686 Vol.11,No.3/September 2019/Advances in Optics and Photonics Tutorial Indeed,in general the beam will change shape as it propagates,and it is not itself the "eigenfunction"of the mathematical problem (though it is easily deduced from the ac- tual eigenfunctions in simple communication problems).In this SVD way of looking at communications,the jth communications mode is a pair of functions-s)in the source or input space,and in the receiving or output space.Explicitly,therefore, communications modes are pairs of functions-one in the source space and one in the receiving space. They are a set of communications mode pairs of functions-a pair ls)and o),a pair ls2)and lg2),and so on.To find these functions,we perform the SVD of the coupling operator Gsg between the volumes or spaces.For the communications prob- lems we consider first,this Gsk is effectively the free-space Green's function for our wave equation. 1.3b.Mode-Converter Basis Sets When we change from thinking just about waves in free space to trying to describe a linear optical device,we can consider how it scatters input waves to output waves [Fig.1(b)].By analyzing this also as an SVD problem,in this case of a device (or scattering)operator D,we can similarly deduce a set [11]of input source functions si))that couple one by one to a set of output wave functions{));these two sets of functions are the mode-converter basis sets. In this second case,we want to describe the device as one that converts from a specific input mode lus to the corresponding output mode l),and so on,for all such mode pairs;again,as in the case of communications modes,we think in terms of pairs of functions here,one in the source or input space,and one in the receiving or output space.We can consider these as mode-converter pairs-a pair si)and Ri),a pair lus2)and oR2),and so on,just as in the communications modes.In this way of look- ing at a linear optical device [6], any linear optical device can be viewed as a mode converter,converting from specific sets of functions in the input space one by one to specific corresponding functions in the output space,giving the mode-converter pairs of functions. The device converts input mode lusi)to output mode R),input mode lus2)to output mode 2),and so on.In this case,though the mathematics is similar to the communications modes,this is more a way of describing the device,whereas the communications modes are a way of describing the communications channels from sources to receivers.For the device case,we may not have anything like a simple beam between the sources and receivers,but we do have these well-defined functions or "modes"inside the source space or volume and inside the receiving space or vol- ume.We could also view the mode-converter basis sets as describing the communi- cations modes“through”the device. In an actual physical problem for a device,there are ways in principle in which we could deduce the mode-converter pairs of functions by experiment [7,12]without ever knowing exactly what the wave field is inside the device.Then we could know the mode-converter pairs as eigenfunctions without knowing the "beam";this point em- phasizes that it can be more useful and meaningful to use the pairs of functions in the source and receiving spaces as the modes of the system rather than attempting to use the beam through the whole system as the way to describe it. 1.4.Usefulness of This Approach There are several practical and fundamental reasons why these pairs of functions are useful.Indeed, in general the beam will change shape as it propagates, and it is not itself the “eigenfunction” of the mathematical problem (though it is easily deduced from the ac￾tual eigenfunctions in simple communication problems). In this SVD way of looking at communications, the jth communications mode is a pair of functions—jψSji in the source or input space, and jϕRji in the receiving or output space. Explicitly, therefore, communications modes are pairs of functions—one in the source space and one in the receiving space. They are a set of communications mode pairs of functions—a pair jψS1i and jϕR1i, a pair jψS2i and jϕR2i, and so on. To find these functions, we perform the SVD of the coupling operator GSR between the volumes or spaces. For the communications prob￾lems we consider first, this GSR is effectively the free-space Green’s function for our wave equation. 1.3b. Mode-Converter Basis Sets When we change from thinking just about waves in free space to trying to describe a linear optical device, we can consider how it scatters input waves to output waves [Fig. 1(b)]. By analyzing this also as an SVD problem, in this case of a device (or scattering) operator D, we can similarly deduce a set [11] of input source functions fjψSjig that couple one by one to a set of output wave functions fjϕRjig; these two sets of functions are the mode-converter basis sets. In this second case, we want to describe the device as one that converts from a specific input mode jψSji to the corresponding output mode jϕRji, and so on, for all such mode pairs; again, as in the case of communications modes, we think in terms of pairs of functions here, one in the source or input space, and one in the receiving or output space. We can consider these as mode-converter pairs—a pair jψS1i and jϕR1i, a pair jψS2i and jϕR2i, and so on, just as in the communications modes. In this way of look￾ing at a linear optical device [6], any linear optical device can be viewed as a mode converter, converting from specific sets of functions in the input space one by one to specific corresponding functions in the output space, giving the mode-converter pairs of functions. The device converts input mode jψS1i to output mode jϕR1i, input mode jψS2i to output mode jϕR2i, and so on. In this case, though the mathematics is similar to the communications modes, this is more a way of describing the device, whereas the communications modes are a way of describing the communications channels from sources to receivers. For the device case, we may not have anything like a simple beam between the sources and receivers, but we do have these well-defined functions or “modes” inside the source space or volume and inside the receiving space or vol￾ume. We could also view the mode-converter basis sets as describing the communi￾cations modes “through” the device. In an actual physical problem for a device, there are ways in principle in which we could deduce the mode-converter pairs of functions by experiment [7,12] without ever knowing exactly what the wave field is inside the device. Then we could know the mode-converter pairs as eigenfunctions without knowing the “beam”; this point em￾phasizes that it can be more useful and meaningful to use the pairs of functions in the source and receiving spaces as the modes of the system rather than attempting to use the beam through the whole system as the way to describe it. 1.4. Usefulness of This Approach There are several practical and fundamental reasons why these pairs of functions are useful. 686 Vol. 11, No. 3 / September 2019 / Advances in Optics and Photonics Tutorial