[lee] chapter 6 - the prism coupler_ocr version

8/14/2019 [Lee] Chapter 6 - The Prism Coupler_ocr version

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The Prism oupler6.1 11\TRODUCTIONOne of rhe primary diagnostic tools for evaluating the propagation characteristics ofdielectric waveguides is the prism coupler. In its simplest form the coupler makes useof a high-refractive-index prism placed in close proximity to a slab dielectric waveguideas shown in Figure 6.1. When an optical beam passing through the prism is incidentupon its bottom at an angle exceedmg the critical angle the evanescent fields thatextend below the prism base penetrate into the waveguide. These fields as we showare capabk of transferring power between the incide11t beam and a waveguide mode.By appropriate choice of the angle of incidence and proper coupler design a siguiticantportion of the power in the inddem beam may be transferred into a single chosenwaveguide mode. Further by reciprocity if a second identical prism is placed in closeproximity to the waveguide at some distance away from the input prism eac h propagating mode will be coupled out of the guide at an angle that is characteristic of thatparticular mode. By measuring the output angles a determination can be made ofwaveguide film refractive index and thickness.

To analyze the operation of the prism coupler it is first necessary to understandwhy it is possible for power to be transmitted through the gap between the prism andwaveguide even though the incident beam exceeds the critical angle at thls prism-airinterface. This phenomenon is referred to as frustrated total internal reflection.

6.2 FRUSTRATED TOTAL INTERNAL REFLECTIONThe problem of transmission of a light beam through an air gap is a special case ofplane-wave t r a n s m i s s ~ o n through two dielectrlc interfaces as shown in Figure 6.2. Forthe prism coupler configuration :-egions 4 1 and 2 correspond respectively to theprism a ir gap and ~ ~ a v e g u i d e film.

Let us assume initiaHy that a plane wave is incident from region 4 at an angle isuch that the critical 2 ngle is not exceeded at either the upper or lower interface. Thisrestriction will be subsequently removed. To calculate the transmission coefficient useis made of the so-called ray summation technique. The incident and transmitted planewaves are considered to be made up of an infinite number of pencil beams or rays afew of which are shown in Figure 6.3. Depending on whether the incident wave is TEor TM the amplitude of each ray is chosen to be proportional to the electric or magneticfield strength respectively. For convenience the amplitude of the rays corresponding

147


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1"-M r i l l ~ PRISM ( O U P L l ~ R

Figure 6.1 Geometry for prism coupler.

to the incident field are normalized to unity. What we wish to determine is how eachincident ray will contribute to any arbitrary single transmitted ray. Let us, therefore,examine the contributions to the transmitted ray T . We number the incident raysconsecutively I, 2 3 . . . The phase of each incident ray is referenced relative toany arbitrary constant phase plane such as the one indicated by the dashed line inFigme 6.3. From this t1gure, it is observed that ray I contributes directly to thetransmitted ray T, by propagating along path 1-1 , being partially transmitted andreflected at the upper and lower dielectric interfaces. Let the phase shift resulting frompropagation along 1-1 be given by J 0 Further, Jet the respective transmission coefficients associated with upper and lower interfaces be defined by T4 and T,2 Theseare obtained by direct evaluation of the TE or TM wave transmission coefficients fura single dielectric interface derived in Chapter 3. Recalling that each incident ray hasbeen normalized to unity, the contribution from ray I to output ray T, is then givenby

(6.1)To obtain the contribution to the output from ray 2 we note from Figure 6. 3 that

this ray does not contribute to T, along a direct path. Instead, the por:ion of the raywhich is transmitted through the upper interface traveJs along the zigzag path 2 2 -

:; z

Figure 6.2 Transmission of a light bc


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Bysymmetry, ~ < > must also be the relative phase shift between any two adjacent:cdent rays. Thus the contribution to the output pencil beam T, from each subsequentY 3, 4, Isobtamed by multtplymg the previous ray's contribution by the factorgiVen m expresswn 6.2. The total contribution to the transmitted ray T, is thereiore

Td T41TI2eJ yields

l - av1 - a

Provided that lal < 1 then in the lintit as N-;. oo, d --;. 0 andlim S vN ~ w 1 - a

(6.5)

We, therefore, obtain for the transmission coefficient for a plane wave propagating

I IHJS-11{/\ JEU TO i\L INTU\N/\L l < l . l U ~ C I I O N 1 ~ 1

tlownward through the gap

(6.6)

To further show explicitly the dependence ofT, on the gap thickness, we note that thephase shift 0 can be broken into two parts, a portion in region 4 along the path 1- I'and a portion in region l along l '-1 . The phase shift for the first portion is simplyminus the product of the wavenumber k4 with the- segment length l1_J', or

To obtain the phase shift along the second segment reference is made to Figure 6.5.The geometry shows that the segment I' -I can be represented as the sum of twosmaller paths

''- " (g cos e, + liz sine,)and, therefore, the corresponding phase shift is

< >. - " - k, g cos e, + sin e,)

where we have made use of Snell s law to relate k4 to k 1 The total phase shift Po istherefore given byPo ,_,, + ,_,,

1 g - k4 11_ 1 + liz sin OJBecause the reference phase plane in region 4 may be chosen arbitrarily, the secondterm in the above expression may be made equal to a multiple of 21T by appropriate\ / '4;.;

/g / '1

. // 0 , . t l j / /'2Figure 6 5 Geometrical construction for calculating the phase shift along the path segment l1 1.


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152 THE PRISM COUPLER

choice of the length 111 . Thus, for such a suitably chosen reference plane < Jo - k gand the transm1sswn coetfic1ent for ray Tct is given by

(6.7)Note that although the transmission coefficient was obtained explicitly for only a singletransmitted ray: the same result must apply to all transmitted rays because of symmetry.

Let us x a ~ m e this express1on now for the case in whkh the critical angle is ex


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154 THE PRISM COUPLERas

If we define

2

2

tan(< J22) = a. 1 k _tan(< J4 2 ) = a J k4 ,

and note that k2/ k4.shows that (k2.Jrx.")l(k4, /a 1) then straightforward algebraic manipulation

(6.12)where

Tand

6.3 W


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I56 THE PRISM COUPLER

where < >, is the phase shift due to total internal reflection from the lower boundarv ascalculated m Chapter 3. The reflection at the upper boundary will be represented interms of its magnitude and phase

Thus n may be rewritten as

where

R, IR le

T 1 - IR, I eM1 - IR,Ie

(6.13)Note that < > represents the relative phase between successive- ray contributions. Thepower reflection coefficient at the upper boundary ra = i 11 12 , must y conservationof power be related to the power transmission coefficient upward through the gap by

Therefore,

(6.14)

Let us adjust the prism coupling gap so that ' is almost zero. This can always beaccomplished smce m the hm1t that g approaches infinity we have total internal reflection and t, equals zero. In this limit the binomial approximation may be used togive

n( I - t ) 12 = I - tu 2 u

Furt3er, provided also that lfn/2)t,l 1 for ail n then by Taylor series,

so that Eq. 6.14 may be rewritten as

[_ (ni2)t efrl< > lA = T I' I - (1 e ~ > J (6.15)

W ~ V E G i l l E EXCITATION USING THE PRISM COUPlER 157

Additionally, let the angle of incidence e, be adjusted in such a manner that the phaseterm is close to a multiple of 2 IT. That is

where

Noting that

then algebraic manipulation of expression 6.15 yields

Since by assumption both S and t are small, the product term o t,/2) in the aboveexpression for A may be neglected and

(6.16)where

Let us investigate how A varies with distance along the prism. J\ote that fromFigure 6.8, the distance z from the left end of the prism is given by (n - l)ll.z, sothat increasing values of n correspond to increasing distance from the left end of theprism. We first examine fbe n-dependent term multiplying iL by considering it to bethe sum of two phasors. To the term le we subtract a second term } 6 -\ti/2)' whosedirection and magnitude depend on n and o. Note that as n increases, the magnitudeof the second phasm decreases. Thus for various values of n the phasor sum R mightlook as shown in Figure 6 9a The tip of the resultant vector versus increasing n wouldfberefore, look as shown in Figure 6.9b and lA,Iwould exhibit an oscillatory behavioras shown in Figu,-e 6.10. Thus, when the phase difference between adjacent raycontributions is not exactly 2 11 the field amplitude builds up under the prism in anoscillatory manner with some rays adding constructivelyand some adding destructively.Aoo is observed to e the saturation value fcir an infinitely wide beam. However whenfbe phase term S = 0, the osciilatiom cease, giving an exponential buildup with N(Figure 6.11). In this situation each additional pencil beam adds in phase with the onebefore it, asymptotically building up to the value A for large beamwidths.


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Rn ~------::

n = 4 . . ;. ;.

n==5.---J

n = 6 ----=----.,-----;.--1a)

Figure 6., a_) Phasor representation of Eq. 6.16 for several values of n = I, 2, _ N. (b) Trace of thetip of the resultant vector R versus n.

Let us examine the dependence of A on the parameter o. The ratio IAoo (b 7 O)j"lAx (o = OJI' is a measure of the relative power coupled into the waveguide and isg1ven by

IAx O 7 0) IA,(o = OJ [t + (2ott,J J 6.17)which is plotted in Figure 6.12. Since the transfilissivity through the coupler air gapmay, in theory, be made as small as desired, the above curve can be very sharplypeaked about o = 0 and o an, therefore, be forced to be arbitrarily small for significantpower buildup within the guide.

W A V H i l J I I > I ~ EXCITATION USJNG TilE Pl


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160 THE PRISM COUPLERRelative amplitude

Figure 6 12 Ratio cf coupled intensity under non-phase-matched conditions to [hat under phase-matchedconditions as a function of mismatch parameter B

p where e corresponds to the bounce angle for the pth waveguide mode:

The required angle of incidence 6 for the incident beam inside the prism coupleris related to P by Snell's law (phase matching)

6.20)Thus, by adjusting e properly, we can couple to any propagating mode desired. R e l a t i o n ~6.20 is an extremely important one. It states simply that in order to couple to a particularwaveguide mode we must use an incident beam which is phase matched to that modeThat is,

6.2ljUnder this condition the incident and guided waves are said to be synchronous. Thefunction of the prism now becomes clear. Notice that in absence of a prism having apermittivity grealer than that of the waveguide film, the phase-matching criterion couldnot be satisfied by an incident plane wave. For example, a p lane wave incident on thewaveguide film in absence of the prlsm c;:mp1er would have a corresponding diagram

n=

n=N iguR 6.13 Effective cancellation of rays. contributing to outputwhenQ 0

WAVUHliiJE bXClTATION LSING THE PRISM COU I.ER 161

n= n=N

J< igure 6 14 Summation of rays rontributing to ootput whea 3 = 0.

as shown in Figure 6.15. Because for guided waves k -s w < k then it is clearfrom Figure 6.15 that no angle of incidence can satisfy relation 6.21. However, if theprism permittivity"< equals or exceeds that in the film we can have the situation shownin Figure 6.16 and phase matching is always possible.To determine the coupling length necessary to transfer appreciable power into aparticular rr:ode of the waveguide recall tha t under phase-matched conditions (8 = 0)the buildup of beam intensity goes as

or(6.22)

where

Thus, the distance required for the amplitude to build up to li of its saturation value

k t : : : W ' v ~

AirFilm

Figure 6 15 Without high-refractive-iEdex: prism coupler, phase makhingof incident beam to guided waveis impossible.


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PrismFilm

Figure 6.16 Introduction of prism pcnnits. pha }e matching to guided wave.

A is given by2i\.zt, (6.23)

The parameter i\ z can be related to the waveguide dimensions and mode propagationanglewl hc the gu1de e, usmg Ftgure 6.17. Note that to properly relate the ray spacingto gmde geometry m our ray approach we must incorporate the effect of the Goos-Hanchen shift discussed in Chapter 2, replacing the physical thickness d with theeffective thickness d, . This dimension was shown in Section 3.6 to be

ddf = d + d, + d, (6.24)where

forTE waves

for TM wavesand

.t

WA vH;UUE EXCITATION USING TH i I'RISM COlll l.lm I(.J

Figure 6.17 Proper computatio:J. of spacing D z between adjacent rays requires use of 'effective'' waveguidebvundaries resul:ing from Goos-HaencheJ. sbtft.

from the geometry then,6.25)

An explicit form for the lransmissiviJy tlJ for a wave traveling upwarC through the airgap is readily obtained from the transmissivity expression for downward propagationt,. As is observed from comparison of Figures 6.18a and 6.18c, the two coefficientsare measures of the power transmitted in op}}Jsite directions along the same ray path.

Intuitively we would expect that the amount of power transmitted across an interfaceshould not depend upon the direction of that flow. To demonstrate this rigorously, weobserve that the geometry of Figure 6.18a is obtainable from that of Figure 6.\Sc bythe interchange of the roles of regions 2 and 4 as well as the incident and transmittedangles e and a . Correspondingly, the expression for t, is, theretore, immediatelyderived from that fort, given in Eq. 6.12 by interchange of the indices 2 and 4 as wellas the angle 9, and fl . If these interchanges are perfonned, it is easy to show that theresult remains invariant. Thus, fer example, we obtain forTE waves from expressions

1) t t

8; 4

l 1

2

1) 1)a) b) c)

Figure 6.18 Geometry for computing transmissivity through the air gap. a) Downward path for computingtd; b, c) equivalent upward paths fur computing 1,.


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164 THE PRISM COUPLER6.12, 6.23, and 6.25 a lie buildup distance in the weak coupling limit given by

(6.26)

For many typical applications, coupling lengths are on the order of sevecal thousandAwhich for visible radiation corresponds to distances on the order of a few centimeters.

We have demonstmted that under phase-matched conditions the waveguide amplitudeinitially grows as the prism length L is increased. It might be hypothesized therefore,that by making L sufficiently large, all of the incident radiation could be coupled intothe guide. This is, in fact, not the case since radiation is not only coupled into thewaveguide through the air gap between prism and film but is also coupled out. Whenonly a small percentage of the incident power has been transferred into the waveguide,the dominant direction of power flow is from the incident beam into the guiding fllm.However. when a sufficiently large percentage of incident power is coupled into thefilm the reverse process dominates. Steady state results when these two processesbalance, leading, as we shall show, to a maximum theoretical coupling efficiency ofabout 80 .

We shall compute the maximum coupling efficiency using an extension of the raytechnique. Let us assume TE incidence, as shown in Figure 6.19. The magnitude ofthe incident power density in the prism si is simply

Incidentbeam

Pnsm

I= 0 z= LFigure 6 19 Geometry fo calculating power coupled into the guidetl-'ave mode.

WAVECiUIDE EXCITATION US[NG '11-IE I'RISM COUPLER 65

where

and the electric field amplitude lEI is normalized to unity for convenience. The totalincident power P is equal to the product of Si with the cross-sectional area of theincident beam -or

L cos eiP = SW =' '14Consider next the total power flow from left to right across the waveguide entrance

at the plane = L As shown n Figure 6 1 9 ~ this power is contained in an upwardand downward bundle of rays each having electric field amplitude A.v. The total powerdensity incident on the entrance aperture from each bundle is

The total power P catried by these two bundles into the waveguide is equal to theproduct of the power density S, and the waveguide aperture seen by the bnndle, d 'sin ep Nt.lte that the effective waveguide thickness dcffdiscussed in Section 3.6 is usedrather than the true thickness d. Multiplying by two to account for the two bundlesyields

Making use of Eq. 6.16, P.v can be written under phase-matched conditions as

Noting that Ed lu and that

k - ITd k dl4

[lee] chapter 6 - the prism coupler_ocr version

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