MgO:LiNbO3crystal (L0.5mm, N1).
Frames (c) and (d), 2 mm long MgO:LiNbO3composite crystal
with N6segments. Frames (e) and (f), 3.5 mm long
MgO:LiNbO3composite crystal with N15 segments. The
numerical simulations use Eqs. (1) and are made for undepleted
pump [left frames, (a), (c), and (e)] and depleted pump [right
frames, (b), (d), and (f)]. In the depleted case the input photon
intensities of the pump and the signal are chosen to be equal.
We have included an up to 10% random error in the thickness of
each domain compared to the values in Table 1.
May 15, 2014 / Vol. 39, No. 10 / OPTICS LETTERS 2961

acceptance bandwidth Δλ1∕2. Here, Δλ1∕2is defined as
the FWHM at the pump level Ipfor which one has the
first maximum (η1) in the undepleted-pump regime
for Δk0. Table 2demonstrates that the required inten-
sities rapidly decrease with the increasing number of
segments N: for example, the intensities required for
the 15-segment composite crystal is by a factor of 6 lower
than for a single-domain bulk crystal. We found numeri-
cally that, for a fixed total crystal length, the bandwidth
of the nonlinear conversion scales roughly as the square
root of the number of domains used. This makes the
composite crystals very useful when the maximum pump
power is limited, e.g., by the damage threshold of the
nonlinear crystal.
In summary, we have used the analogy between the
frequency conversion equations and the time-dependent
Schrödinger equation in order to build composite crys-
tals for frequency conversion in analogy with the popu-
lation-inverting composite pulses in NMR and quantum
optics. We have used the sign-alternating dual-compen-
sating composite pulse sequences by Shaka and Pines
[12,13], which are particularly suitable for frequency con-
version because they require only sign flips of the non-
linear optical susceptibility between the neighboring
crystal segments. The resulting composite crystals
deliver robust and broadband frequency conversion both
in the undepleted and depleted-pump regimes. Although
we have given an example of SFG in a crystal and con-
figuration allowing noncritical phase matching, the pre-
sented concept is valid also for critical phase matching or
for configurations where the primary phase-matching
mechanism is already QPM with short period domains.
With only minor changes, the technique can be also
extended to other nonlinear interactions, such as second-
harmonic generation, difference frequency generation,
or parametric generation and amplification. The pro-
posed technique may be implemented by using domain
poling of rather long aperiodic domains or by simply
stacking several crystals of specific thicknesses. We
note that the concept of segmented composite crystals
can be alternatively developed by using properly chosen
detuning phases between different crystal segments
implemented by the standard QPM technique [19]. One
may also extend the use of the composite approach to
the quantum regime of spontaneous parametric down-
conversion in analogy with broadband entangled-photon
generation [20,21].
This work is supported by the Bulgarian NSF
grant DMU-03/103 and the Alexander-von-Humboldt
Foundation.
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Table 2. Pairs L;Ipof the Total Crystal Length L
(in mm) and the Pump Intensity Ip(in GW∕cm2)
Required to Achieve a Predefined Value of the
Spectral Acceptance Bandwidth Δλ1∕2(FWHM)
for Different Composite Crystals from Table 1
Δλ1∕2(nm) N1N6N11 N15
20 (0.5; 1.5) (4.5; 0.6) (12; 0.4) (20; 0.25)
40 (0.2; 5) (2.5; 1.5) (5.5; 1) (7; 0.8)
60 (0.13; 8) (1.5; 3) (3.5; 2) (4.5; 1.5)
2962 OPTICS LETTERS / Vol. 39, No. 10 / May 15, 2014