if we have a beam that consists of many
spectral components there's various
dispersive elements available which we
can use to split those spectral
components apart for instance using a
grading we can disperse the light across
a CCD or e4 spectrometry or across a
spatial light modulator for things like
spectral pulse shaping filtering switch
and cetera you can also use the grading
in the opposite way to combine multiple
spectral components into a single beam
for polarization there's also many ways
in which you can split a beam up into
its orthogonal polarization components
like pausing beam splitters by referring
to crystals like cutting so for both
wavelength and polarization there's a
lot of options available some have been
around for centuries that allow us to
manipulate a light beam in the
wavelengths and polarization basis what
about the spatial properties of light
just getting the trivial cases where you
just want to split the beam up into a
bunch of non overlapping sections when
you can just use an array of points to
sample it like a CCD camera or fiber
array or if you have a beam that
consists of many spatial components like
say the modes of a fiber and you want to
split them apart I've been the consists
of many orthogonal spatial components
where all the spatial components are
more or less on top of each other right
overlapping in space for that task there
are limited options and there options
that have only been invented relatively
recently so one is a photonic plant
photonic planet is a multimode 2 single
mode transition it's a tapering of an N
mode multimode fiber on one end into n
single mode fibers on the other end
that's a full 3-dimensional refractive
index profile it's a lossless device
whereby a spatial mode put into one end
gets split up and decomposed into its
individual spatial components as
diffraction-limited spots on the other
end the photonic lantern was originally
invented for astronomy where light is
collected by the telescopes into a
multimode fiber before being sent to
single mode spectrometers the all fiber
construction of a photonic Lantern is
both its main benefit and also in some
applications its main drawback
ultimately the type of devices you can
make with a photonic Lantern are limited
by the kinds of fiber
you can actually manufacture the second
option is a multi plane light conversion
system em PLC and that's what I'll be
talking about today it's a bit like a
discrete version of a photonic lantern
light propagates through successive
phase plates separated by free space
propagation which allow light to be
transformed from one spatial basis to
another like the photonic lantern this
device is also just phase manipulation
so it's lossless but it's capable of
more easily implementing arbitrary
transformations as it's not constrained
by the same limitations of what you can
actually manufacture as a tapered fiber
device like the photonic lantern is
calculating the phase masks required to
implement this conversion can be done by
using the masks to match the phases of
the input and output mode sets at each
plane so ultimately if we want this
device to actually work we need the
modes in one direction to match the
modes in the opposite direction need to
be the same so to start with we
propagate each of the input modes from
the source to the first plane then we
propagate the output modes backwards
through the system to meet them at that
same first plane to calculate the phase
mask we should use for that plane we
perform the spatial overlap between each
pair of input and corresponding output
modes and we sum all of those overlaps
fields for each pair of input and output
modes together then we take the phase of
that superposition and that's our new
phase mask for that plane then we
propagate to the next plane and do the
same thing overlap the forward and
backward modes sum their overlaps
together and then take the phase of that
suit position as the new phase mask and
like that you work your way through the
device through each plane
then when you get to one end you flip it
around and propagate back the other
direction following the same process so
we can see from this demo here that even
after just one pass back and forth it's
already started to match up quite well
there's lots of little tweaks and
optimizations and additional constraints
you can apply and should apply to the
algorithm to make it work well but this
is the basic idea in practice the fewer
facemask planes you can get away with
using the better for several reasons
first it's because makes it smaller
cheaper and easier to actually make the
thing and second because these MP LC
devices are theoretically lossless but
in practice the loss is going to scale
with the number of optical surfaces that
the light has to travel through so let's
say that each plane has 0.1 DB of loss
that's pretty decent but if you've got
100 planes now you've lost 10 DB and 90%
of your light so how many planes do you
actually need for a given transformation
well it depends what the two spatial
basis are that you're trying to convert
between some are easier than others in
the worst case if you're converting
between two bases each a set of n
orthogonal modes then at most you'll
need to end planes but if you want that
system to work over a more extended
bandwidth then you'll need to add more
planes to make that transition smoother
and more adiabatic because the more you
diffractive in with your masks the more
wavelength dependent it's going to
become so you want to be smooth
eventually if you added an infinite
number of planes you'd have a completely
adiabatic device that's kind of like a
photonic Lantern now which naturally has
high bandwidth so we can see that if
we're converting a basis of like 45
modes and we were to need 90 masks plus
maybe some extra ones for increased
bandwidth then suddenly most of our
lights gone right even for this
theoretically lossless device but we
also know that not every basis
transformation requires that many planes
and not every basis transformations
number of planes even scales with the
size of the basis so for a simple
example if we want to convert between
the XY
in real space spaces and the k-space
angle basis and that we just need a lens
right or in fact you don't even need a
lens you can just use free space
propagation and that gives it to you for
free so in that instance you need
depending on how you define a plane
either no planes one plane or - plane
but that's a fixed number that's
independent of the size of your basis so
in fact this basis is completely
continuous anyway well here's another
easy one we've got an array of spots and
we just want to rearrange those spots
for that you just need two planes one
plane to tilt the spot in the right
direction and then one plane to stop the
spot from going any further once it
actually gets there again as long as
you've got enough pixels the number of
planes required doesn't scale with the
number of spatial modes it's always just
- that's a pretty easy one because all
the spatial modes don't overlap at all
in either plane so they're completely
separate but here's another example
where we're converting between two
orthogonal basis where all the different
spatial components overlap so we've got
our mate Gaussian set can be converted
to a leg a Gaussian basis just using a
pair of cylindrical lenses one lens to
start the transformation and one to stop
it and keep the beam round again this
doesn't scale with the number of modes
it's always just two planes it works
because all the modes regardless of how
many modes there are all need the same
thing for this transformation to occur
they all need a phase shift to be
introduced between the X and the y
component in a cylindrical lens focusing
on one axis but not the other gives one
axis of growth a shift and the other one
nothing so not every basis
transformation requires a full - n
number of phase planes and some don't
even depend on n in this work we're
interested in perhaps the most common
transformation type required in both
optical telecommunications and also
astronomy splitting the modes of a
multimode fiber into an array of single
mode fibers but within that we've got
flexibility regarding exactly what maps
to what right we just want a lossless
unitary transformation everything else
is negotiable we are necessarily care
what the Shea
of the single-mode fiber array is
geometry of the single-mode fiber array
ultimately it'll be easier in practice
if it's a linear array or rectangular or
square array or some sort of densely
packed array like a triangle or a
hexagon but we can always just add a few
extra planes at the end to remap the
shape of the array itself if need be as
long as the overall solution is low loss
and low plane count for a given array
geometry we also don't care which fibers
map to what modes we don't care about
the ordering of the array so we don't
care about array shape or geometry and
we also don't care about the order so
those are design degrees of freedom we
also generally don't care what
orthogonal set of modes the single mode
fiber maps - it could be something like
like a Gaussian which closely resembles
the fiber eigenmodes
but it could also just be a complete
scrambling of modes astronomers don't
care what the mode mapping is and in
telecommunications we generally don't
care either because over long lengths of
transmission fiber the modes are going
to couple anyway regardless to how you
put them into the fiber in the first
place so with this in mind we set out to
try to make the laziest possible MP LC
device a device where each phase mask
needs to do as little as possible
converting from an array of single mode
fibers to a mode basis which was as
close as possible to that original array
of spots but or an orthogonal and
complete set so we start off with the
spots arranged in this concentric circle
pattern this geometry was originally
used in mode division multiplexing and
before ultimately being replaced by a
photonic lanterns it's an array geometry
which minimizes the mode dependent loss
for a given number of modes each one of
those spots excites a superposition of
modes in the fiber basis which looks a
lot like a spot but isn't actually a
Gaussian it's got some other features to
it the closer you get to the edge of the
fiber core the less gauss in those spots
look and also the more loss you get but
they're always quite localized spot
like distributions so this fiber array
to fiber mode transformation isn't
lossless and you know some spots have
more loss than others we don't want that
we want a lossless complete orthogonal
set so let's make one using this as a
starting point to do that we can take
the coupling matrix between the spot
basis and the fiber basis you know the
overlap between each mode and each each
spot we can then use a singular value
decomposition to split that non unitary
matrix the coupling matrix into three
components are unitary transformation of
the input basis you are unitary
transformation of the output basis V and
a non unitary diagonal matrix of the
singular values these represent the
transmission coefficients for each modal
component through the system and these
are what we want to be unitary won't all
be the same if you look at these
singular values we can see that we've
currently got about seven or eight DB of
mode dependent loss going from the spot
array to the fiber modes so mode
dependent losses the ratio between the
maximum and minimum singular value so we
just force all those singular values to
be 0 DB no loss and then put the matrix
back together again so now we've got a
coupling matrix that looks a lot like
the old matrix except now it's
orthogonal we've got that original
coupling matrix and we converted it to
the nearest orthogonal basis now if we
use this new basis we get localized
spotlight modes but they form a complete
orthogonal lossless set which is great
because it should be easy to convert
between our 2d single mode fiber array
and the spotlight modes because they're
almost the same and they don't really
over that much in space right they're
almost independently localized so let's
have a look at a solution here we have
45 modes implemented with just less than
n divided by 2 planes so 22 planes the y
axis here is the propagation direction
file left column is the sum of the
intensities of all modes just like a
summary second from the left is the
phase masks that do the transformation
and the rest of the x-axis illustrates
each of those 45 modes individually as
they evolve through the system all right
so let's have a look at the evolution of
these spots through the whole system
you
another perhaps more intuitive
look at it is to arrange the evolution
of the modes into the shape of the
original rate in all these you can see
that the spots get focused and refocused
once near the start and also once near
the end that doesn't necessarily have to
happen you know these aren't unique
solutions it kind of depends on what
your starting mask scenario was but that
is a solution which works quite well and
it keeps the light from falling off the
edge of the mask over the course of the
22 planes where it keeps it keeps it
focused but the evolution is also
somewhat disappointing and unsatisfying
it does go through all those planes does
all these sort of fancy theatrics on the
way I need to return to something which
looks almost the same as what it did at
the start there are also other solutions
which keep the spots much more spot like
throughout the entire transition but
don't necessarily give you any less
planes or any less loss so if we look at
the performance of the transition we can
see it does have good insertion loss
when social loss is the average singular
value from before it represents the
average loss over all possible spatial
paths through the system it's also got
decent mode dependent loss for a45 mode
device of about 2.2 DB and quite good
bandwidth and flatness because you know
each mask isn't doing that much so the
device is performing well better than
most MP LC devices would for a spot to
fiber mode conversion but ultimately
there's an even better mapping that we
can employ what we can do is just take
the positive quadrant of a grid of spots
in the XY plane and map the XY positions
to the XY indices of the hermite
gaussian modes this mapping greatly
reduces the number of planes required
for the conversion which we'll see in a
second but it's also an excellent
mapping in the context of graded-index
multimode fiber because each hermite
gaussian mode consists of a
superposition of near degenerate fiber
eigenmodes
so ideally each hg mode each hermite
gaussian mode consists of a
superposition of fiber modes
is almost exactly the same propagation
constant and group delay so modes within
these mode groups couple heavily in a
multimode fiber anyway
almost immediately so for high quality
graded index fiber exciting a hermite
gaussian mode is about as mode selective
as you're going to get in practice
anyway it's also good because a
triangular fiber array is something
that's not too difficult in practice
it's a densely packed array so that you
could just use something like a slice of
a hexagonal fiber array looking at a
solution example here we can see that
the transformation has been performed
quite well
using just seven planes same design
parameters as the previous example for
the spot basis except I have doubled the
resolution of the masks in x and y
that's not a necessary point just that
if you've got less planes it frees up
some computational complexity and memory
to spend on resolution if you wish but
does make a big difference it's not a
number of pixels thing so if we zoom in
we can take a closer look through the
short transition for each of the first
few fiber modes and then for all 45 of
the modes
in terms of performance it's a bit worse
insertion loss and mode dependent loss
and not quite as wide bandwidth compared
to the quasi spot example from before
this is a quick little demo simulation
you could optimize that a bit more send
to the response a bit better for the
mode event loss but ultimately the point
here is that we've used less than a
third of the number of planes 45 modes
in seven planes and got a sort of
comparable level of performance if we
have a look at the transformation for a
45 mode example of 15 mode example and a
6 mode example or with the same basic
design parameters other than the number
of modes in the set we can see similar
features of the transformation so it
starts off as the array becomes this the
formation or a bit like X like if
there's only a few modes with stronger
intensity on the XY axis and at the
access origin the stem flips to the
other side starts to have this more X
like distribution with light in all four
quadrants that evolves into this thing
which looks a bit more like a tick or a
bird depending on the solution with a
sort of semi circular body in one
dominant quadrant two rectangular legs
slash wings on the top and bottom
quadrants and then little head in the
left quadrant so the details can change
a bit depending on the exact solution
again it depends on things like the
starting conditions the number of planes
design parameters etc but it always has
that same general form to it so what
exactly is it doing I don't have a
rigorous mathematical answer for you
unfortunately but it seems to be
exploiting the symmetry of the modes to
achieve the transformation in a very
limited number of planes if we look at
the symmetry of a hermite Gaussian mode
it's symmetric in both x and y if you
look at the locations of the dominant
lobe of Alamut gaussian mode they're
organized in this triangular semicircle
like pattern
you
which looks a lot like our original
triangular array shape and also the
shape the array ends up being
manipulated into during the transition
so if that array is duplicated in x and
y that gives you the four main lobes of
the hermite gaussian mode my guess at
this point is that's what it's
exploiting to enable it to perform the
conversion with very little dependence
on the number of modes and the number of
planes required right all modes need the
same thing they need to be duplicated in
X and they need to be difficult in Y
because all the modes need the same
thing you need less planes okay so we
started here with a triangular array
which sometimes will be fine again it's
just a triangular slice of a densely
packed hexagonal fiber array but even
better if it was a 1d fiber array
because that makes it obviously they're
very easy to come across you can get
arrays 1d arrays which are very tightly
Specht in their tolerances and it also
makes it easier to get little micro lens
arrays for collimating the spots on to
the first plane but the important point
here is that we've got a good solution
as our starting point so now it's not a
big deal to remap from a triangular
array to a linear array using additional
face masks and there's two ways you
could go about that you could do sort of
two separate problems do a set of masks
for the triangular to hermite gaussian
transition then another set of masks for
the linear two triangular transition and
then just paste the two solutions
together and optimize it a bit more
alternatively you can just evolve the
shape of the array as the simulation
itself is running so here I've just
started off from the triangular array
got a decent solution and I've opened up
that array into a line keeping the
symmetry of the array the same but all
the way through until you get to the
last set of modes in the center so the
hermite gaussian modes where the M and
the N indices are the same so these are
the sets of modes which don't have a
symmetric 90 degree rotation version of
themselves like the other modes do
they're all on their own right so that's
just a final optimization step
and where you break that symmetry at the
last possible moment to get your final
linear array and then you can perform a
few extra little tweaks etc to try to
eke out some better performance but
ultimately the solutions are going to be
looking at here I use 14 planes rather
than 11 its largest to do with the
manufacturer of the faceplates itself
but you know more planes gives us a bit
better performance but it also helps
more with the bandwidth so more planes
means it becomes a bit more adiabatic to
smoother transition smoother phase masks
less diffraction per plane which means
more bandwidth and we can see in the
solution here for the 14 plane
implementation that most of those extra
planes actually just end up being used
in the spot remapping section of the
transition rather than her might
gaussian conversion part of the
transition it's actually the spot
rearrangement part which hurts you the
most bandwidth wise as that's the bit
where the light is trying to be steered
across a bigger angle why are you trying
to steer the light from out wide right
into the center of the beam and it's
usually is out of spot from the edges of
the arrays you know they have to travel
the furthest distance which is also the
biggest constraint on your bandwidth so
we can see that given the option the
algorithm is actually distributed more
planes to the spot transition rather
than the hermite Gaussian transition to
give us more bandwidth and we can see
the hermite Gaussian transition itself
has that same basic form it's just a bit
more spread out over a couple of extra
masks again an interesting way to look
at the evolution of the light through
the system is to arrange each modes
journey through the device into the
shape of the original triangular array
itself although now we're actually
starting from a line but still all right
so you can see it starts off it's all
mostly just rearrangement of the spot
array itself then we get into the
business end we can see that all the
modes focus in two different sized and
different XY portions of that same B
shape with the bright spot on the the
axes origin
the outer X&Y dominant lobes kind of fly
off by themselves and then wait for the
rest of the mode to diffract out and
meet them to form the complete mode in
terms of the performance itself we can
see that the insertion loss is good
about 1.5 DB of insertion loss that's
pretty flat over 100 nanometers and we
also get very good mode dependent loss
this is 1.3 DB to 1.7 DB across that
same hundred nanometers of bandwidth and
again 1.3 1.5 1.7 DB of mode apparent
loss for a 45 mode device even
theoretically is very is very good
especially given the few number of
planes it requires all right that's it
so we've shown that using either a quasi
spot like basis or particularly using a
hermite gaussian basis from a triangular
fiber array we can significantly reduce
the number of planes that are required
for a multi plane light conversion
system