Spatially nonreciprocal Bragg gratings based on surface plasmons: Supplementary Online
Material
In this appendix we study a generic Bragg grating using coupled mode theory and elucidate the conditions required for nonreciprocal operation. Fig. S1 shows a sketch of a periodic dielectric slab grating.
Fig. S1: Contra-directional coupling in a Bragg grating. The complex permittivity varies periodically along z with period of Λ.
Assuming coupling between the two identical counter-propagating modes, the net electric field in the grating is expressed as
E(x,y,z)=U(x,y)[A(z) e-jβBz +B(z) e+jβBz], where U(x,y) is the normalized transverse mode distribution, A(z) and B(z) are respectively the
amplitudes of the forward and backward propagating modes, and βB=π/Λ is the Bragg constant. The coupled mode equation for the Bragg
grating is given as:
(A1)
dA
  j l B  j A
dz
dB
 j  l A  j B
dz
where δ=β-βB is the detuning parameter. κ±l is the coupling coefficient which is defined as:
k 2    l ( x, y ) U ( x, y ) dxdy
 l  0
2
2
U ( x, y ) dxdy
2
(A2)

where k0=2π/λ0 is the free space propagation constant, β=2π/λ is the propagation constant in the grating and Δε±l is the ±lth Fourier component
of the periodic permittivity, Ref[1].
The transmission matrix for the coupling due to the ±lth Fourier component is derived from the coupled equations in Eq.A1.




cosh( L)  j sinh( L)
 j l sinh( L)


A(
L
)
A(0)








 T
 T  
 l


 B ( L) 
 B(0) 
j
sinh( L)
cosh( L)  j sinh( L) 




(A3)
Where σ2=κlκ-l-δ2. Using the transmission matrix given above, the reflectance and transmittance from the right and left sides are obtained as:
T
Rr  21
T11
Rl 
Tr 
2
T12
T11
1
T11
(A4)
2
2
T T T T
Tl  11 22 12 21
T11
2
It can be shown that T11T22-T12T21=1, thus Tr=Tl or the transmittance is always reciprocal. However only if T21=T12 will the reflectance be
reciprocal. It is obvious from the transmission matrix in Eq.A3 that we need κl=κ-l* in order to have T21=T12
Fourier analysis shows that the coupling coefficients for a Bragg grating with modulation of only the real permittivity always satisfy κl=κ-l*.
Therefore reflectance will always be reciprocal for these gratings. However if there is significant modulation of imaginary permittivity
(gain/loss) then the coupling coefficient will not be symmetric and nonreciprocal reflectance will occur.
Ideal non-reciprocity occurs when κl≠0 and κ-l=0 (or vice versa). This requires that the refractive index of the grating have a form similar to
Eq.1, which implies that the modulation of the real and imaginary components of refractive index must be equal (Δn= Δk), and that there
must be an exact 90˚ phase difference between them. Using the Hamiltonian operator, we demonstrate that the realization of these 2
conditions corresponds to the breaking threshold of PT symmetry.
The Hamiltonian of a Bragg grating is defined using the coupled equations in Eq.A1 as:
  j
 A
d  A
 H H 
dz  B 
 B
 j  l
 j l 

j 
(A5)
The eigenvalues of the Hamiltonian are obtained as λ = ±σ = ± (κl κ-l - δ2)1/2. We will explore the eigenvalues at the Bragg wavelength where
δ=0 and eigenvalues simplify to λ = ± (κl κ-l*)1/2
If κl=κ-l* (i.e. for a grating with a real modulation of refractive index only), the eigenvalues of the Hamiltonian will be real 𝜆 = ±|𝜅𝑙 |. This
state corresponds to unbroken PT symmetry.
If one of κl or κ-l is zero, then the eigenvalues coalesce to λ=0 which corresponds to the threshold for PT symmetry breaking. This state is also
referred to as a PT symmetry phase transition or exceptional point. At this state ideal non-reciprocity is attainable.
If κl≠κ-l* the eigenvalues form a complex conjugate pair which corresponds to a PT symmetry broken state. The coupling coefficient will not
be symmetric, which in turn causes asymmetric reflectance, but it is not ideal non-reciprocity which occurs precisely at the breaking
threshold.
References:
[1]. L. A. Coldren, Scott. W. Corzine, “Diode lasers and Photonic Integrated Circuits”, (John Wiley and Sons, Inc, 1995).