Now we are well placed to numerically simulate the beam propagation in our light-matter system. Here two different beam propagation sections are to be distinguished, parts where the field experiences diffraction only due to free space propagation in the host medium (a distance z0 from the beam waist
w0 to the first sphere with diameterds and a distancez1 from the end of the
first sphere to the second sphere) and parts where the field encounters the varying refractive index distribution due to the spheres (in the first sphere
Figure A.2: Geometrical layout of the beam propagation algorithm: Distancez0from the beam waist
w0 to the first sphere with diameterds and distancez1 from the end of the first sphere to the second
sphere. The waists are separated byDf assuming mirror symmetry aroundDf/2 for the center of the
array withDthe centre separation. A) signifies the homogeneous and B) the inhomogeneous part in which the field propagates.
and the second sphere), see figure A.2.
The propagation of the field can therefore be split in A) homogeneous parts (refractive index is constant to nh) and B) inhomogeneous parts (refractive
index can be nh or ns dependent on the position of the field~r). Both beam
propagation parts will be dealt with in the following section, for simplicity I will solely focus on the forward propagating field (+)10.
A.4.1
A) propagation in homogeneous medium
n
hTo numerically propagate a paraxial field amplitude E+(x, y) in the z-
direction from one plane (z = zi = 0 at w0) to the next (z =z0). Solutions
have to be found to the wave equation, from A.10 where ∆n2 =n2h−n2h = 0 so the wave equation becomes for the forward propagating field:
∂E+ ∂z = i 2k∇TE+ | {z } dif f raction +0 (A.13) with ∆n= 0
An algorithm is used that decomposes the filed into plane waves, where each component travels at a different angle in k-space11 or wave vector space.
Each component is propagated individually by adding a phase shift due to
10As will be shown later due to the symmetry of the system only one beam propagating
past two spheres is calculated.
• The electric field is Fourier transformed and thereby decomposed into individual plane waves at the start planez =zi = 0.
˜ E(kx, ky, z = 0) = 1 (2π)2 Z Z E(x, y, z = 0)ei(kxx+kyy)dk xdky (A.14) ˜ E(kx, ky, z) = spectrum ˜
E(kx, ky, z) is the transverse spectrum of the electric field and can be seen as
the field amplitude of a plane wave with wave vectors in the transverse plane
kx and ky.
• The component in the z-direction kz for each of these plane waves can
be simply calculated by:
kz = q k2−k2 x−k2y (A.15) k= 2πnhλ k2z+k2x+k2y= n2ω2 c2 =k 2
Here we do not apply the Fresnel approximation13 which assumes that the
field does not change much in the transverse plane|kx,y|<< k (or plane wave
approximation) for enhanced accuracy of the model. Assuming a monochro- matic field with wavenumberk in a medium of refractive index nh.
• When the field is propagated a distance ∆z = z0 in the positive z-
direction each of the plane wave components experiences a phase shift ∆φ = kz∆z. The amplitudes of the individual components in the
transverse planez =z0 = ∆z are related to the initial plane through:
˜ E(kx, ky, z =zi+∆z) = ˜E(kx, ky, z =zi)e(−i √ k2−k2 x−k2y∆z) (A.16) ∆z= propagated distance ˜ E(kx, ky, z) = spectrum
12This algorithm was first proposed by [139].
• By taking the inverse Fourier transform the electric field distribution, after a propagated distance ∆z = z0, is obtained. This represents an
exact solution to the wave equation.
E(x, y, z =zi+∆z) = 1 (2π)2 Z Z ˜ E(kx, ky, z =zi)e−i(kxx+kyy)e(−i √ k2−k2 x−k2y∆z)dxdy (A.17) zi= 0
The Fourier Transformation is carried out by using a discrete fast Fourier Transform algorithm (FFT) which is readily obtained in MATLAB14. Here
A.17 simply reduces to:
E(x, y, z =zi+∆z) =iF F T F F T[E(x, y, z=zi)]∗ e( −i√k2−k2 x−k2y∆z) | {z }
f requencytransf eref unction
(A.18)
F F T = Fast Fourier Transform
However care has to be taken to choose the right sampling parameters, e.g. grid size and resolution. The grid size has to be big enough so the transformed electric field is not cropped by the boundaries15.
A.4.2
B)
propagation
in
inhomogeneous
medium
n(~r) =n
s+ ∆n
To propagate the field within the sphere the split-step Fourier method is used to account for phase shift and diffraction within the sphere for a in- homogeneous or varying refractive index distribution due to the sphere. By segmenting the sphere into individual cylindrical slabs j, see figure A.3. The field distribution is calculated at the beginning (1) and at the end (2) of each slab (j) of radius r, see figure A.3. The algorithm for the split-step method follows from [137] to:
14cMathWorks.
the first slab. Right: j indicates the number of each slab which has got an individual radius ofrjand a
width of ∆zthe actual sphere radius intersects each slabjat ∆z/2 to obtain a volume approximation of the sphere. E(x, y, zj+1) = b A+Bb E(x, y, zj) E(x, y, z+ ∆z) = e(Ab+Bb)∗∆zE(x, y, z j) E(x, y, zj+ ∆z) ≈ eAb∗∆zeBb∗∆zE(x, y, z)(A.19) b A= diffraction operator b B= inhomogeneous oper- ator
Where (from A.10) Ab = i
2k∇TE+ is the linear differential operator that ac-
counts for beam diffraction andBb =ik0∆n 2(~r)
2nh E+is the space dependent or in-
homogeneous operator. Here the inhomogeneous operator adds an additional phase shift to the diffraction operator of ∆n(~r) where the beam encounters the sphere. This additional phase change is applied at the beginning of each cylindrical slab (marked (1) in figure A.3) in MATLAB to:
E(x, y, z) = E(x, y, z)e ik0 n2s−n2 h 2nh θrj ∆z (A.20) k0=2λπ k= 2πnhλ
where θrj is a Heaviside step function which is zero outside each slab j and
unity within the slab of radius rj16. The field is then propagate with A.20 a
step length ∆z to plane (2) see figure A.3.
Here two constraints apply to the step length ∆z and the sampling grid in x,y for the scalar field, with step size ∆x and ∆y:
• The sphere should be accurately sampled so the first and the last slab is resolved within the sampling grid in x and y. The radius of the first slab is r2
1 = rsphere2 −(rsphere −∆z/2)2 where the radius of the
Sphere intersects with the slabs at ∆z/2 (see figure A.3(right)). Using
rsphere >>∆z we obtain a minimum stepsize of:
∆z > 1r sphere∆x2.
• Given a grid spacing ∆x= (xmax/nx) withxmaxthe grid size andnxthe
number of points. We want the phase change to be small to capture the outline of the sphere with the optical field distribution. Hence the step size needs to be sufficiently small to resolve the refocusing effect of the sphere. From A.13 we obtaink2
T∆z/2k =kmax2 ∆z/2k <1 with kmax =
(π/∆x) is the maximum transverse wavevector and k = 2πnh/λ. The
constraint for the stepsize ∆z is: ∆z < 4nh∆x2
πλ
Next I continue with the calculation of the forces that act on the spheres.