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The University of Manchester Research
Dual-Frequency Defect-Mode Lasing in AperiodicDistributed
Feedback (ADFB) CavitiesDOI:10.1109/LPT.2016.2561079
Document VersionAccepted author manuscript
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Citation for published version (APA):Folland, T., &
Chakraborty, S. (2016). Dual-Frequency Defect-Mode Lasing in
Aperiodic Distributed Feedback(ADFB) Cavities. IEEE Photonics
Technology Letters, 28(15),
1617-1620.https://doi.org/10.1109/LPT.2016.2561079
Published in:IEEE Photonics Technology Letters
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IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. XX, NO. X, XXXXX 1
Dual-Frequency Defect-Mode Lasing in AperiodicDistributed
Feedback (ADFB) Cavities
Thomas G. Folland and Subhasish Chakraborty
Abstract—In this letter we aim to provide the first
in-depthstudy of a multiband hologram filter response under the
influenceof gain. We find that these aperiodic DFB gratings, which
areessentially computer optimized digital holograms, can be
inverse-designed to possess multiple defect-like modes analogous to
thosepresent in quarter-wave phase shifted gratings. We
attributethis phenomenon to the interaction between multiple
photonicband gaps in aperiodic DFB cavities. Further examination
ofthe aperiodic DFB gratings reveal that the defect-mode
lasingsolutions are insensitive to variations of the grating
feedbackstrength (κ). This result in particular, which has
importantsignificance to minimize error in fabrication, is
confirmed againstpublished experimental data for an aperiodic DFB
laser.
Index Terms—Lasers, Distributed Feedback Lasers,
LaserResonators, Semiconductor Lasers.
I. INTRODUCTION
CONTROLLING the color of light emitted by a laserrequires
careful design of the resonant cavity used.Although the traditional
target for resonator design is singlefrequency lasing, reliable
multicolor emission is also appeal-ing for other applications. For
example, coherent frequencygeneration requires the photomixing of
two lasers in a non-linear medium to generate spectrally pure light
at a sum ordifference frequency [1]. By generating both frequencies
inthe same cavity the common-noise effect will significantlyreduce
the linewidth of generated light, thereby relaxing phasestability
requirements on the laser. Distributed feedback (DFB)gratings are
one of the most conventional ways of achievingfrequency selective
mode control. The conventional uniformDFB grating reflects light at
twice the grating period (Λ)through Bragg scattering, leading to a
photonic stop-bandat the Bragg wavelength (λB = 2Λneff , where neff
is thegrating effective index) [2], [3]. Distributed feedback
lasersbased on uniform gratings show a mode spectrum with
twodegenerate modes with identical threshold gain values at thestop
band edges [4], which may result in dual color emission.The spacing
between the two band-edge lasing modes and theircorresponding
threshold values are intrinsically linked throughthe grating
feedback strength (κ), so it is difficult to controlthe mode
spacing without affecting laser performance. Further-more small
perturbations to the DFB cavity, such as non-zero
Manuscript received Feburary 17, 2016; revised April 14, 2016;
acceptedApril XX, XXXX. Date of publication May XX, XXXX; date of
currentversion June XX, XXXX. TGF acknowledges the North-West
NanoscienceDoctoral Training Centre, EPSRC grant EP/G03737X/1
T. Folland and S. Chakraborty are with the School of
Electricaland Electronic Engineering, The University of
Manchester,Manchester M13 9PL, U.K. (e-mail:
[email protected];[email protected];).
Digital Object Identifier 10.1109/LPT.XXXX.XXXXXX
a)
b)
Periodic
Single Phase Shift
Aperiodic
0.90 0.95 1.00 1.05 1.10Frequency (fB)
|FT|
c)
Fig. 1. Illustration of spatial domain holograms a) - c) show
hologramswith an increasing number of spatial frequency components,
represented bya uniform grating, a grating with a quarter-wave
phase-shift and an aperiodiclattice.
facet reflectivity, can break the threshold degeneracy
betweenband-edge modes [4], leading to single mode emission.
A common way to induce robust single frequency emissionfrom a
DFB laser is to break the real-space uniformityby introducing a
quarter-wave phase-shift or defect into thestructure; this leads to
a defect-mode within the photonic stopband at λB. However, with a
single quarter-wave phase-shift,designers are restricted to only
one defect-mode. Relaxing thegrating spatial uniformity further
allows almost any desiredspectral response to be achieved,
resulting in the formationof an aperiodic lattice [5]. The recently
developed discretelytunable aperiodic DFB (ADFB) laser is a
strikingly successfulexample of such an approach [6]–[8], in which
multiple defectsare incorporated into a grating using aperiodic
lattice engi-neering to simultaneously enable precise frequency
selectionand tuning. Unlike other approaches [9]–[12], the
aperiodiclattice design is not the output of a deterministic
mathematicaloperation; rather it is a computer-optimized digital
hologram.The hologram contains a multitude of phase-shifts, the
preciselocations and sizes of which are set such that they
operatecollectively to provide a well-defined set of spatial
frequencycomponents, as revealed by the Fourier transform (FT) of
thehologram [5], [6]. To illustrate the physical structure of
suchaperiodic lattices, in Fig. 1 we present uniform,
quarter-waveshifted and ADFB gratings of the same length (see
method-ology for synthesis). Each grating has a distinct FT
response,with one, two and three spatial frequency components
foruniform, quarter-wave shifted and ADFB gratings respectively,as
indicated on the right-hand side of Fig. 1.
The key advantage of the ADFB technology, to defineand control
resonances at multiple wavevectors with highresolution in the first
Brillouin zone, has been discussed indetail elsewhere [5], [6]. The
ADFB grating plays two roles:
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IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. XX, NO. X, XXXXX 2
(i) it provides mode selection photonic bands at
user-definedfrequencies, and (ii) it enables electronic switching
betweenthese bands when integrated within a Fabry-Perot (FP)
lasercavity with a current-controllable spectral gain shape.
Thesecond point has been much discussed in our prior work[8], where
we employed time-domain modeling to calculatethe lasing mode
spectra of the compound system (i.e. thecombined FP and grating).
However, a detailed calculationof the ADFB filter response in the
presence of gain was notperformed, and as such the grating FT (Fig.
1) does not directlyreveal the frequency and threshold of lasing
modes. In thispaper we study the multiband ADFB (i.e. digital
hologram)filter response under the influence of gain. The
methodology,which exploits the transfer matrix method (TMM) to
calculatethe ADFB reflectivity and group delay filter functions,
issimilar to techniques described in references [12]–[14]. Weshow
that the ADFB gain-reflection response contains
infinite-amplification (finite output for zero input) singularities
atwhich the laser oscillation condition is satisfied. At
thesesingularities the device enters a slow light regime whichcan
result, for example, in dramatic increase of
light-matterinteraction within a gain medium [3]. The ADFB
singularitiescan be designed to behave like defect-modes, the
frequenciesof which can be directly inferred from the grating FT
(unlikeband edge modes). Indeed, the introduction of gain to
theADFB filter function has been instrumental in enabling us
topredict modal amplification within a graphene controlled
laser[15].
II. METHODOLOGYThe aperiodic lattice (Fig1) is inverse designed
by setting
a target filter function κ̃( f ), [8] and using a binary
searchalgorithm with simulated annealing (SA) to optimize
thelattice; here κ̃( f ) contains two high transmission defect
stateswithin a wide photonic stop band [5]. The SA algorithm
startsby initially selecting a random grating structure to form a
triallattice. The spectral response of the trial lattice is
calculatedusing the FT method described elsewhere [5], [6], [8],
givingan initial spectral response. The difference between the
targetspectral characteristics and the actual spectral
response(i.e, theFT of the trial lattice), describing the ‘error’,
is then calcu-lated and used as the cost function for the SA
optimizationprocess. On each subsequent iteration, the algorithm
modifiesa randomly selected grating element, generating a new
trialgrating and re-computing the cost function. The new
trialgrating is then evaluated in an annealed, probabilistic
fashion.Specifically, if the recomputed cost function falls below
apredetermined cost function value then the trial grating
isaccepted and the process continues for further optimization.
Toensure that the search process avoids becoming trapped in
anylocal minima at the early stages of the optimization
process,small increases in the cost function are also accepted
accordingto Maxwell-Boltzmann classical probability statistics. As
thesystem ‘cools’ the probability of accepting positive increase
incost function reduces, and the system tends towards a
globalminimum in cost space, producing an optimal grating
design.
To calculate the forward problem (i.e. the filter response of
ahologram under the influence of gain) accurately we exploit a
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a)
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