Effect of disk birefringence on a differentialmagneto-optic readout
Ivan Prikryl
The performance of a magneto-optic readout system can be degraded by a phase shift between two basicpolarization components of a readout beam. One source of this phase shift is undesired birefringence inthe substrate of an optical disk. The effect of disk birefringence on the signal-to-noise ratio of a differentialmagneto-optic readout is evaluated. It is shown that this effect may be represented by two parameters J2and J3 that depend only on the birefringence and the numerical aperture of a beam illuminating the disk.Previous papers on this subject either presented involved, open-form expressions for the differentialmagneto-optic readout or assumed that the beam illuminating the disk is collimated. Here simpleclosed-form formulas are derived by using parameters J2 and J3 . The effects of system conditions that caninteract with the disk birefringence, such as an imbalance between the differential data detectors, are alsoconsidered.
Key words: Optical data storage, birefringence.
Introduction
The magneto-optic (MO) memory system uses anobjective lens that converts an incoming collimatedradiant beam into a beam that is focused on a MOactive layer of an optical disk. The objective lens has alarge numerical aperture (NA) to minimize the radi-ant spot size on the MO layer. The MO layer is on theback surface of the disk substrate. The substratefunction is to protect the MO film from dirt, scratches,and oxidization. Unfortunately some disk substratesexhibit a birefringence that undesirably affects theMO readout.
It is well known that a mutual retardation, i.e., aphase shift, between two orthogonal polarizationcomponents of a light beam that is reflected from theoptical disk lowers the signal-to-noise ratio (SNR) ofthe MO readout. Descriptions of retardation effectsthat are caused by various factors, including diskbirefringence, can be found in the literature. 6 For abeam that is focused on the disk the retardation thatis introduced by the disk birefringence varies acrossthe beam. Consequently retardation-dependent char-acteristics, which define the SNR of the MO readout,need to be treated as functions of pupil coordinates of
The author is with the Optical Products Division, AppliedMagnetics Corporation, 18950 Base Camp Road, Monument, Colo-rado 80132.
the objective lens. The descriptions that are used inRefs. 1, 2, and 3 disregard the retardation variationacross the beam. This is equivalent to disregardingthe convergence of the illuminating beam and thedivergence of the beam that is reflected from theoptical disk. The interaction of disk birefringencewith a real focused beam is investigated in Refs. 4-6.References 4 and 5 provide both theoretical andexperimental results while Ref. 6 studies a focusedbeam experimentally only. Here we also study diskbirefringence effects with a real focus beam. Theexpressions for the MO readout that are derived inthis paper, however, are presented in a clearer formthan those presented in previous papers. Under cer-tain simplifying assumptions we have derived closed-form relations for a focused beam that are as simpleand revealing as those derived previously for a colli-mated beam. In contrast, the relations that arepresented in Refs. 4 and 5 are in a general open formthat provides little insight if not supported by numer-ical data. Moreover our analysis goes beyond thescope of earlier papers by investigating the diskbirefringence under the condition of unbalanced dif-ferential detection.
The investigation of the effect of general birefrin-gence on an uncollimated beam represents a complexproblem. 6 Fortunately modern processes for injec-tion molding of the disk substrate result in an opticalanisotropy in the plane that is parallel to the disk thatis at least 1 order lower than the anisotropy in a planethat is perpendicular to the disk.2 6 Thus it is reason-
able to approximate the actual disk birefringence byvertical birefringence, i.e., uniaxial birefringence withthe optic axis perpendicular to the disk. Because theassumption of vertical birefringence provides a realis-tic model that tremendously simplifies the analysis,we restrict our study to this case. The model ofvertical birefringence may also be used to describe thebirefringence that is induced by thermal changesduring disk operation, assuming that the illuminateddisk area is not too close to the disk hub or the diskedge.
It is shown that a degradation of a focused readoutbeam that is caused by the disk birefringence may berepresented by two parameters J2 and J3 . By usingthis representation, useful simple formulas for theevaluation of the differential MO signal as a functionof disk vertical birefringence and other relevantparameters are derived. These formulas demonstratehow disk birefringence decreases the MO readoutamplitude and how it can increase detector shot noiseand modify the effectiveness of common noise rejec-tion in the differential readout. It is shown that theeffectiveness of common noise rejection can be notonly reduced but improved by the disk birefringence.An expression for the SNR in the presence of diskbirefringence is also derived.
Finally a few specific numerical examples are pre-sented to illustrate better the effects of disk birefrin-gence on the MO head performance under differentdesign and alignment conditions.
Differential MO Readout Amplitude
To analyze the interaction of an actual uncollimatedbeam with a disk, one can decompose this beam into aspectrum of plane waves and study the individualplane waves independently. The decomposition to thespectrum of plane waves is also extended to theincoming and returning collimated beams. The size ofthe spectrum is defined by the system aperture. Todetermine the total power that is detected by dataphotodiodes we evaluate the power that is carried byeach plane wave and then we integrate over the wholespectrum. Phase relations between individual planewaves are irrelevant if one is not interested in theintensity distribution across the beam.
To simplify the problem we neglect the effects ofthe disk groove and the finite size of the written spot.We believe that here would a gain of simplicityoutweigh the loss of accuracy. The groove diffractionwould introduce an asymmetry that would greatlycomplicate the boundary conditions between the an-isotropic substrate and the MO grating. The finitespot size would require the use of integral transforma-tions between the field on the MO layer and theincident and reflected spectra of plane waves. Thiswould interfere with the simple Jones matrix descrip-tion. The analysis below is based on neglecting thediffraction.
Any plane wave that is reflected on the MO layerand that emerges from the disk substrate has s- and
p- polarization components that are mutually shifted.The phase shift between these polarization compo-nents depends on the angle of incidence. If we assumethat the substrate birefringence is vertical, the planeof incidence for any plane wave is always the principalplane of the anisotropic substrate so that the ordinaryand extraordinary plane waves are always in the stateof pure s and p polarizations, respectively. Conse-quently there is no cross talk of energy between theordinary and extraordinary plane waves that is due tobirefringence during reflection on the disk MO layer.Furthermore the angle of reflection equals the angleof incidence not only for the ordinary plane wave butalso for the extraordinary one. Under these circum-stances the phase shift between s- and p-polarizationcomponents of the plane wave emerging from the diskis twice the phase shift that is gained after onetraversal of the substrate. If the angle of incidence onthe disk substrate that is parallel to the plane z = 0 ise, the following equation which is derived in AppendixA, holds:
8(e) = (kh/n,)[n,(n 2 -sin2E)"2-n(n - sin2
E)"/2
] (1)
where k = 2r/ A is the wave number, h is thesubstrate thickness, nO is the index of refraction forany ordinary wave, and nz is the index of refractionfor the extraordinary wave that propagates parallel tothe disk. n, is one principal index of refraction and theother two are nx = n, = n0.
The disk Jones matrix 0'(e) for a plane wave that isreflected from the disk can be written as
exp[i8(e)]9'(E) = r(e) e)(E)exp[iT(E)]
O(e)exp[iT(E)]
exp[-i8(E)] I'(2)
where 8(E) is the phase shift that is defined by Eq. (1),O(E) is the Kerr rotation, T(e) is the Kerr ellipticity,and r(E) is the amplitude reflectance of the MO activelayer. The matrix 9'(E) was obtained from the rela-tion
(3)
where
( )(E)exp[iT(E)] TE)= r(E)t O)(E)expirE 1 (4
is the approximate expression of the Kerr-effect Jonesmatrix, and
I exp[ib(E)/2] 0
YE 0 exp[-ib(e)/211(5)
is the Jones matrix that introduces the birefringentretardation 8(e) for one pass through the substratelayer. Obviously if there is no disk birefringence then8(e) = 0, the matrix 2(e) is the unit matrix, and thedisk matrix 9'(e) = %'(e).
Not only the angle of incidence E but also anaximuthal angle + must be used to identify uniquelyeach plane wave from the spectrum of plane wavesilluminating the disk. The collimated beam illuminat-ing the objective lens is linearly polarized so that themagnitudes of the p- and s-polarization componentsof the convergent beam illuminating the disk aredependent on the azimuthal angle +. Generally thecomplex amplitude of the convergent beam can alsobe a function of both angles E and +. The linearpolarization of the collimated beam has its origin inthe laser diode, which is always used as a light sourceand usually is improved by refractions in a prismassembly.
Let the plane x-z of the fixed coordinate system x, y,and z be associated with the aximuthal angle + = 0and let the collimated beam illuminating the objectivelens be linearly polarized in the plane x-z, i.e., let theplane x-z be the plane of the laser diode junction.More precisely, we assume that the polarization ofeach plane wave from the spectrum of plane wavescharacterizing the collimated beam that illuminatesthe objective lens can be approximated by a nonzero xcomponent and a zero y component. Then the p- ands-polarization components of the plane wave illumi-nating the disk at the angle of incidence e and theazimuthal angle + can be found with the help of therotator matrix
fcos(4,) -sin(,)]6RO = sin(4, cos(4,. (6)
Analogously if the p- and s-polarization componentsof a plane wave reflected from the disk are known, thefixed x- and y-polarization components of this planewave can be obtained after passing through theobjective lens by using the transponse matrix T orthe inverse matrix R'' to the rotator matrix M. Itholds that R(-O) =R T(O) = '(
By incorporating the polarization asymmetry of thebeam that is illuminating the disk into our investiga-tion, the Jones matrix of the system, which consists ofthe disk and the objective lens, can be written:
.2(e, 4) = 3(4,)'(eXl T(4,) (7)
This gives
where a(E, +) is the complex plane-wave amplitude ofthe beam illuminating the disk.
Before the returning collimated beam reaches thepolarizing beam splitter (PBS), which splits the beaminto two differential data channels, it may acquire aphase shift af between its x and y components and theintensities of these components may be modified. Thex- and y-polarization components x(E, ) and y(E, ) ofthe plane waves just before the beam enters the PBScan be expressed by
[x(E, 4,), y(e, 4,)] = [a(e, ), 0(e, 4, (10)
where the Jones matrix,,ff is defined by
(11)
where I tXj and t,1 are the modules of the amplitudetransmission coefficients for the return path from thedisk to the PBS.
To find the signals in the two separated datachannels we first define the Jones matrices of theinputs to these two channels. These inputs are de-fined by the amplitude coefficients of the PBS for agiven angle of incidence, which is typically 45°. Theyalso depend on the angle Q between the beam polariza-tion plane x-z and the plane of incidence on the PBS.The nominal value of the angle is 45°. Let the beamthat is transmitted by the PBS enter the first datachannel and let the reflected beam enter the seconddata channel. Further let tp and t be the transmissionamplitude coefficients of the PBS for the p and spolarizations and for the given angle of incidence onthe PBS active surface. rp and r are the correspondingreflection amplitude coefficients. By introducing thecomplex parameters q = t /t and q = r /r8, theextinction ratios Q and Q of the PBS that areassociated with the first and second data channels areQ = tst3'*/(tptp*) = qlql* and Q2 = rprp*l(rsr *) = q,respectively. The symbol * denotes complex conjuga-tion. An ideal PBS would have Q1 = Q = r = t = 0and the modules I tp and rs = 1. The nonzerocoefficients t and rp, i.e., the nonzero ratios Q1 and Q2,
represent undesired amplitude leakage between thetwo data channels.
Figure 1, which shows a schematic of the differen-tial MO readout system, helps to identify better theassociation of the individual system components withthe basic parameters that are used in the derivation.
The Jones matrix of the input to the first channel
where I(E, 4) = a(E, 4)a*(e, @) is the intensity of theplane wave (, 4) illuminating the disk, R(E) =r(e)r*(E) is the disk reflectivity, T = tptp* is thetransmittance of the PBS for thep polarization, R. =r.rS* is the reflectance of the PBS for the s polariza-tion, T, = ttx* and Ty = tty* are the transmittancesfor the x- and y-polarization components along thepath from the disk to the PBS, and
If M(e, ) and I 2(e, @) are the intensities of the planewave (e, @) in the transmission and reflection datachannels, respectively, then the part of the electricalsignal of the differential MO readout that is induced
whereg, andg2 are the electrical gains (in mA/mW ormV/mW) of the transmission and reflection channels.By substituting Eqs. (22) and (23) into Eq. (24) we get
The value of the electrical signal of the differentialMO readout that is induced by the whole spectrum ofplane waves can be obtained by integrating Eq. (25)over a circular pupil of the objective lens. We selectsin(e) and + to be the pupil polar coordinates and thenwe can write
v = fefrdULPepiI v(e, 4,)sin(e)d[sin(e)]d,
= 1/2 f - f u(e, 4,)sin(2E)dEd4,, (29)
where ENA is the NA angle of the beam illuminatingthe disk, i.e., ENA = sin-' (NA). To obtain the differen-tial MO readout as a function of time V = V(t), onehas to express the Kerr parameters and T asfunctions of time O(E) = (t, E) and T(E) = T(t, E), andthen use the integral in Eq. (29) with the integrandfunction of Eq. (25) rewritten as follows:
V(t, e, 4,) = I(e, O,)R(E) [A(e, 4,) + B(t, e, 4,)A + C(t, , O,)r]. (30)
Although the above formulas can be used for anumerical evaluation of the differential MO readoutwithout additional modification their forms are tooinvolved and do not provide a clear picture of therelations between the readout and the individualinput parameters. A significant simplification can beachieved if the nonzero intensity I(E, 4)) of the planewaves from the spectrum of plane waves illuminatingthe disk is assumed to be constant and if the weakdependence of the Kerr parameters and T and thedisk reflectivity R on the angle of incidence aresuppressed. By using these simplifications (whichmay slightly exaggerate the effect of the disk verticalbirefringence), we can simplify Eq. (29) for the differ-ential MO readout to
illuminating the disk, a, , and F are given by Eqs.(26)-(28), and J 2 and J3 are defined by the integrals
J2 = 1(NA)2fNA sin2 [8(e)]sin(2e)de,
J3 = 1(NA) 2 f NA cos[8(e)]sin(2e)dE.
(32)
(33)
For the case of zero disk birefringence, J2 = 0 andJ = 1. With increasing birefringence the value of theintegral in Eq. (32) is increasing up to values 0.5,then it starts to oscillate about this value with a smalldecreasing amplitude until it stabilizes at 0.5. On theother hand the value of the integral in Eq. (33) isdecreasing with increasing birefringence up to 0,then it starts to oscillate about the zero value with asmall decreasing amplitude until it stabilizes at thezero value. For realistic values of the disk birefrin-gence, however, these integrals never reach the oscil-lation regions.
The derived formula of Eq. (31) is valid for thedifferential readout. For the single-channel readoutone can find a similar expression,
V5(t) = PR~ot_(1 - J12) + [E2(t) + J/2]
- rs(t)J 3 cos(T - o)), (34)
where the quantities a%, s, and r, for the first andsecond data channels are defined by Eqs. (26), (27),and (28), respectively, and where zero is substitutedfor the data channel gains g2 and gl. For an averagedsingle-channel signal Fs = F/2, and a and P3s are
ots = TgRlJsin 2 (f + Q cos2 (fl)]
+ g1T,[cos2 (f2) + Q, sin2 (fl)]}/2, (35)
Ps = Tg 2Rj[cos2(fl) + Q sin2(n)]
+ gTP[sin 2 (fl) + Q cos2(n)]1/2. (36)
In Eq. (31), the terms that are associated with aand 13 define the magnitude of an undesired dc offset,which carries a noise, while the desired differentialMO signal is given by the term that is associated withparameter F. Similarly, in Eq. (34), the terms that areassociated with as and fs define the mean level of thesingle-channel signal and the MO information isagain given by the term that is associated with F. Theoffset terms that are associated with parameters aand ax8 represent the energy that is carried by the xpolarization and the offset terms that are associatedwith parameters 13 and P3s carry the energy of the ypolarization. The r terms can be identified as interfer-ence terms. Assuming that (t) equals either or-0, depending on the magnetization direction of theilluminated disk domain, we can express amplitude a(mean-to-peak value, not peak-to-peak value) of thedifferential MO readout as
and for the undesired differential dc offset b we get
b = PR[(a + p02 ) + J2q( - a)/2]. (38)
The amplitude a, of the single-channel MO readout is
as = PR(r/2)E)J3cos(T - a)l, (39)
and the mean signal value bs in the single channel is
b = PR[(as + p5E)2) + J 2 (pS - as)12]. (40)
One can see that the disk birefringence has threeeffects on the differential MO readout. First theparameter J3 decreases the MO readout amplitude.Second the parameter J2 affects the signal-to-common-noise ratio by modifying the absolute value of thedifferential offset b. This is true, however, for anonideal differential system only. The ideal differen-tial system has a = a = 0 so that the offset b is zeroregardless of the birefringence.
Finally the parameter J2 increases the detectorshot noise in each channel by increasing the dc levelb,. The bias bs can never be zero and is typically atleast 1 order higher than the MO amplitude.
Now we look more closely at how birefringenceaffects amplitudes a and as and biases b and b.Equations (37) and (39) show that a and as arelinearly proportional to integral J3. Since we alreadyknow the behavior of J3 with increasing birefringencewe thus know the behavior of the amplitudes.
The effect of birefringence on a nonzero differentialoffset b is not so transparent. The absolute value ofoffset b, depending on the values of parameters a and13, can either decrease or increase with increasingbirefringence. If we bear in mind that integral J2 cannever be negative and if we neglect the small termwith )2 in Eq. (38), we find that the birefringencedecreases the nonzero absolute value of the offset b ifa > 0 and simultaneously (1 - 4/J2) a < 13 < a or ifa < 0 and simultaneously (1 - 4/J 2)x > 3 > at. In allother cases the absolute value of nonzero b is in-creased. From this we can make up the followingsimple and useful rule: assume only realistic magni-tudes of o and 13; then, whenever o = , the birefrin-gence reduces the absolute value of offset b. FromEqs. (26) and (27) it can be deduced that this canrepresent the case in which the nonzero differentialbias b is attributed to an angle fQ that does not equalexactly 45°. On the other hand, if bias b is created by anonzero extinction ratio Ql or Q2 rather than by amisaligned angle Q, birefringence increases the abso-lute value of offset b.
The single-channel parameters s and P3s are al-ways positive. This means that the bias bs increaseswith birefringence as long as P3s is greater than as. P3sis greater than as for any practical MO head design.The MO head always has the transmittance Ty, whichpredominantly determines Ps as - 1, and the transmit-tance Tx, which predominantly determines as as nomore than 0.33.
From a physical point of view, the birefringencecauses a transfer of radiant energy among the threeterms of Eqs. (31) and (34), i.e., among differentpolarization terms. A reduction of energy in theinterference term causes a decrease of the MO ampli-tude. Any transfer of energy between the x and ypolarizations results in a modification of the offsets band b,.
Until now we assumed that the beam spot on thedisk is infinitely small and we did not worry aboutdiffraction from the disk structure. In reality there isa finite beam spot size on the disk and beam diffrac-tion on the disk grooves and written domains. Theseeffects decrease the readout amplitude and the totalenergy of the return beam and may tend to homoge-nize the retardation across the beam. As a roughapproximation, one can simulate the loss of energyand the decrease of amplitude by using the followingsubstitutions:
R RLG, E > E5s, (41)
where parameter G ( < G < 1), represents thediffraction loss factor (G = 1 if there is no diskgroove and therefore no groove diffraction loss) andparameter p,, (0 < p, < 1), represents the signalloss that occurs because the beam spot size on thedisk is not infinitely small (s -- 1 for an infinitelysmall spot). The parameter ps is dependent on thespacing of the written data. Estimates of parameterswG and s can be obtained from a separate analysisthat solves the diffraction on the disk but does notnecessarily incorporate disk birefringence. The homog-enization of the retardation across the beam is linkedto the evaluation of parameters J2 and J3 . The changeof these parameters, however, might be too subtle tojustify a rough correction.
Signal-to-Noise Ratio
An ultimate performance of the MO readout systemcan be judged by the magnitude of the SNR. Thesignal is the differential MO signal and the noise7consists of laser diode noise, disk noise, shot noise,and electronic noise. The laser diode and disk noisesrepresent common noise because they have nearlyidentical values in both data channels.
Useful information is carried by the ac componentonly of the differential readout. The differential sig-nal power that is associated with the ac component isgiven by a2 /2 [see Eq. (37)].
Quantum effects and, particularly, optical feedbackin the MO readout system are responsible for laserdiode noise.8 9 It is customary to characterize thisnoise by the relative intensity noise (RIN), which isdefined as the detected noise power per hertz ofbandwidth to the detected average signal power.Detected power means electrical, not radiant, power.If the RIN of the laser diode for a given opticalfeedback is RINL then the detected laser noise powerNL in the differential MO readout can be expressed as
It is also convenient to use the RIN to characterizethe disk noise that results from a variation of disk andbeam parameters as the disk is rotated. If the RIN forthe disk is RIND the detected disk noise power ND inthe differential MO readout can be written as
ND = RIND(b2 + a2 /2)BW.
I(43) 1
.Note that for a single data channel, the laser and disknoise powers would again be given by Eqs. (42) and(43) only with a and b replaced by as and b8, respec-tively. Now since a 2as, bs > as, and, in the idealdifferential channel, b = 0, we easily find that thedetected laser and disk noise powers in the singlechannel are - 2(bs /a)2 times greater than in thedifferential readout. Thus the SNR for the differen-tial readout is - 8(bs /a)2 times greater than the SNRin the single channel, assuming the presence of laserand disk noises only. For a typical ratio value bs /a 20, the SNR in the differential channel is 35 dBgreater than in the single channel.
Shot noise arises from the probabilistic nature ofphotons that generate electrons inside the detector.The well-known formula for shot noise power Ns isNsH = 2qi,,BW, where q = 1.6 x 0-1' C is the chargeof the electron and idc is the dc through the detector.Shot noise power in the differential channel, which isthe combined noise power from detectors in both datachannels, is
NSH = 4q(bs2 + as212)"/2BW. (44)
Electronic noise is noise in the absence of radiationthat is incident on the detectors. It comprises detectorjunction noise and preamplifier noise. Electronic noisepower NE can be expressed with the help of noiseequivalent power (NEP) since NE = (RNEP)2BW,where R is the detector responsivity. NEP representsradiant power, not electronic power. Electronic noisepower NE in the differential readout will be
NE = 2(RNEP) 2 BW, (45)
if NEP is the NEP of each data channel.By using the above relations and assuming that
a = a/2, we derive the following expression for theSNR of the differential MO readout:
1/SNR = 4(RIN/4)[1 + 2(b/a)2]
+ (qla)[/2 + 4(b5 /a)2]1"2 + (RNEP/a)2}BW, (46)
where RIN = RIN + RIND and the birefringence-dependent quantities a, b, and b, are as defined above.The first, second, and third terms in Eq. (46) arerelated to the common, shot, and electronic noises,respectively.
Quantitative Analysis
To demonstrate the findings of our analysis better,we apply the expressions that were derived above to
bkrerknn (n. - n.)Fig. 2. Dependence of the differential MO signal normalizedamplitude on the disk vertical birefringence. The vertical birefrin-gence is defined by the difference n, - n0 of the principal indices ofrefraction. The variable parameter is the NA of the objective lens.
some specific cases of differential readouts. The re-sults of these numerical examples are summarized infive figures.
Figure 2 shows how increasing the disk birefrin-gence and the numerical aperture (NA) of the objec-tive lens decreases the normalized amplitude of thedifferential signal. The normalized amplitude is deter-mined by the integral J and the birefringence isdefined by the difference between the principal indi-ces of refraction n_ and n.
An increase in the shot noise in the differentialreadout with increasing disk birefringence is shownin Fig. 3. The shot noise increase is given essentiallyby the increase of the single-channel dc offset b.Since birefringence increases the integral value J2and, in real systems, s is always greater than as,birefringence also increases the bias b. In Fig. 3 theshot noise is normalized by the birefringence-indepen-dent electronic noise whose value has been selected asequal to the shot noise for the case of zero birefrin-
r 4.0
a03.0
2.0
I .
Vb'bboo ' o.0o2 0.0004 0.0000brefringe (n. - n.)
0.0010
Fig. 3. Dependence of the shot noise (normalized by the electronicnoise) on the disk vertical birefringence. The vertical birefringenceis defined by the difference n - n of the principal indices ofrefraction. The variable parameter is the transmittance T of the xpolarization (the polarization that is parallel to the laser diodejunction) between the disk and the PBS. The transmittance T isassumed to be 1.0. The NA of the objective lens is 0.55.
Fig. 4. Dependence of the differential MO signal (normalized bythe common noise) on the angular misalignment of the PBS. ThePBS angle is 450 for perfect alignment. The variable parameters arethe disk vertical birefringence n, -n,, and the transmittance T, ofthex polarization (the polarization that is parallel to the laser diodejunction) between the disk and the PBS. The transmittance T isassumed to be 1.0. The NA of the objective lens is 0.55.
gence. Two values of the parameter T, (the transmit-tance for the polarization that is parallel to the laserdiode junction) were selected for Fig. 3. The otherparameters that were used to determine the curves ofFig. 3 were set as follows: T = 1.0, R = T = 1.0(i.e., Q, = Q = 0), fl = 45°, g, = g2, and NA = 0.55.We assume that transmittance Ty is constant so theeffect of the birefringence is greater for smaller valuesof Tx.
It is obvious from Eq. (46) that the disk birefrin-gence can change the common-noise-to-differential-MO-signal ratio only if the differential offset b isnonzero. It follows from Eq. (38) that this can happenonly if at least one of the two parameters a and isnonzero. This means that the disk birefringence canaffect the ratio of the signal to the common noise onlywhen the data channels are not perfectly balanced.
0SE
1.00PBS trnstnce for p-polarIzatkon
Fig.5. Dependence of the differential MO signal (which is normal-ized by the common noise) on the PBS transmittance T, for the ppolarization. The PBS reflectance R, for the s polarization isassumed to be 1.0. The variable parameters are the disk verticalbirefringence n, - no and the transmittance T, of the x polarization(the polarization that is parallel to the laser diode junction)between the disk and the PBS. The transmittance T, is assumed tobe 1.0. The NA of the objective lens is 0.55.
10brefrnwc (n. - n.)
Fig. 6. The differential MO signal-to-total-noise ratio as a func-tion of the disk birefringence n - n for a typical MO readoutsystem with a NA of 0.55. The variable parameters are the angularpositioning fl of the PBS and the PBS transmittance T, for the ppolarization. The PBS reflectance R, is 1.0 and the transmittancesT, and T between the disk and the PBS are 0.1 and 1.0,respectively.
The dependence of the differential-MO-signal-to-common-noise ratio on the PBS angle fl, i.e., on theangular misalignment of the PBS, is shown in Fig. 4.Its dependence on the PBS transmittance Tp whileR = 1.0, i.e., its dependence on the extinction ratioQ2, is plotted in Fig. 5. Both Figs. 4 and 5 show thisratio for two values of birefringence (n_ - n,, = 0.0003and n - no = 0.0) and two values of the transmit-tance T, (T, = 0.1 and T, = 0.3). An objective lensNA of 0.55 is used. The other parameters that areused for the evaluation of the results plotted in Fig. 4are T= 1.0, R = T = 1.0 (i.e., Q = = ),g 1 = 2,RINL = -115 dB, RIND = -110 dB, and BW = 30.kHz. The other parameters that are used to obtainthe results of Fig. 5 are T = 1.0, R = 1.0 (i.e.,Q = 0), fl = 45°, g, = g2 , RINL = -115 dB, RIND =- 110 dB, and BW = 30 kHz. From Fig. 5 one can seethat the birefringence reduces the signal-to-common-noise ratio if the imbalance between the data chan-nels is caused by a nonzero extinction ratio. HoweverFig. 4 shows that birefringence improves this ratio ifthe imbalance is due to the PBS angular misalign-ment.
Finally Fig. 6 shows how the SNR, when the noiseis the total noise, may depend on disk birefringence intypical MO systems. The results in Fig. 6 are obtainedfrom the following parameter settings: NA = 0.55,read pupil power P = 1.5 mW, disk reflectivity R =0.2, T, = 0.1, T, = 1.0, R, = 1.0, detector responsivityR = 0.5 mA/mW, NEP = 3.0 pW/(Hz)112 , g = 2 = 1mA/mW, RIN = -115 dB, RIND = -110 dB, andBW = 30 kHz. fl and Tp were varied as shown. We cansee not only a reduction in the SNR with increasingbirefringence but also how system imperfections, inour case the PBS rotation and extinction ratio, maypartially compensate for each other's unwanted ef-fects and thus alleviate the reduction of the SNR.
ConclusionPreviously published studies that dealt with diskbirefringence in the MO readout system either pre-
sented complicated open-form expressions or viewedthe beam that interacts with the disk as a collimatedbeam. Our analysis offers simple, close-form formulasfor a realistic model that assumes the interaction of afocused beam with a disk with vertical birefringence.We demonstrate that although the disk vertical bire-fringence always reduces MO signal amplitude, itdoes not always increase the noise level. Sometimes,depending on other system parameters, noise may bedecreased. Our numerical analysis indicates that ifthe vertical birefringence n - n is kept below
0.0003 it will not affect the SNR as much as othersystem parameters that typically are not so tightlycontrolled.
Appendix A
A plane wave passing through a layer of thickness hand an index of refraction n,, under an angle ofrefraction E. acquires the phase shift
6 = knoh cos(e,), (Al)
where k is the wave number. Similarly, the phaseshift of a plane wave passing through a layer ofthickness h and an index of refraction n under theangle of refraction is
ti = kneh COS(ee).
refraction n = ne for the extraordinary wave. It isevident that the latter index is the function of thewave normal of the extraordinary wave. To define thedirection of the normal of the extraordinary wave byits angle of refraction e we write
s.2 + s = sin2 ee = 1-cos 2 ee, s, = -sin E = cos2 E. (A7)
By substituting Eq. (A7) into Eq. (A6) we get
sin2E, = (nz2/ne2 )(ne2 - n0
2)(n 2 _ n2),
cos E, = (n 2I /n 2 )(n 2 - n0
2)I(n2 -n 2).
(A8)
(A9)
Now we assume that the input medium is air. Bycombining Eq. (A8) with the law of refraction for theextraordinary wave we get
sin(e) = n. sin(e.), (A10)
and then we obtain
sin2 e = n,2 (n02 - n, 2)/(n2 -nO2); (All)
we combine Eq. (All) with Eq. (A9) to get
ne cos(e,) = (n. /n,)(n.2 - sin2 ) 2. (A12)(A2)
The phase shift difference between these two wavesafter one pass is obviously
Snell's law for the ordinary wave,
sin(e) = n. sin(e), (A13)
8 = kh[nO cos(E80 ) - n, coS(e,)].
In an anisotropic layer of thickness h, n and n canrepresent the indices of refraction for ordinary andextraordinary waves propagating in the layer underangles of refraction E and e0, respectively. We areinterested in expressing n cos(E.) and n cos(ee) asfunctions of the anisotropy of the layer and the angleof incidence E on the layer.
We start by writing the Fresnel equation of wavenormals for waves in an anisotropic medium:
S02
(1/n2
_ /n 82 )(11n2 - 1/n 2
)
+ S 52(1/n
2 1 -/n 1 2)(1/n2 - 1/n 2 )
+ s 2(1/n2 - 1/n02)(1/n2
- 1/n52 ) = 0, (A4)
where s, s, and s are the direction cosines of thewave normal. If we assume that the birefringence isvertical and the layer is perpendicular to the z axis wecan put n = n = n and Eq. (A4) breaks into twoconditions:
2 2n = n, (A5)
s02(1n 2 - 1/n2) + s 2(1/n2 - l/n 2)
+ s 02(11n2 - 2/n)) = 0. (A6)
Equation (A5) defines the index of refraction for theordinary wave and Eq. (A6) defines the index of
nO cos(E8) = (n02
- sin2 ) 2. (A14)
Finally by substituting Eqs. (A12) and (A14) into Eq.(A3) we obtain
8(E) = (kh/n,)[n,(n 2_ sin2
) 2- n,(n 2
- sin2)/
2] (A15)
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