Beam Position Transient Response in the Controlled Optical Beam Beam Position Transient Response in the Controlled Optical Beam Waveguide Waveguide

Beam position transien ts in self-aligning beam waveguides are studied. The control system senses the beam position at each lens and i n troduces a correction to the transverse position of the. pre ceding lens. The Laplace Transform of the beam position response at some (k + n)th lens as a result of a disturbance of the kth lens is found. The special case of the confocal waveguide using t he in tegrator with gain as a control results in a time function of the beam position at t h e lenses equal to the product of a decaying ex ponen tia l and Laguerre Polynomials. The second-order controlled waveguide has been simulated with the results showing that over shoot can be con trolled by increasing the damping term in the con trol. An alternative control structure is presented which isolates t he disturbed elemen t and its correction from t he remaining controls resulting in the simple exponential decay of the path perturbation using t h e integrator with gain as the control.

An additional Appendix (C) has been added wh ich ou t lines a n al t er native to the form of con trol developed i n t h e main body of the work.This ma t erial was arrived a t too la t e for i nclusion i n the origi nal ma nuscript bu t is of sufficient si gn ifica nce to warra nt i t s i nclusion here.The advan tage of these wavelengths for commun ications systems, extreme bandwidth and high information ca pacity, will be important for future transmission systems owing to the ever-increasing demands placed on links by computer data, video, business and personal transmissions.
An i nherent problem i nvolved in a beam waveguide composed of discrete elements (lenses or reflectors) is i nstability of the transmission path .This is due to random displacements of the elements leading to eventual departure of the beam from the guide axes.< 4 , S) This paper analyzes the performance of linear self-aligning beam waveguides.The linear correction system has been described and shown to be stable in time.<6 ) An understanding of the transient response of t h is system is necessary to determine its suitability and practicality .

The system
The beam waveguide is composed of discrete gu idance elements spaced uniformly along the guide axis.The guidance elements are 1 lenses or are equivalent to a lens of focal length (f).They are separated by distance (d), (Figure 1).
The trajectory of the beam center in the guide is described by ray optics.The transmission matrix for a lens relates the paraxial ray position and slope, leaving an individual lens, to position and slope entering.For a lens whose optical center is not displaced from the guide axis,(7) 1) [ where the ray positions rout and rin and the ray slopes r~ut and ri_n are measured relative to the guide axis.
If the optical center of the lens is displaced an amount S, the emergent ray position is modified as follows; The transmission through the basic unit cell of the waveguide is found by relating the ray at the nth reference plane located immediately before the nth lens to the ray at the (n + l)th reference plane immediately before the next lens;

General description and analysis
To control the beam position at each lens, the system monitors the beam position at each reference plane.This signal is used to correct the alignment of the previous lens.With the system initially at rest, the Laplace Transform Sn, of the total displacement of the i  or, in matrix form;

H(S)i~
13) Solving this relationship for the ray position component of the vector R we have•(S) , where the Um(a) are the Chebyshev Polynomials of the second kind Using the recursion relation for Un_3(a), and combining terms th we obtain the Laplace relationship between a disturbance at some k th ( 9) lens and the ray position response at the (k In order to find the ray position response as a function of time it is necessary to expand the Chebyshev Polynomial using the expansion sunnnation The step response is of interest because it is identical to the response that results from initial misalignment.It is likely that the control system will not operate continuously but will be turned on periodically and will respond to initial misalignmen t.
The response given by equation 22 is plotted in Figure 2 out th to the 13 lens.The beam position is bounded and decays rapidly to the axis.This result was cross checked by simulating the guide on a digital computer.

24)
The case of the con trol function . n Ra y posi t i on respon se a t the seve n .oddlen ses ( u p t o the th ir t e e nth ) • fo llow ing a un i t s t e:P .

d i s t ur ba nce o f t h e zero t h len s for a c on t rol fun c t i on o f H(s)
-G/s.  in an overshoot which is bounded at all lenses.

Conclusion
Self-aligning beam waveguides employing feedback to stabilize the beam position at each lens can be designed so that overshoot in the transient response is bounded.Large transients and excessive departure of the beam from the guide axis can be eliminated by increasing damping.The response of a heavily damped second-order control system approaches that of an integrator with gain.In a confocal guide, this res ~lts in a transient response to a step displacement which is the product of a decaying exponential and Laguerre Polynomials.This response decays rapidly to zero and has an overshoot that is less than the initial beam displacement at each lens.

II APPENDIX A INTRODUCTION AND REVIEW
The well recognized "connnunication explosion" which has been going on for some time will soon reach the point where available transmission equipment will not be capable of handling the ever in-  The structure is composed of a series of identical thin lenses of focal length (f), separated by distance (d).The effect of a thin lens on a paraxial ray may be expressed in one dimension as( 7) where R -[:] The unprimed term is ray position and the primed term is ray slope relative to the optical axis of the lens.This expression relates ray position and slope at an output reference plane immed-

19
In like manner, the transmission of a straight section of a homogeneous medium is given by (7) where L is the length of the medium traversed.
In the beam waveguide the lenses may be displaced from the transmission axis, adding a correction term to the output ray slope and the lenses will be separated by straight sections.Combining To find the transient beam position at an arbitrary lens entrance plane as a result of a disturbance of some previous lens from the axis we assume that the waveguide elements are initially aligned (all lenses are centered on the reference axis and the beam is on axis throughout the structure) and introduce a transverse displacement of a single lens.
By iteration of equation 7 with the assumed initial conditions;

FCS)
Block d i agr am of t he bas ic unit cell of t h e controlled beam wavegu i de.  and 4. Figure 3 shows the response at the eleventh lens after the disturbance for three values of the damping factor, S = 0.8, 1.0 and 1.2.
These curves demonstrate that for light damping <s = 0.8), the overshoot exceeds the initial disturbance.The response at the first lens is the classic second order system response but as we progress down the waveguide the nature of the response becomes increasingly oscillatory with an overshoot that increases with distance from the disturbance.
By increasing the damping factor to 5 = 1.2, the beam at the eleventh lens is returned to the axis with overshoot remaining less than the initial displacement.Figure 4 shows the response at the odd numbered lenses up to the eleventh for a damping factor of 3 = 1.2.Note that increasing to a very large value yields a con- trol which approximates the integra t or with gain resulting in a beam displacement which remains bounded by the maguitude of the initial disturbance at all lenses.
BL.l.Simula t ion of t he con focal wave gui de with H(s) -G/s.

4)
We now h ave two possibilities.If there is no disturbance th of the k lens we have from iteration of equation l; 5) rn+l (s) -r 1 ( s) nand therefore the total displacement term in e quation 4 goes to zero, meaning that the kth lens will not be moved erron eously by the con trol.
On the other hand, if the kth lens is disturbed, Dk(s) will have some magnitude and will affect ray position at the (k + l)th lens.
Tracing the ray from the (k -l)th lens to the (k + l)th lens we have; As is evident from equation 5, this response will be seen at all odd numbered lenses beyond the disturbed elemen t.Therefore, due to the control co n~iguration, we now have a system i n which the correction is not iterated by the un it cells following that in which the initial disturbance occurs.The end result of this control scheme is that a tran sverse lens displacement is isolated from the structure and corrected without unnecessary motion of the other lenses on the waveguide.The control is now able to distinguish between errors in beam position caused by the disturbance of the controlled lens a nd those errors generated by a lens associated with a previous control.
'the confocal waveguide with non-iterative control has been simulated on a digital computer with results confirming the conclusion of isolated control indicated above.The simulation program appears at the end of this appendix as CL.l (Guide3).
Beam waveguides provide a low loss medium for the transmission of energy a nd i n formation at infrared and optical frequencies.Cl, 2 , 3)

( 1 -2 1 2
(d/f)H(s)) In this equation is inserted the particular form of H(s) to be evaluated and the ratio of spatial separation of lenses to focal length, (d/f).The Laplace Transform can be inverted for particular values of d/f and transfer function H(s).This results i n a time domain response at any lens expressed in a closed sunnnation.4. Special cases of the time response Owing to t h e form of t h e time response of t h e ray position (equation 18) a case of innnediate interest is t h e con focal waveguid e wh ere focal length a nd spatial separation of the lenses are In this case only one term of the sunnnation remains, t hat term for which n-l-2m= O, and in this case the response of the ray position at an arbitrary {k + n)th lens for a disturbance at t h e kth lens is n- case the con trol function is taken to be a n i n tegrator with gain ; 21) H(s) = -G s and the input is a unit step disturbance at the kth lens, we find for th the time domain response at the (k + n)

2 -
also of practical interest.It represents a second order control correcting at each lens.The results of simulating a guide employing this con trol function at each lens are shown in Figures 3 a nd4.In both of these figures the response is to a step disturba nce at the zeroth lens.The simulations show that undesirable overshoot results when the control system is underdamped.The response at the

7 Figure 3 .
Figure 3.Ray pos i t i on .response a t t he el~ve n th lens f or t h e se c ond order con t rol; wn =l.O.

Figure 4 . 8 .
Figure 4.Ray position response at t h e firs t six odd lenses (up t o t h e eleventh )for the second order con t rol wi th C =1.2.
creasing traffic.Computer data, television and rapidly expanding personal and business transmissions (both audio and video with the implementation of the "picture 'phone" now in service in some areas) are already taxing the ingenuity of connnunications engineers to provide sufficient bandwidth to accommodate present traffic, and future requirements will present even greater challenges to technology unless a transmission system with vastly increased information carrying capacity is exploited.~ Laser carrier transmission appears to fill this need and methods are currently available both for modulation and demodulation of the coherent beam output of the basic device.The literal "missing link" at the present time however, is a suitable and dependable transmission path to assure uninterrupted propagation of the modulated beam over long distances from transmitter to receiver.Atmospheric transmission is not feasible due to scatter and random disturbances disrupting the beam path (haze, rain, dust, etc.).

14
Optical fibers are useful for short distances but the high attenuation factor eliminates this technique for applications requiring transmission over long paths without the use of an excessive number of amplifiers or repeaters.An alternative which is not subject to the objections noted is the beam waveguide.Originally developed for the millimeter and submillimeter wave range and then extended to the optical region o f the spectrum, the beam waveguide is based on the principle of periodically restoring the field distribution of the propagated beam of energy by means of physical elements which function as phase transformers.These devices are dielectric elements with a quadratic profile which reset the cross-sectional phase distributi6n in the beam at regular intervals along the guiding structure.The individual elements are, for optical wavelengths, positive lenses or pairs of reflectors which (1, 2, 3) yield a phase correction identical to a single lens.In terms of propagation losses, the beam waveguide is a most attractive means of transmitting optical energy.Diffraction losses -3 at the apertures are negligible (10 db/lens) for lenses of only 10 rmn.radius.The iteration loss per lens is therefore determined by inherent reflection and absorption which can be reduced to less than 0.1 db/lens with current technology.<i 2 ) An inherent problem in the implementation of the beam waveguide as a transmission link is the stability of alignment of the phase transforming elements to the propagation axis.If the elements remain in position with the lens optical centers on the transmission axis the beam is perfectly iterated with minimum loss.Misalignment of a single lens results in a beam path which oscillates about the axis wi t h no increase in loss provided the lens apertures are sufficiently large to accommodate the deviations in beam position.More serious lens displacements lead to an increase in overall attenuation of the beam, and the beam path becomes unstable for .(4, 13) periodic spatial displacements of the lenses.• An automatic alignment system for the waveguide which corrects lens disturbances along the structure in a series of small movements has been developed and tested.In this control a small (0.05 mm) corrective step is applied at each lens on the waveguide based on the direction of beam deviation from the transmission axis at the following lens.After each lens in the guide has been moved one step, the entire process is repeated.The beam path returns to the axis after a sufficient number of correction cycles have occurred to return all lenses to the axis.A system which operates continuously on each lens is mentioned but no evaluation other than possible increased speed of response is presented.(14)The continuous servo control has been analyzed in the steady state and shown to >provide an asymptotically stable s y stem which returns the propagated beam to the guide axis for initial displacements which do not completely disrupt the transmission path .( 6 ) Such disturbances are to be considered as catastrophic failures and are not subject to correction by any form of control.Before the controlled beam waveguide can be regarded as a viable alternative to existing transmission methods however, information concerning the transient performance of the system with disturbances in element alignment must be obtained.Some method of stabilization which returns the beam to the waveguide axis in a reasonable period of time with a nominal expenditure of control energy is desirable in terms of maintaining satisfactory iteration of the beam and minimal attenuation of the propagated energy.This thesis derives the as yet unpublished transient response relationship between beam position and transverse dis t urbances of the structure elements.APPENDIX B DERIVATION OF THE TRANSIENT RESPONSE For a detailed derivation of the transient response of the controlled beam waveguide we first require an expression showing how the beam path in the guiding structure is affected by transverse lens displacements from the propagation axis.We assume that the beam diameter is sufficiently smaller than the lens apertures to allow us to ignore the effects of diffraction and therefore enable us to use paraxial geometric optics to trace the path of an ideal ray through the waveguide.

4 )Figure B. 2 .
Figure B.2.Block diagram of t h e uncon t rolled beam waveguide.
f k we find the response of beam position at a lens entrance plane n unit cells beyond the element (the kth) which is disturbed; 12) = To simplify this expression we first rewrite the matrix T(s) by extracting a common scalar factor; d / f)H(s) ( )

for n odd 1 7 (
a = and the Um(a) are the Chebyshev Polynomials of the second kind; (d / f))Un_ 2 (x) ---~~~~~~~ -un_3(x) (1 -(d / f)H(s))l-(d/f)H(s)) 172 n-2 un-2(x) (1 -(d/f)H(s))l / 2 -Un_3(x) (d/f)H(s))l/2 Using this expansion in equation 12 and performing the indicated matrix multiplication we arrive at the final vector relation; n+l Dk(s)(l -(d~f)H(s)~(d/f) 2 -(d/f) U (x) -un_3(x) (1 -(d/f)H(s))l/2 n-2 ( llf) 1 -(d/f)(l + H(s)) (1 -(dtf)H(s))l/2Un-2(x) -Un_3(x)The ray position response relative to the transmission axis at the (k + n) /f)H(s))l/2 n-2 n-3 } Now, by applying the recursion relationship for the Chebyshev Polynomial ( 15 ) in equation 21 we reduce the ray position response to a compact expression; n+l 23) rkt-n(•) -( 1 -(d~f)H(s) rT (d/f) un-l(x) Dk(sd/f)H(s) k This equation gives us a closed form surrnnation for the ray position response at a lens entrance plane located n unit cells beyond a transverse disturbance of the kth element in the guiding structure.Using this relation it is now possible to evaluate the response of any desired configuration of guiding structure and control.The pertinent parameters (lens focal length to separation ratio (d / f), and control form H(s)) may be inserted in equation 26 and the inverse Laplace Transform taken to arrive at the time domain expression for response for the structure being tested.The confocal waveguide ((d/f) = 2) is particularly interesting due to the fact that in this case only one term of the sunnnation in equation 26 remains (that term for which n -1 -2m = O), yielding illustrating that the beam passes through the transmission axis at all even numbered lenses after the disturbance in this case.For this configuration an analytic expression for the time domain response may be found for a control in the form of an ideal integrator with gain 28) H(s) -G/s which, when used in equation 27 and assuming a unit step disturbance the kth lens, results in 1) 2 e-2Gt tl(n;l) m=O m -2Gt)m n odd In order to reduce this expression to something less formidable we note that the Laguerre Polynomial of integer order is given by (ll) time domain ray position response at any arbitrary lens entrance plane located an odd number (n) of unit cells beyond a lens disturbed by a transverse unit step.This response for t he first six odd numbered lenses (up to the thirteenth) is plotted in Figure 2. Digital computer simulation of the controlled waveguide using the integrator with gain as a correction for the disturbance has been carried out as a test of the analytic results.Disagreements between > the computer output and the derived expression were negligible and attributable to digital integration error.The simulation program used is located at the end of this appendix (Listing BL.1--Guidel).In an effort to de t ermine the form of the time domain response for a class of physically realizable controls ra t her than ideal cases the waveguide was simulated using a correction function 35) H(s) = -tJ2 n a second order system operating on the transverse position of each lens.The simulation program used (Guide2) is listed as BL.2.Two sets of resultant unit step response curves are shown in Figures 3 Figure C .1.th The displacement at any k lens now will be the sum of the initial disturbance and a correction involving the sum of ray

1 -
e v a lue of r ay position found in equation 7applied to the displaced lens resulting i n a ray vector at the lens following t h e disturbance of 2H(s)If we now assume a control in the form of the integrator withH(s) = -G 2sand an input disturbance of a unit step, we have