Polarisation control of optically pumped terahertz lasers

0.31 MB·English

by  G. Slavcheva

Polarisation control of optically pumped terahertz lasers
G. Slavcheva1,2 and A. V. Kavokin3
1Blackett Laboratory, Imperial College London,
Prince Consort Road, London SW7 2AZ, United Kingdom∗
2Mediterranean Institute of Fundamental Physics, Via Appia Nuova 31, 00040 Rome, Italy
3Spin Optics Laboratory, St. Petersburg State University, 1,
Ulianovskaya, 198504, Russia and School of Physics and Astronomy,
University of Southampton, Highfield, Southampton SO17 1BJ, United Kingdom
Opticalpumpingofexcitedexcitonstatesinsemiconductorquantumwellsisatoolforrealisation
ofultra-compactterahertz(THz)lasersbasedonstimulatedopticaltransitionbetweenexcited(2p)
and ground (1s) exciton state. We show that the probability of two-photon absorption by a 2p-
excitonisstronglydependentonthepolarisation ofbothphotons. Variationofthethresholdpower
3 forTHzlasingbyafactorof5ispredictedbyswitchingfromlineartocircularpumping. Wecalculate
1 thepolarisation dependenceoftheTHzemissionandidentifyphotonpolarisation configurationsfor
0 achieving maximum THz photon generation quantumefficiency.
2
n PACSnumbers: 78.67.-n,78.67.De,71.35.-y,78.45.+h,78.66.Fd,79.20.Ws,78.47.da
a
J
Introduction.-Excitonsinnanoscalesemiconductorma-
9
2 terials exhibit low-energy excitations in the range of the
exciton binding energy, analogous to inter-level excita-

] tions in atoms, yielding infrared and terahertz (THz)
l
l transitions. Thus excited exciton ladder states represent
a
h a natural system for generating THz radiation and co-
- herence. The demand for development of new compact
s
e and efficient coherent terahertz radiation sources is cur-
m rently rapidly increasing, due to ever growing range of
. very diverse technological applications in the relatively
t
a little-explored THz spectrum of radiation [1]. Towards
m this goal recently a new scheme of a microcavity based
- polaritontriggeredTHzlaser(THzverticalcavitysurface
d
emittinglaser(VCSEL))hasbeenproposedbyoneofthe FIG.1: (Coloronline)Schematicenergy-leveldiagramoftwo-
n
authors[2],wherebythe2pdarkquantumwell(QW)ex- photontransitionsto2pexciton statesinaQW.Theground
o
c citon state is pumped by two-photon absorption using a (1s) and excited (2p dark) discrete (bound) excitonic states
andtheexciton (unbound)continuumstatesareshown. The
[ cw laser beam.
pumpingfrequency,ωpump ishalfthatofthe2pexcitonstate.
1 In this Letter we theoretically demonstrate polarisa- ωTHz is the center frequency of the emitted THz pulse; Γ0 -
v tion control of THz emission and of the quantum effi- ground (1s) state exciton spontaneous emission rate.
4
ciency for THz photon generation. We consider a THz
0
VCSEL proposed in Ref.( [2]), where the pump beam is
9
Two-photon p-exciton absorption.- The quasi-2D
6 split in two. Each of the split beams goes through a po-
. lariser, so that the two photons pumping the 2p exciton (Q2D) exciton wave function at the Γ point is given for
1
narrow QWs by [4]:
0 do not necessarily have the same polarisation. We show
3 that by rotating one of the polarisers one can switch on v
1 and off the THz laser. Ψλ(re,rh)= √0SUλαβ(ρ)Φαc (ze)Φβv∗(zh)uck(re)u∗vk(rh)eik||.R||
:
v Using crystalsymmetry point grouptheoreticalmeth- (1)
i where v is the unit-cell volume, S is the QW area, R
X ods [3] we calculate the polarisation dependence of the 0
is the centre-of-mass (c.o.m.) co-ordinate, r=r r is
optical transition matrix element for two-photon exci- e h
r −
a tonicabsorptioninGaAs/AlGaAsquantumwells,aswell the relative motion co-ordinate, r(ρ,z) and ρ=ρe−ρh
is the in-plane relative motion co-ordinate, the z axis
asoftheintra-excitonic2pto1sTHztransitionradiative
is taken normal to the QW layers. α,β are subband in-
decay rate. This enables us to calculate the polarisation
dicesandΦα istheα-subbandenvelopefunctionofthe
dependence of the quantum efficiency for THz photon c(v)
generationandthusidentifymaximumefficiencyregimes conduction(valence)band; Uαβ(r r )istheenvelope
λ e− h
of operation. The optically pumping scheme to a 2p ex- function of the 2D exciton associated with subbands α
citonstatebytwophotons,eachofhalftheenergyofthe of the electron and β of the hole; λ = (n,m) is the 2D
2p exciton state, is shown in Fig. 1. excitonquantumnumber,labellingthediscreteexcitonic
2
states (n = 1,2,..., m < n); uck,uvk are the periodic where EG is the direct interband energy gap, the total
| |
parts of the Bloch wave function for conduction and va- matrix element can be written as:
lence bands, correspondingly; the exciton c.o.m. wave
e2
vector,k|| ≈0,isontheorderofthephotonwavevector. Vfi=√Smµ¯ c2A1A2[hc|εˆ1.p|viIδ(1,2)+hc|εˆ2.p|viIδ(2,1)]
Consider the case of allowedconduction-to-valence band ξ
(8)
dipole opticaltransitionatthe Γ point. Incubic crystals
Let us define a 2D reduced Coulomb Green’s function:
the conservation of parity upon absorption of two pho-
tons requires the final excitonic state to have the same Uαβ(ρ)Uαβ∗(ρ′)
parity as the valence band, therefore the final exciton is G(ρ,ρ′)= λ λ (9)
E Ω
λ
in a p-state. The TPA probability is given by: λ −
X
WTPA = 2~π |Vif|2Sf(E) (2) wfrohmerethEeλcoisndthuecteioxnc-ibtoanndhyeddgroegaenndicΩen=ergyEGas+m~eωaαsu<re0d.
−
Xif A closed form of the reduced Green’s function for an
unscreened exciton in 3D (N-D) has been derived in [7]
where S is the final density of states and the momen-
f
([8]) and in the 2D limit of interest is given in [9]:
tum,p,matrixelement,V ,betweentheinitialandfinal
if
states is given by [5]:
1 −2ρ/κ a∗ 4ρ ρ
G(ρ,0)= e α B ln γ+3 4
Vfi= me22c2A1A2 hf|Eε1.p|Elihl|ε~2ω.p|ii+hf|Eε2.p|Elihl|ε~1ω.p|ii 2π (cid:20)− (cid:18)καa∗B(cid:19)− − (cid:18)καa∗B(1(cid:19)0(cid:21))
Xl (cid:20) l− i− 2 l− i− 1 ((cid:21)3) where a∗ is the 3D exciton Bohr radius, γ is the Euler’s
B
reflectingtheorderofabsorptionofthefirstphotonwith constant and κ2α = EGE−B~ωα with EB ≡ Ry∗ – the exci-
polarisation vector ε , energy ~ω and vector potential tonbindingenergy. Introducingcylindricalco-ordinates
1 1
A and the second - with polarisation vector ε , energy and the overlap integral of the subband envelope wave
1 2
~ω and vector potential A . Since the TPA is a two- functions:
2 2
stepprocess,one shouldsumoverallintermediate states
|li with energy El. The first matrix element has been Iαβ =hΦαc (z)|Φβv (z) = dzΦαc∗(z)Φβv (z) (11)
calculatedbyElliott[6]inthe 3Dcaseandforthequasi- Z
(cid:11)
2D case here considered reads: and momentum matrix element along z-direction:
hl|εˆα.p|ii=√SΨ∗λ(0)hc|εˆα.p|vi (4) P = Φα(z) p Φβ(z) = dzΦα∗(z)~ ∂ Φβ(z)
=v Uαβ(0)Φα(z )Φβ∗(z ) c p v αβ h c | z v c i ∂z v
0 λ c e v h h | | i (cid:12) (cid:11) Z (12)
(cid:12)
where εˆ , α = 1,2 is the photon polarisation vector, the general expression for the sum over intermediate
α
Ψ (0)istherelativemotionexcitonwavefunction,given states can be recast as:
λ
by Eq.(1), evaluated at r=0, and the interband matrix
element hc|εˆα.p|vi is given by: Iδ(α,β)=−i~vS02 Iαβ d2ρUδαβ∗(ρ)(εˆβ.ρˆ)∂G∂(ρρ,0)
hc|p|vi= v10 d3ru∗c(r)~i∇uvˆz(r) (5) +PαβR d2ρUδαβ∗(hρ)(εˆβR.ˆz)G(ρ,0)i (13)
cZell where ρˆ= ρ andˆz are unit vectors.
|ρ|
The second matrix element entering Eq.(3) is between ForVCSEL configurationandnormalincidence geom-
etrywechooseεˆ zandthereforethesecondterminEq.
hydrogenic-type exciton states and can be written as: β
⊥
(13)vanishes. ThederivativeoftheGreen’sfunctioncan
1 f εˆ .p l = 1 d3rΨαβ∗(r)(εˆ .p)Ψαβ(r) beeasilycarriedoutusingEq.(10). We takeforthe exci-
mh | β |i µ¯ξ Z δ β λ tonrelativemotionwavefunctionthe 2Dhydrogenatom
= 1 d3rUαβ∗(ρ)Φα∗(z )Φβ(z )(εˆ .p)Uαβ(ρ)Φα(z )Φβ∗(z ) wave function for bound exciton states [10], [11]:
µ¯ δ c e v h β λ c e v h
ξ Z
(6) |m|
ewxhceitroenmmiasstshaelofrnegeξe-ldecirtercotniomnaisnstahnedQµ¯Wξ ispltahneer.edInutcreod- Unαmβ(ρ)=Nnm a∗B n2ρ− 21 ! e−a∗B(nρ−12)
ducingaspecialnotationforthematrixelement,summed
2ρ (cid:0) (cid:1)
over the intermediate states, according to: L2|m| eimφ, n=1,2,3,..., m <n
× n+|m|−1 a∗B n−21 ! | |
(14)
I (α,β)= d3rUαβ∗(ρ)Φα∗(z )Φβ(z )(εˆ .p) (cid:0) (cid:1)
δ δ c e v h β 1/2
Z (7) where N = (n−1−|m|) and Lα(x) –
×Xλ UEλαλβ+(ρE)GUλα−β∗~ω(0α)Φαc∗(ze)Φβv(zh)Φαc (ze)Φβv∗(zh) associatendmLagu(cid:20)eπrar∗Be2(pno−ly12n)3o[m(n−ia1l+s|[m12|)]!.]3(cid:21) n
3
FIG. 2: (Color online) 3D surface plots of thenormalised excitonic TPA against polar angles of εˆ1 and εˆ2 in theQW plane at
specificphaseshiftsδ1,δ2. (a)co-linearly polarised photonsδ1 =0,±π; δ2 =0,±π;(b)1st linear–2nd σ+ circularly polarised
photon δ1 = 0; δ2 = π2; (c) 1st σ+ circularly polarised photon – 2nd linearly polarised photon δ1 = π2; δ2 = 0; (d) σ+– σ+
or σ−– σ− co-circularly polarised photons δ1 = ±π2; δ2 = ±π2; (e) σ+– σ− or σ−– σ+ counter-circularly polarised photons
δ1 =±π2; δ2 =∓π2; (f) co-left-elliptically polarised photonsδ1 =−π6; δ2 =−π8.
Introducing polar co-ordinates (ρ,ϕ) we obtain for fortheTPAprobabilityto2p-excitonstatesin[s−1m−2]:
TPA to 2p-exciton states with n=1,m= 1:
± K C2 ~2 16E2
I2,1(α,β)= 3πi√34π~κIαβa∗3E Jp,2(κα) (15) W2(p2)= 27TπP3A 21M2Iα2β(cid:18)a∗B4(cid:19)S2cp1,hh1(2EG−BE2p)2h1+(εˆ1(.1εˆ82))2i
α B B
whereSc1,hh1 is the final2p-excitondensity ofstatesper
2p
whereE = ~2 istheexcitonbindingenergyandthe unit areafor a heavy-holeexciton (c1-hh1)andthe coef-
integral,BJ 2(µk¯ξa)∗B2= 9(143+36ln(32))a∗3κ3. ficient KTPA for an infinite quantum well, is given by:
p,2 α − 2048 B α
Substituting in Eq. (8) the excitonic two-photon ab- 128πe4A2A2v2
K = 1 2 0 (19)
sorption matrix element is obtained: TPA ~µ¯2m2c4SL2E2
ξ z B
V = v02 e2 A A 4~Iαβ 1 J c p v (16) The photon polarisation vectors εˆ1,εˆ2 with polar angles
fi √S mc2 1 23πi√3πEBa∗B2 eff,2h | | i ϕ1,ϕ2 and phase shifts δ1,δ2 correspondingly, lie in the
QW plane. 3D plots of the exciton TPA probability are
where we have defined effective matrix element for cubic
shown in Fig. 2 for different polarisations of the two
crystals, using the invariance of the interband matrix el-
pumping photons.
ementM = c p v undercrystalpointsymmetrygroup
h | | i We suggest adding an external THz cavity at the VC-
transformations [14], [5], [3], [15]:
SEL output that will filter out the linear polarisation of
the emitted THz radiation, and will thus constitute our
1
Je2ff,2= 2(εˆ1×εˆ2)2|Jp,2(k1)−Jp,2(k2)|2 reference frame, fixing the direction of our co-ordinate
1 system x-axis. We shall assume that the generated THz
+ 1+(εˆ.εˆ)2 J (k )+J (k )2 (17)
2 1 2 | p,2 1 p,2 2 | mode is X-polarised. By inspectionof Fig. 2 one cansee
= Ch12 (εˆ εˆ)2i k2 k2 2+ 1+(εˆ.εˆ)2 k2+k2 2 that maximum (5-fold) increase of the two-photon ab-
2 1× 2 1− 2 1 2 1 2 sorptionratewithrespecttoYY polarisationisachieved
where Cn1 =−9(14(cid:12)(cid:12)3+203468ln((cid:12)(cid:12)32))h. i(cid:12)(cid:12) (cid:12)(cid:12) o f2o(ra)l)i.neTarhlye tXwXo-,pXhoX¯to,nX¯aXb,sXo¯rXp¯tiopnolararitseedcapnhiontcornesas(eFbigy.
Ourpumping schemeenvisagestwophotonseachwith a factor of 3 for linearly-circularly or circularly-linearly
half the energy of the 2p-exciton state: ~ω1 = ~ω2 = polarised photons (Fig 2(b,c)); by a factor of 2 for both
~ω = E22p and k12 = k22 = k2 = 2E2GE−BE2p, therefore the circularly polarised (Fig. 2 (d,e)), by a factor close to 5
first term in Eq. (17) vanishes and from Eq. (2) we get (but always less than the one for linear polarisation) for
4
elliptically polarised photons (Fig. 2 (f)). Our results UsingEq.(18)andEq.(21),aftersomealgebraonecan
show that changing polarisation from YY to XX, pass- obtain for the normalised quantum efficiency:
ingthroughcircularlyandellipticallypolarisedpumping,
onIenctaran-evxacriytotnhiecl2apsing1tshrtersahnoslidtiobnyparofabactboilritoyf.-5W. ecal- η=e2~2L1zM44m2 21n423a+∗B2S36Iα2lnβ 32 2E24(pE(22pE−GE−1Es)23p)2 SS12ccsp11,,hhhh11!P(εˆ1,εˆ2,εˆ)
→ (24)
culate next the polarisation dependence of the 2p 1s (cid:0) (cid:0) (cid:1)(cid:1)
→ where the polarisation dependence is given by:
photon intra-excitonic transition rate, generating THz
emission(Fig. 1). Weareinterestedintheopticaltransi- cos2(ϕ)
tion matrix element between initial two-fold degenerate P(εˆ1,εˆ2,εˆ)= 1+(cosϕ cosϕ +cos(ϕ +δ )cos(ϕ +δ ))2
1 2 1 1 2 2
state Ψδ with δ = (n = 2,m = 1) = (2,p) and final (25)
±
state Ψλ with λ = (n = 1,m = 0) = (1,s). The matrix and ϕ is the polar angle of the THz photon polarisation
elementis ofthe secondtype Eq.(6) andfornormalinci- vector.
dencegeometry(εˆ zˆ)andexcitonwavefunctions,given We shall assume that the 2p exciton excited by two-
⊥
by Eqs. (1),(14), we obtain: photon absorption has a lifetime, which is long enough
thatitlosesanymemoryofthepolarisationandphaseof
Mlf =vS02(cid:18) −i8~(cid:19)(cid:18)µm¯ξ(cid:19)Iα2β(cid:18)3√13(cid:19)32π√a∗B24Φ(ϕ)Z dρρ2e−38aρ∗BL22(cid:18)(342aρ∗B0)(cid:19) tWhee eshxcailtlactoionnsi,dseortehmatissitiocnanwietmhitonweitpharatnicyuplaorlaproislaatriiosan-.
wherewehaveintroducedpolarco-ordinatesandthean- tion (either linear or circular) and all possible choices of
gular dependence is given by: Φ(ϕ) = cosϕe∓iϕ. The polarisation of the two pumping photons.
integrationoverρiseasilyperformed,giving: 81 a∗3. Fi- The quantum efficiency polarisation dependence is
512 B
nally, the 2p 1s intra-excitonic optical transition rate showninFig. 3fordifferentpolarisationconfigurationsof
→
for an infinite QW is given by: thetwopumpingphotonsatagiven(linear)emittedTHz
photon polarisation. The plots for circular polarisation
27 ~2
W(2) = KTHz I4 S (E)Φ2(ϕ) (21) oftheTHzradiationlookexactlythesamebutarescaled
2p→1s 512 OPAπ2a∗2 αβ 1s downbyafactorof2(notshown),resultinginmaximum
B
efficiency η =0.5. In additionto the resultspresentedin
where S (E) is the final (1s) state density of states and
1s Fig. 3, we should note that for counter-X X¯ -linearly
the one-photon THz emission coefficient is given by:
polarised pumping photons δ = 0(π);δ = π(0) max-
1 2
(cid:0) (cid:1)
16π e 2 v2A2 imum quantum efficiency η = 1 is achieved for linearly
KOTPHAz = ~ µ¯ c 0L4TSHz (22) polarised (φ = 0,π) and η = 0.5 for circularly polarised
(cid:18) ξ (cid:19) z (ϕ= π)THzemission,unconditionally,foranydirection
4
where ATHz is the THz photon vector potential, ex- of the linear polarisation of the two pumping photons
pressed in terms of the THz emission intensity, ITHz as: in the QW plane. Furthermore, if the THz emission is
A = 2πcITHz 1/2, where n is the refractive index Y-linearly polarised, the quantum efficiency η = 0, i.e.
THz nωT2Hz no THz radiation should be emitted in this case. Fig.
and ωTHz(cid:16)= E2p−~E(cid:17)1s. 3 shows that the maximum quantum efficiency η = 1
The polarisationdependence ofthe THz emissionrate could be achieved within certain regions in the plane for
can be inferred from the angular dependence: for linear linearly polarised THz emission for all combinations of
(e.g. along x-axis) polarisation of the emitted THz pho- linearandcircularpolarisationsofthetwopumpingpho-
ton (εˆ xˆ), Φ2(ϕ) = 1, for y-linear εˆ yˆ, Φ2(ϕ) = 0 and tons. Note that in both (circular and linear THz emis-
|| ||
thereforethereisnoTHzemission,andforcircularlypo- sion polarisation) cases, maximum quantum efficiency is
larised THz photon, Φ2(ϕ)= 21, the corresponding THz achievedalongYY linesforco-linearlypolarisedphotons,
emission rate is half of the one for x-linear polarisation. forY-polarisedfirst(second)photoninthecaseoflinear-
Quantum efficiency.- The quantum efficiency of THz circular (circular-linear) polarisation, or along diagonal
radiation generation can be defined as the ratio of the lines forco- andcounter-circular-circularpolarisationof
THz photongenerationrate and the two-photonabsorp- the pumping photons. We emphasise, however, that al-
tion rateby a 2p-excitonandis proportionalto the ratio though maximum quantum efficiency could be achieved
ofthe squaresofthe oscillatorstrengths, G andg, ofthe both by counter- and co-linearly polarised photons, the
2p 1s and 0 2p transitions [2], which can be quantum efficiency in the former case is constant and
→ | i → | i
expressed in terms of the transition probability [13]: does notdepend onthe directionofthe polarisationvec-
torsintheQWplane,whilemaximumquantumefficiency
G2≡f2p→1s= ω6π2c3εnµ¯3ξeS2W2(p2→) 1s inthe latter case is obtainedsolely forspecific directions
THz (23) of the polarisation vectors in the plane (YY).
g2≡f2p= 6ωπ2c3nε3µ¯eξ2SW2(p2) Inordertoverifythesepredictionsexperimentally,one
2p
can envisage pumping of a QW structure with two laser
where ω = E2p. beams having the same frequency (equal to a half of the
2p ~
5
FIG. 3: (Color online) 3D surface plots of the normalised quantum efficiency of THz photon generation against polar angles
of the pumping photons polarisation vectors in the QW plane at different phase shifts δ1,δ2 at linear, φ=0,±π polarisation
of the emitted THz radiation (a) co-linearly polarised photons;(b) 1st linear-2nd circularly polarised photon; (c) 1st circular
–2nd linearlypolarised; (d)σ+–σ+ orσ−–σ− co-circularly polarisedphotons;(e)σ+–σ− orσ−–σ+ counter-circularlypolarised
photons.
2p-exciton resonance frequency) but different polarisa- WethankProf. E.L.Ivchenkoforvaluablediscussions.
tion. These two beams may be generated by the same AKacknowledgesfinancialsupportfromthe EPSRCEs-
laser but should propagate through different polarisers tablished Career Fellowship grant.
before focussing on the sample. In addition we suggest
includingadelaylinebetweenthetwopartsofthepump-
ing beam, which would provide the phase difference of π
betweenthemtoobtaincounter-linearlypolarisedbeams
∗ Electronic address: [email protected]
forwhichunconditionalmaximumefficiencyispredicted.
[1] M. Tonouchi, NaturePhotonics, 1, 97 (2007)
The intensity and polarisation of the THz light emitted
[2] A. V. Kavokin, I. A. Shelykh, T. Taylor, and M. M.
by the structure could be measured as a function of in- Glazov, Phys. Rev.Lett. 108, 197401 (2012)
tensities and polarisations of the two pumping beams. [3] E. L. Ivchenkoand G. E. Pikus, Superlattices and Other
As reference experiments one can measure the intensity Heterostructures (Springer-Verlag, Berlin, 1997).
of THz emission with one of the pump beams switched [4] A. Shimizu, Phys.Rev.B 40, 1403 (1989)
off. Analysingtheresultsofsuchexperimentsoneshould [5] G. D.Mahan, Phys.Rev.170, 825 (1968)
[6] R. J. Elliott, Phys.Rev.108, 1384 (1957)
bear in mind that the two photons used to generate a
[7] L. C. Hostler, Journal of Math. Phys., 5, 591 (1964); L.
2p-exciton may originate from the same beam as well as
C. Hostler, Phys. Rev.178, 178 (1969)
from different beams. Comparing the spectra obtained [8] S. M. Blinder, J. Math. Phys. 25, 905 (1984)
with both beams switched on with those obtained with [9] R. Zimmermann, Phys. Stat.Sol. (b),146, 371 (1988)
only the first or only the second beam switched on, one [10] M.ShinadaandS.Sugano,J.ofthePhys.Soc.ofJapan,
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different polarisations.
Scientific, 1994)
Conclusions.- We have developed a theory of the two- [12] HandbookofMathematicalFunctions,Ed.MAbramowitz
photon absorption to p-exciton states in QWs and cal- andI.A.Stegun(USDepartmentofCommerce,National
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Math. Ser.55
sition probability, using crystal symmetry point group
[13] Quantum Electronics3rdedition,A.Yariv(JohnWiley),
methods. Weshowthatthetwo-photontransitionrateis
1988
strongly dependent on the polarisation of both photons
[14] M.InoueandY.Toyozawa,J.ofthePhys.Soc.ofJapan,
and our model predicts variation of the lasing thresh- 20, 363 (1965)
old by a factor of 5 by switching from YY to co-linearly [15] V. Heine, Group theory in quantum mechanics, (Dover
XX-polarised pumping. We calculated the polarisation Publications, 1993)
dependence of the intra-excitonic THz emission and the
quantum efficiency for THz photon generation. Maxi-
mumquantumefficiency ispredictedforcounter-linearly
polarised pumping photons and linearly polarised THz
emission. Conditions for achieving maximum quantum
efficiency for different polarisations of the pumping pho-
tons are identified, thereby opening routes for polarisa-
tion control of the THz VCSEL and a range of new ap-
plications entailed from it.