Compensation of spacer-thicknessvariations in a holographic Fabry-Perot filter
Louis Sica, Tin Aye, Indra Tengara, and Bernard Wexler
The fabrication of a solid, holographically recorded Fabry-Perot interferometer that uses plate glass forthe spacer has recently been reported. The component produced sharp, circular Fabry-Perot fringes inspite of its use of a plate-glass spacer. We develop a general theoretical characterization of such acomponent that accounts for its low sensitivity to spacer-thickness variations. We use the Kogelniktheory of volume holograms to calculate the phase change on reflection from the mirrors. This phasechange results from the position of the fringes formed throughout the two holographic media during therecording process. An expression for the wavelength location of the transmission peak versusspacer-thickness variation is derived that agrees with the current experimental information available.
The experimental demonstration of a solid Fabry-Perot interferometer fabricated from plate glass bymeans of holographic recording has recently beenreported.1 2 Researchers created the component bycontacting a mirror to the rear side of the plate glass,previously coated on both sides with dichromatedgelatin, and exposing it in a retroreflection mode sothat both Bragg mirrors were formed at the sametime. High-quality Fabry-Perot fringes were formedby this interferometer after processing. In this pa-per we give a theoretical account of a primary capabil-ity of this type of component, which is the dramaticcompensation of variations in spacer thickness by thephase change on reflection from the holographicallyformed mirrors.
A large amount of research has been done in fieldsrelated to the subject of this paper. Exact theories ofdiffraction from thick dielectric gratings are nowavailable,3-7 and questions parallel in part to thoseconsidered here have been treated in the context ofBragg reflectors on bistable devices8 and surface-emitting lasers.9 Nevertheless, an initial though notexhaustive survey of past research, including texts,10
L. Sica and B. Wexler are with the Naval Research Laboratory,Washington, D.C., 20375. T. Aye and I. Tengara are with thePhysical Optics Corporation, Torrance, California 90505.
Received 19 May 1993; revised manuscript received 3 January1994.
did not reveal analyses specifically adaptable for usehere, especially because our goal was the derivation ofanalytic formulas useful for engineering design ratherthan computer predictions of final performance (whichwere already in hand). Consequently, the Kogelniktheory"1 was appealed to; though it is not exact, itappeared to be sufficiently accurate for our presentpurpose.
To develop a simple theory of the behavior of theinterferometer, we slightly modified the Kogelniktheory for reflectivity from a thick hologram toinclude the effect of an additional phase shift in thecosine dependence of the refractive-index modulation.This modification is outlined in the following section.The next step is a computation of optical path lengthsthrough the holographic media and spacer, beforeand after reflection, to compute the phases of theBragg gratings formed from the interferences of theincident and reflected waves at the time of exposure.From the resulting phase changes on reflection fromthe holographic mirrors, a modified Fabry-Perottransmission formula may be obtained. This for-mula accounts for the behavior of the component.
It is found that, in the absence of shrinkage, theholographic Fabry-Perot (HFP) interferometer recon-structs the mirror-reflected wave used in its creationindependently of spacer thickness. If the compo-nent is used in transmission at an angle differentfrom the recording angle or if shrinkage occurs, a lowsensitivity to spacer-thickness variations arises.The magnitude of this effect is derived from the basicrelations obtained. We do not account for the effect
of absorption in the Bragg mirror material. How-ever, this could easily be added because it is includedin Kogelnik's original theory.
Coupled-Wave Theory
The phase shift of a wave on reflection from ahologram with sinusoidal index of refraction modula-tion is derived from Kogelnik's coupled-wave theory"as presented by Collier, Burckhardt, and Lin (CBL).12Because the development here closely follows that ofCBL, our theory will be quickly sketched, with stepsand definitions differing from those of CBL specified.
The case under consideration is one in which theBragg planes are parallel to the boundary of themedium and in which the hologram is specified by aspatially varying dielectric constant,
E = Eo + El cos(K -r+) (1)
with IKI = 2/d. The medium is also initially as-sumed to have an absorption constant at independentof position. Our purpose in this section is to obtainthe complex reflection coefficient when is nonzeroso that it may be used in the Fabry-Perot trans-mission formula. One may accomplish this by follow-ing the derivation given by CBL, using Eq. (1) with+ • 0 in place of the condition + = 0 as ordinarilyassumed.
As a result of using Eq. (1) for the dielectricconstant and using the absorption constant a inthe differential equation for the electric field, oneobtains
V2a + (12 - 2iotp + 2xp[exp(iK r + i)
+ exp(-iK r - i)]}a = 0, (2)
where P = 2lirno/Xa, X = Trnl/Xa, and no and n1 are theaverage refractive index and amplitude of the space-oscillating part of the refractive index, correspondingto the dielectric constants Eo and E1 in Eq. (1). Asolution for Eq. (2) is sought in the form
where the primes indicate the first derivative withrespect to z, CS = u(/Il, CR = p/ P, and F is defined by
(6a)Ipl2 - 1l2 = P2 - o2 2F.
From Eq. (4), Eq. (6a) may be expressed as
p2 - 2 = 2p.K-K 2 = 2K sin 0 - K2 , (6b)
where '/2 - is the angle between p and K, and theparameter r, important in the development below, is
K2
IF= K sinO 0 -- (6c)
For a grating parallel to the boundary, 0 is the anglebetween the direction of incidence, specified by p andthe boundary. When the Bragg condition is satis-fied,
Ksin 0 = -'
213(6d)
sothatF= 0.One may obtain an equation in R alone from Eqs.
(5a) and (5b) by differentiation:
a( + ) (+irt )R+2 zR"1(z) + \C + )R'(z) + k SR )(Z) = 0,
(7)
which is independent of the offset phase , given inEq. (1). This implies that the phase of the transmit-ted beam is independent of the phase of the grating.After R is found from Eq. (7),the reflected-amplitudemay be found from Eq. (5b). It is clear that one maysatisfy Eqs. (5a) and (5b) for + • 0 by multiplying thesolution S(z, c+ = 0) by exp(if). From the boundaryconditions for a hologram of thickness T, R(0) = 1,and S(T) = 0, S(O) may be shown to be (a = 0 as willbe assumed from now on)
a = R(z)exp(-ip r) + S(z)exp(-ia r), (3)
where R and S are the amplitudes of the transmittedand reflected waves, respectively. Both are in gen-eral complex.
After Eq. (3) is used in Eq. (2), only those terms arekept that satisfy the Bragg relation
a= p - K, (4)
where I p I = P = 2ITfnO/Xa. We find that to satisfy Eq.(2), one must satisfy two differential equations in S(z)and R(z):
csS'(z) + (iF + )S(z) = -ixR(z)exp(i+), (5a)
CRR'(Z) + CoR(z) = -ixS(z)exp(-il), (5b)
S(0) = - exp(-iTr/2 + i)
+ [1 - (-) ] coth(vr2 - 2)1/2
(8)
where g = FT/(2 cos t'o) and v = rnlT/Xa. Theangle io lies between the normal to the Bragg planesand the incident-beam propagation direction. Atnormal incidence at the Bragg wavelength, F = 0 [seeEqs. (6c) and 6(d)] implies that t+ = 0, so
S(0) = exp i(, - 7/2)tanh(XA)- (9)
Thus, to find the phase shift on reflection, one shouldmultiply the wave incident from the left by theexponential exp(-iTr/2 + i). These results have
been derived through the use of the CBL sign conven-tion shown in Eq. (3), in which a wave advancing tothe right is represented by exp(-ikz + it). If theBorn and Wolf 13 representation exp(ikz - it) is used,then the relation of Eq. (9) is replaced by its complexconjugate.
In general, a reflection may occur that is notprecisely at Bragg resonance, with the result that F •
0, and thus another phase term must be included inEq. (9). This can be found from Eq. (8). Defining
= r/vr, and then defining Y and X as
Y = A, X = (1 - 2)1/2 Coth[T(- 2)1/21
(10a)
respectively, we see that Eq. (8) may be written as
exp[- - - i arctan(j) + i (b
S(0) ~(X2 + y 2)1/2 1b
Note that because ultimately the Born and Wolf13sign convention is used, it is the complex conjugate ofEq. (lOb) that we employ below to obtain the phase onreflection inserted in the Fabry-Perot formula. Theproblem to be considered next is the calculation of +for the two holographic mirrors. This depends onthe positions of the interference fringes throughoutthe structure at the time of recording.
Right Hologram Mirror
Figure 1 shows the coordinates to be used in thederivation of the phase angles of the right- andleft-hand hologram mirrors. For the right-hand mir-ror (z = h to z = h + T), optical paths and phases aremeasured with respect to the origin of coordinates at02 and labeled r2. For the left-hand mirror, to theleft of 01, coordinates are measured with respect toorigin 01 and labeled r1. For waves in these holo-graphic media having refractive index n before process-ing, the normals to the incident and reflected wave
x
n n
05 2Ta-- - 03
', z = h z T + hn
Fig. 1. Solid line with arrows indicates sequence of coordinatetransformations to coordinate origins 01, 02, 03, 04, 02, 05,0,. Holographic media lie to the left of 01 and between z = h andz = T + h, whereas the spacer lies between z = 0 and z = h.
fronts are inclined to the z axis and are specified bythe unit vectors
h = yk + t, (11)nref = -yk + at,where a and y are direction cosines with respect toaxes x and z. The propagation vectors are taken tolie in the plane of the figure, so the direction cosineP = 0 (there should be no occasion for confusion withthe magnitude of the p vector defined above). Forwaves in the spacer medium with index n,, a subscripts is added to the wave normals' unit vectors anddirection cosines.
The incident and reflected waves, U2,, and U2n ref(see Appendix A), in the recording medium betweenz = h and z = h + T are given by
2rrin r2U2n = exp A" + nkisi (12a)
U2n ref =_ exp 2 'rinref + + i + ci + 4 refi),
(12b)
where X,,, is the write wavelength, As is the phasecaused by traversing the spacer layer, 4kT resultsfrom reflection from the mirror plus vertical transla-tion, and kref is any phase change on reflection. Thephases k2T and ~,c derived in Appendix A, are given by
27rn-y2T+2T X
Aw
= 2rrn~hus weYs
(13)
The field U+ caused by the superposition of U2n andU2n ref is
2rrinax2 \ [ex in(Z2U+ = expt -TA~ + exitp A
+ exp(- 2rinYZ2 + 2Ti + kefi) - (14)
The sinusoidally modulated component of index ofrefraction generated by exposure and processing isassumed to have the same phase as the interferencepattern I+ = U+ U+, , given by
I, = 2[1 + cos( 2 jT)2 - (22 - Ief)1(15)
Left Hologram Mirror
The incident and reflected fields in the region ofnegative z1 in the left-hand hologram (see AppendixA) are given by
where 4 2h iS Reflected Incident Right VolumeWave Wave Hologram
27rn,2h-y,4 ) 2h = X.xaw
(16c)
The superposition of incident and reflected fieldsequals
U_ = exp A27 )[ exp( 2 z)
+ exp(- 2inyzl + 2hi + ck2Ti + 4)refi)],
(17)
and the intensity I of the interference pattern,which generates the same phase in the resultingindex of refraction modulation, is
I = 2[1 + cos 22rrz, - (42h - 42T - (ref)] -(x,,/2nly
(18)Left VolumeHologram
Sign Convention
As we show above, apart from the diminution of itsamplitude, a wave incident from the left representedby exp(ipz - it) would, after reflection from a bound-ary with a hologram specified by
n(z) = [no + n, cos(Kz + +))], (19)
be representedbyexp(-iuz - ijet - is) + iir/2 + i)B),where NB = arctan(Y/X) and the complex conjugateof Eq. (lOb) is being used. The effect of reflection onthe phase of the wave is thus represented by multipli-cation by the phasor
i=-P = exp - - i + B
2(20a)
If 4) is positive, as shown in Fig. 2(a), the Bragg planesof the hologram (not the surface) have a bump that isconvex toward the left. This results in a bump thatis convex toward the left in the leftward-travelingoptical wave, or a decrease in the optical path tra-versed by the wave. However, if the hologram repre-sented by Eq. (19) is on the left, as indicated in Fig.2(b), the light wave sees positive phase change 4) as adepression or a lengthening of its optical path. Itseffect on the wave is reversed in sign from the samehologram on the right. However, the effect of thephases r/2 and 4 )B, which taken alone correspond to ahologram in the canonical 4) = 0 position, must be thesame whether the hologram is on the left or the right,because it looks the same to a light wave in eithercase, i.e., it causes a phase lag that is the equivalent ofan increase in path. Thus a leftward-propagatingwave will have its change of phase on reflectionrepresented by multiplication by the phasor
PI = exp( + i + i) (20b)
(a)IncidentWave
ReflectedWave
Boundary(b)
Fig. 2. Incident plane wave and resultant reflected wave afterreflection from holographic medium with (a) convex outwardperturbation and (b) concave outward perturbation of Braggplanes.
Fabry-Perot Phase Function
The normalized transmission function of the Fabry-Perot may now be obtained. Following the notationin Born and Wolf,'3 we see that the transmission is
11 + - sin2( " '+ ) (21)
where the finesse is F = rrRl/2 /(1 - R), = 4rnshcos 0/X, h is the thickness of the spacer, n is thespacer index of refraction, and 0 is the angle of thepropagation direction with respect to the normal tothe spacer. The angles 4) r and Al, are the phaseangles of the reflection coefficient caused by reflectionfrom the right and left mirrors. Positive phaseangles have the same effect as an increase in the
spacer thickness h. From Eqs. (20a)(18), one obtains
r
4)r = + B + 4 )2T + refb
'rr
), (20b), (15), and
1 = + HUB - 2T - 4) 2h - ref,
xawd = 2ny
Finally,as
(22a)
(22b)
(22c)
using Eqs. (lOb), (13), and (16c), we define CF
(>(H. X A 4 r+i X 2'rn 5h cos 0 or
2 - 2
- S + arctan(Y/X). (23)
Equation (23) is the main result of the previousanalysis.
Consequences
Antiresonance Behavior
To understand the consequences of Eq. (23), we find ituseful first to consider the case in which the illuminat-ing wave is incident at the same angle and wavelengthas the original write wave. In this case cos 0 = ys andX = XA,. This corresponds to the Bragg condition forthe mirrors so that F = 0, r = 0, Y = 0, and thereforearctan(Y/X) = 0. Thus,
0.47 0.48 0.49 0.5 0.51 0.52 0.53
Fig. 3. Plot of the analytic expression for the HFP transmission ofEq. (21) developed in this paper. The parameters of the structureare X,, = 0.5 pLm, h = 10.031847 pim, n= 1.57, n = 0.1, T =2.90322 pLm, no = 1.55, and n = 1.55. The absorption is assumedto be zero.
same filter structure. Good agreement between thetwo treatments is to be expected in view of the recentresearch of Kim and Garmire,4 which shows thatmatrix methods and coupled-wave methods yieldessentially equivalent results for Bragg reflector struc-tures over a considerable range of parameters.
Filter Compensation at Normal Incidence
In practice, the holographically fabricated Fabry-Perot interferometer may suffer from slight shrink-age of the Bragg mirrors because of processing. Weinclude this in the above treatment by multiplyingthe grating spacing d by a small factor s slightly lessthan 1 so that
sxw2nry (26)
CF(0, X.) =
and the transmission is a minimum:
T1
(24)
(25)
1+2
This result has a simple interpretation in terms ofcommonplace concepts of holography, namely thatfor the recording geometry used, a hologram is beingmade of an object wave retroreflected from the mirror.Consequently, if after development the medium isilluminated with the reference wave at the sameangle as during recording, the corresponding objectwave will be reconstructed. The significant fact isthat, for this operating condition, the fabricatedFabry-Perot interferometer has an exact antireso-nance that is independent of the thickness of thespacer layer.
An illustrative plot of Eq. (21) that uses Eq. (23) isshown in Fig. 3 for the data given in the correspond-ing caption. The results of the analytical theorydeveloped above are indistinguishable in a plot fromthose computed from a matrix model applied to the
By carrying out the recording at an angle to thenormal when fabricating a HFP interferometer sothat y = s, we see that the final spacing is the same asif recording had been carried out at the same wave-length at normal incidence without shrinkage. Aswe show below, small changes in the direction cosine,y that either overcompensate or undercompensatethe effect of shrinkage allow us to shift the filter froman antiresonance to a resonance condition at thewrite wavelength at normal incidence.
At normal incidence or = FT/2, vr = rlrnT/X,,, andF = K - K2/23, where ,3 = 27rno/Xa and K = 27r/d.Evaluating F by substituting values for K, A, and dfrom Eq. (26) results in
r = 2r 1 - n_ -sX" sXno}
(27a)
We may now use this expression to obtain Y = (,Iv, or
2nyXa ( aYXan 1 s,\ - I,n18xw sxw/
(27b)
after the additional simplifying assumption that n =no, which implies that the average refractive index
before processing equals the average after processing.It will further be assumed that n8 = n, which impliesthat y = ys.
The behavior of Eq. (23) and ultimately the trans-mission, Eq. (21), for small variations in xa around Xwmust be obtained. First the effect of these variationson Y will be evaluated so that the result may be usedin Eq. (23). Note from Eq. (6d) that the Braggcondition at normal incidence implies that
K= 23, (28a)
from which
XB = sXw/'Y-
Inserting this result into Eq. (27b) yields
2nll_ a A\n,XB XB
(28b)
(29)
into which one may now introduce Xa = Xw + d toobtain
this condition to Eq. (31), with dX and h allowed tovary so that we obtain the variation in wavelengthnecessary to maintain resonance corresponding tochanges in spacer thickness h. Expanding Eq. (31)to first order in differentials dX and dh yields
2Trnh 27rndh 2rrnh dX Tr 2rrnhy 2rndhy1 _ _ _ ___
X,,, x X,, X 2 X X,
+ R_ Y,+ FR,2n d 1 2 X,, d X=pr 3b+ +1T T(1fn B B Xwp r. (32b)
After using Eq. (32a), we find that this becomes
2irn(1 - y)dh dX [2rnh _ 2n 1 2XwAw ~ ~ ~ p Aw 1 -tPn B B/
(32c)
where a term of order (dX/XB)2 has been dropped.Finally, using Eq. (32a) to obtain a value for2'rrn(1 - y) and defining h/,w = Nh, we obtain dh/xWas
Y = o + ndI 2 _ dX )Y=Yo+-- XB XB (30a)
where
E 2n x\ 1-X.\nY o i l XB(30b)
To simplify the development of the sensitivity esti-mate, we assume that the filter transmission ofinterest is located near the peak of the Bragg reflectiv-ity so that Y2
<< 1. In this case one may set 6 = 0 inEq. (1Oa) so that X may be determined from
[2 nNh2 antR 2n Xw 2X,, _) dh dX 2 f nl B ( B 1
[2Trn - 1vR -- 1nXx -x ( 2P-l RpTFnXB 1 w)]
(33a)
To simplify this expression we take advantage of thefact that XA x XB and Rp 1. In this case, for a HFPinterferometer, Eq. (33a) becomes
d (2rnNh2 + -Nh)X,, X,, [(2p - ),m
g'R = ~1 = 1
coth(71T) -
(33b)
(30c)
Equation (23) can now be evaluated for wave-lengths xa close to both the write and the Braggwavelengths:
C(0(, dX) = 2rrn 5 h iT 2'rn 5h-y
4>(, d =X, + dX + 2 X, RpY
2n dx 2X, dx\+ R,-X |1 -X X) (31)ni B XB XB,
where arctan(Y/X) Y/X has been used. If a filtertransmission resonance occurs at the write wave-length dX = 0, then
27rnh(1 - y) IT 2n X. (1 ,,
(32a)
where p is any integer or zero. We may now apply
The following parameter values may be experimen-tallyrealized: n = 0.01,n = 1.5,Nh = 10,n 1 = 0.01,and p = 0. If d = 2 A and , = 5000 A, thendh/x, = 1.
A similar development for a standard Fabry-Perotinterferometer yields
dh dx= Nhy' (34)
For the parameter values given above, dh/x = 0.004.Thus, the standard Fabry-Perot interferometer is250 times more sensitive to spacer-thickness varia-tions than the HFP interferometer for the aboveparameter values. The values yielded by Eqs. (33b)and (34) are consistent with experimental spacer-thickness tolerances found in Fabry-Perot filters andthe observed variations in passband location over thesurface of experimental holographic filters. Furtherexperimental details on this subject will be given in afuture publication.
An expression for the compensation of the holo-graphic Fabry-Perot interferometer has been derivedfrom Kogelnik's coupled-wave theory of volume holo-grams. The dramatic capacity of such structures tocompensate spacer-thickness variations is related totheir reconstruction of the previously recorded re-turn wave; thus such holographic structures mostnaturally function as notch filters in transmission.Fortunately, shrinkage together with variation of therecording angle from that of normal incidence per-mits the creation of transmission filters. However,because the condition of use is not identical to that ofrecording, compensation of spacer imperfections is nolonger perfect. An expression relating variations infilter peak position to variations of spacer thickness isderived in terms of material and recording parameters.For the parameters used here, the holographic Fabry-Perot interferometer is 250 times less sensitive tospacer-thickness variations than a Fabry-Perot inter-ferometer of conventional construction.
Appendix A
The derivation of Eqs. (12a), (12b), (16a), and (16b) iscarried out in this section. Refer to Fig. 1 for thecoordinates and geometry relevant to the calculation.The method consists of the computation of the phaseof a plane wave that propagates at an angle to the zaxis from the left holographic medium of index n,through the spacer of index n, into the right holo-graphic medium, reflects from the mirror, and re-traverses the media to the left while interfering withthe incident wave. In regions where the interfer-ences have to be accounted for, both the incident andreflected waves must be written with respect to thesame coordinate axes. The origins for these arelocated on the boundaries with the spacer. Thewaves are written with respect to successive labeledorigins of coordinates in the order 01, 02, 03, 04, 02,05, and 0,, along an arbitrary ray at a changingheight above the z axis. The subscripts on the waveamplitudes indicate the current medium, currentcoordinates, and whether the wave is incident orreflected. This is also indicated by the subscripts ofthe position vectors r.
The incident wave Uln(rl) in the left hologram is
Uln(r,) = exp(2 A ) (Al)
In the spacer this wave becomes
Ul,(r,) = exp( 2 fs r1 )* (A2)
respect to 02 necessitates propagating to 02 overdistance 15 = h/y, producing
U25(r2 ) = exp42nlh, (r2 + A
2irinsfs r2 2Trinsh= exp X' T XwW'S ' (A3)
where
27Tn5 h
_ " ' Y (A4)
After entering the right-hand hologram the wavebecomes
I2-rrinn rU2n(r2) = exp Ar, + i4s)
which is the result of Eq. (12a).Writing this wave with respect to 03 one has
U3. (r3) = exp[2min - (r3 + Ins) + i ]
where
27rinT1n =Away
The reflected wave written with respect to 03 is
U 3 n ref (r3 )
= ex~12frrinhref r 3 + 2ITinTexpX W1Y + i)ref)X
(A)
(A6)
(A7)
(A8)
and with respect to 04 is
[2inlfref~ (r4 + 1,,href)
U4 n ref (r4 ) = exp r
27rinT1+ 2W + is + ire(2Trinnfref r4 2Trin2T \
= exPk A~ + A- + id)s + Zref I,
(A9)
where )ref is any phase change caused by the reflec-tionatz = T + h.
Points in the wave field must now be written withrespect to 02, which involves a negative verticaltranslation of coordinates:
2 'i~rif r - 2T -( .
X"wU2n ref (r2) = exp
The boundary condition along the vertical boundaryimplies that na = na,. Writing this wave with
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