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Modeling photon pair generation by second-order surface nonlinearity in silica nanofibers

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Abstract

In this paper, we present a design of an all-fiber source of correlated photon pairs based on standard telecommunications tapered fibers. We examine the generation of correlated photon pairs using parametric process ${\chi ^{(2)}}$ in silica tapered optical fibers. This nonlinear process is ensured thanks to surface dipole and bulk multipole nonlinearities. The process of photon creation is modeled by taking into account the vector aspect of the propagation of the optical field in a silica nanofiber. The phase matching is provided by propagating the pump field in one spatial mode while generating a photon pair in another spatial mode. The generation efficiency of photon pairs depends on diameter uniformity of the nanofiber after the manufacturing process. We size this nanofiber for a good optimization of photon pair generation efficiency, and we report that the tolerance in diameter uniformity is $\Delta d = 2\;{\rm nm}$ for a generation rate of photon pairs estimated to ${N_{\textit{ph}}} \approx 22\;000\;{\rm pairs}/{\rm s}$, for 1 W power pump and a nanofiber length of 1.1 mm. Deposits on the nanofiber can be used in order to relax the manufacturing constraints on diameter to maximize the generation rate of photon pairs. As an example, the use of polytetrafluoroethylene on the nanofiber applied as a cladding whose thickness is infinite makes it possible to relax the constraints on the nanofiber diameter. For the same $\Delta d = 2\;{\rm nm}$, a generation rate of photon pairs estimated to ${N_{\textit{ph}}} \approx 78\;000\;{\rm pairs}/{\rm s}$ for 1 W power pump and a nanofiber length of 2.4 mm is predicted.

© 2021 Optical Society of America

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Figures (8)

Fig. 1.
Fig. 1. Principle of SPDC process for photon pair generation by modal phase matching in a tapered silica nanofiber.
Fig. 2.
Fig. 2. Axis system in silica nanofiber.
Fig. 3.
Fig. 3. Refractive indices of the five lowest-order modes at 1550 nm (in dashed lines) and 775 nm (in full lines) versus the radius of a naked nanofiber. Phase matching in this radius range only exists between $H{E_{11}}$ at 1550 nm and $T{E_{01}}$ or $T{M_{01}}$ or $H{E_{21}}$ at 775 nm.
Fig. 4.
Fig. 4. Structures of the in-plane components of the electric fields for the different modes. The axes ( $x, y$ ) are purely arbitrary in a perfect circular fiber. The dashed white circles represent the surface of the naked silica nanofiber. The background color indicates the strength of the in-plane electric field component. For this example, the nanofiber radius is 0.5 µm, $H{E_{11}}$ wavelength is set at 1550 nm, and the pump modes are at 775 nm.
Fig. 5.
Fig. 5. Spectral density for 1 W pump power generated in the fundamental mode $H{E_{11}}$ for a 100 µm long silica nanofiber in air: (a) for a $T{M_{01}}$ input mode; (b) for a $H{E_{21}}$ input mode. The dashed red curves represent the location of the phase-matched wavelengths (see text).
Fig. 6.
Fig. 6. Number of photon pairs generated per second in a silica nanofiber suspended in air for different lengths and for a 1 W input power: (a)  $T{M_{01}}$ input mode; (b)  $H{E_{21}}$ input mode.
Fig. 7.
Fig. 7. Spectral densities for a 1 W pump power generated in the fundamental mode $H{E_{11}}$ of a 100 µm long silica nanofiber embedded in PTFE: (a)  $T{M_{01}}$ input mode; (b)  $H{E_{21}}$ input mode. The dashed red curves represent the location of the phase-matched wavelengths (see text).
Fig. 8.
Fig. 8. Number of photon pairs generated per second in a silica nanofiber of different lengths and for a 1 W input power in a silica-PTFE architecture: (a)  $T{M_{01}}$ input mode; (b)  $H{E_{21}}$ input mode.

Tables (1)

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Table 1. Surface and Quadripolar Second-Order Susceptibility Components ( p m 2 / V ) [34,35]

Equations (55)

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{ E ~ 0 ( r , ω ) = i μ 0 ω H ~ 0 ( r , ω ) , H ~ 0 ( r , ω ) = i ε 0 n 2 ( r , ω ) ω E ~ 0 ( r , ω ) ,
F ( r , t ) = 1 2 π F ~ ( r , ω ) e i ω t d ω ,
E ~ 0 η ( r , ω l ) = e i β η l z e η ( r , ω l ) N η l ,
H ~ 0 η ( r , ω l ) = e i β η l z h η ( r , ω l ) N η l ,
1 2 e η ( r , ω l ) h ζ ( r , ω l ) . e z d A = { N η l i f η = ζ 0 i f η ζ ,
E ~ ( r , ω l ) = η a η ( z , ω l ) e i β η l z e η ( r , ω l ) N η l = η a η ( z , ω l ) E ~ 0 η ( r , ω l ) = η E ~ η ( r , ω l ) ,
H ~ ( r , ω l ) = η a η ( z , ω l ) e i β η l z h η ( r , ω l ) N η l = η a η ( z , ω l ) H ~ 0 η ( r , ω l ) = η H ~ η ( r , ω l ) ,
{ E ~ ( r , ω l ) = i μ 0 ω l H ~ ( r , ω l ) , H ~ ( r , ω l ) = i ε 0 n 2 ( r , ω l ) ω l E ~ ( r , ω l ) i ω l P ~ NL ( r , ω l ) ,
F c ( r , ω l ) = E ~ 0 η ( r , ω l ) H ~ ( r , ω l ) + E ~ ( r , ω l ) H ~ 0 η ( r , ω l ) .
z F c ( r , ω l ) . e z d A = . F c ( r , ω l ) d A .
z F c ( r , ω l ) . e z d A = 4 a η ( z , ω l ) z .
. F c ( r , ω l ) = i ω l e i β η l z e η ( r , ω l ) N η l . P ~ NL ( r , ω l ) .
a η ( z , ω l ) z = i ω l 4 e i β η l z e η ( r , ω l ) N η l . P ~ NL ( r , ω l ) d A .
a μ ( z , ω k ) z = i ρ S F G a ν ( z , ω i ) a ν ( z , ω j ) e i Δ β z ,
a ν ( z , ω i ) z = i ρ i D F G a μ ( z , ω k ) a ν ( z , ω j ) e i Δ β z ,
a ν ( z , ω j ) z = i ρ j D F G a μ ( z , ω k ) a ν ( z , ω i ) e i Δ β z ,
ρ S F G = ω k 4 e μ ( r , ω k ) N μ k . ~ NL ( r , ω k ) d A ,
~ NL ( r , ω k ) = e i ( β ν i + β ν j ) z a ν ( z , ω i ) a ν ( z , ω j ) P ~ NL ( r , ω k ) ,
{ ρ j D F G ρ i D F G = ω j ω i , ρ S F G = ρ i D F G + ρ j D F G .
P ~ NL ( r , ω k ) = P ~ ( 2 , B u l k ) ( r , ω k ) + P ~ ( 2 s ) ( r , ω k ) .
P ~ ( 2 , B u l k ) ( r , ω k ) = ε 0 γ [ E ~ ν ( r , ω i ) . E ~ ν ( r , ω j ) ] + ε 0 δ 1 2 ( [ E ~ ν ( r , ω i ) . ] E ~ ν ( r , ω j ) + [ E ~ ν ( r , ω j ) . ] E ~ ν ( r , ω i ) ) ,
{ χ rrr ( 2 s ) , χ rzz ( 2 s ) = χ r φ φ ( 2 s ) , χ zrz ( 2 s ) = χ φ r φ ( 2 s ) , χ zzr ( 2 s ) = χ φ φ r ( 2 s ) .
P ~ ( 2 s ) = δ ( r r 1 ) [ P ~ 1 ( 2 s ) + P ~ 2 ( 2 s ) + P ~ 3 ( 2 s ) + P ~ 4 ( 2 s ) ] ,
{ P ~ 1 ( 2 s ) ( r , ω k ) = ε 0 χ rrr ( 2 s ) e r ( E ~ ν ( r , ω i ) . e r ) ( E ~ ν ( r , ω j ) . e r ) , P ~ 2 ( 2 s ) ( r , ω k ) = ε 0 χ rzz ( 2 s ) e r ( ( E ~ ν ( r , ω i ) . e z ) ( E ~ ν ( r , ω j ) . e z ) + ( E ~ ν ( r , ω i ) . e φ ) ( E ~ ν ( r , ω j ) . e φ ) ) , P ~ 3 ( 2 s ) ( r , ω k ) = ε 0 χ zrz ( 2 s ) ( e z ( E ~ ν ( r , ω i ) . e r ) ( E ~ ν ( r , ω j ) . e z ) + e φ ( E ~ ν ( r , ω i ) . e r ) ( E ~ ν ( r , ω j ) . e φ ) ) , P ~ 4 ( 2 s ) ( r , ω k ) = ε 0 χ zzr ( 2 s ) ( e z ( E ~ ν ( r , ω i ) . e z ) ( E ~ ν ( r , ω j ) . e r ) + e φ ( E ~ ν ( r , ω i ) . e φ ) ( E ~ ν ( r , ω j ) . e r ) ) ,
N ph = G ν ( L , ω ) d ω ,
G ν ( L , ω ) = g 2 L 2 s i n c ( ( Δ β 2 ) 2 g 2 L ) 2 ,
g 2 = ρ i D F G ρ j D F G P p u m p w i t h P p u m p = a μ ( z , ω k ) 2 .
g 2 = λ i λ j ( λ i + λ j ) 2 | ρ S F G | 2 P p u m p .
d a H E 11 e v e n ( ω i ) d z = i a μ ( ω k ) ( ρ i , e v e n , e v e n D F G a H E 11 e v e n ( ω j ) + ρ i , e v e n , o d d D F G a H E 11 o d d ( ω j ) ) e i Δ β z ,
d a H E 11 o d d ( ω i ) d z = i a μ ( ω k ) ( ρ i , o d d , e v e n D F G a H E 11 e v e n ( ω j ) + ρ i , o d d , o d d D F G a H E 11 o d d ( ω j ) ) e i Δ β z ,
d a H E 11 e v e n ( ω j ) d z = i a μ ( ω k ) ( ρ j , e v e n , e v e n D F G a H E 11 e v e n ( ω i ) + ρ j , e v e n , o d d D F G a H E 11 o d d ( ω i ) ) e i Δ β z ,
d a H E 11 o d d ( ω j ) d z = i a μ ( ω k ) ( ρ j , o d d , e v e n D F G a H E 11 e v e n ( ω i ) + ρ j , o d d , o d d D F G a H E 11 o d d ( ω i ) ) e i Δ β z .
ρ i o r j , o d d , e v e n D F G = ρ i o r j , e v e n , o d d D F G .
ρ i o r j , e v e n , e v e n D F G = ρ i o r j , o d d , o d d D F G = 0.
ρ i o r j , e v e n , o d d D F G = ρ i o r j , o d d , e v e n D F G = 0 ,
ρ i o r j , e v e n , e v e n D F G = ρ i o r j , o d d , o d d D F G 0.
ρ i o r j , e v e n , o d d D F G = ρ i o r j , o d d , e v e n D F G = 0 ,
ρ i o r j , e v e n , e v e n D F G = ρ i o r j , o d d , o d d D F G 0.
d a H E 11 e v e n ( ω i ) d z = i a μ ( ω k ) ρ i , e v e n , e v e n D F G a H E 11 e v e n ( ω j ) e i Δ β z ,
d a H E 11 o d d ( ω i ) d z = i a μ ( ω k ) ρ i , e v e n , e v e n D F G a H E 11 o d d ( ω j ) e i Δ β z ,
d a H E 11 e v e n ( ω j ) d z = i a μ ( ω k ) ρ j , e v e n , e v e n D F G a H E 11 e v e n ( ω i ) e i Δ β z ,
d a H E 11 o d d ( ω j ) d z = i a μ ( ω k ) ρ j , e v e n , e v e n D F G a H E 11 o d d ( ω i ) e i Δ β z .
d ( a H E 11 e v e n ( ω i ) + a H E 11 o d d ( ω i ) ) d z = i a μ ( ω k ) ρ i , e v e n , e v e n D F G ( a H E 11 e v e n ( ω j ) a H E 11 o d d ( ω j ) ) e i Δ β z ,
d ( a H E 11 e v e n ( ω i ) a H E 11 o d d ( ω i ) ) d z = i a μ ( ω k ) ρ i , e v e n , e v e n D F G ( a H E 11 e v e n ( ω j ) + a H E 11 o d d ( ω j ) ) e i Δ β z ,
d ( a H E 11 e v e n ( ω j ) + a H E 11 o d d ( ω j ) ) d z = i a μ ( ω k ) ρ j , e v e n , e v e n D F G ( a H E 11 e v e n ( ω i ) a H E 11 o d d ( ω i ) ) e i Δ β z ,
d ( a H E 11 e v e n ( ω j ) a H E 11 o d d ( ω j ) ) d z = i a μ ( ω k ) ρ j , e v e n , e v e n D F G ( a H E 11 e v e n ( ω i ) + a H E 11 o d d ( ω i ) ) e i Δ β z .
L max 2.8 Δ β r δ r max ,
T M 01 i n p u t , Δ β / r = 2.6 10 12 m 2 ,
H E 21 i n p u t , Δ β / r = 5.1 10 12 m 2 .
T M 01 i n p u t , L m a x 1.1 m m s o t h a t N ph 22 000 p a i r s / s , H E 21 i n p u t , L m a x 0.55 m m s o t h a t N p h 2 700 p a i r s / s .
T M 01 i n p u t , Δ β / r = 1.2 10 12 m 2 ,
H E 21 i n p u t , Δ β / r = 1.4 10 12 m 2 .
T M 01 i n p u t , L m a x 2.4 m m s o t h a t N p h 78 000 p a i r s / s ,
H E 21 i n p u t , L m a x 2.0 m m s o t h a t N ph 15 000 p a i r s / s .
χ rrr ( 2 s ) = h χ rrr ( B u l k ) .
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