Simulating the scratch standards for optical surfaces: theory Eric G. Johnson, Jr. I show how to simulate the scattering generated by a scratch on the surface of high-quality optics and their elements. This is accomplished by first describing how the present cosmetic scratch standards tend to be used in the optics industry. Second, I derive from first principles, using the scalar model for electromagnetic radiation, the first-order scattering coefficients for the far-field radiation due to a particular scratch pattern. There are approximations made to get these coefficients. The results allow construction of a set of secon- dary scratch standards. These are a pattern of rectangular grooves that can be made precisely reproducible during the manufacturing phase. Appropriate selection from this set can provide the same range of scatter- ing power and character as is present in the current scratch standards, which are not easily reproducible. Because the method for construction of these new secondary standards is nonrandom, to guarantee the re- producible construction between these standards it is necessary to restrict the observation range 5-10° from the direct beam. 1. Introduction The scratch standard is the most common descriptor of optical surface quality. It is used by making a visual comparison between a secondary standard and the op- tics under evaluation. This activity is done where the optics are being manufactured as well as at other sites; therefore there is a need for a large number of secondary standards. The current method of manufacture of secondary standards has significant problems of re- producibility. This theoretical paper shows that an etched pattern of rectangular grooves can be used to simulate the far-field radiation intensity of an arbitrary scratch. The primary advantage of such grooves is that it is then possible to generate a scattering pattern that will be exactly the same for all the secondary standards. The primary standards can then be used to define the scratch number that is assigned to each secondary standard. The etched pattern suggested here is a series of rectangular grooves with a width -1.0 ,um and a depth of either 0.55 Aum or <0.1 gzm. The form and in- tensity of the radiation pattern are controlled by the number and depth of these grooves and by their relative spacing. This paper goes from first principles to derive the required formulas necessary to simulate the scat- tering pattern. This paper has three additional sections: (Sec. II) the definition of the problem, (Sec. III) the summary derivation of a simplified equation showing the key factors creating the conditions for radiation scattered The author is with U.S. National Bureau of Standards, Electro- magnetic Technology Division, Boulder, Colorado 80303. Received 7 March 1983. or diffracted from a scratch, and (Sec. IV) a discussion of results of the scattering model using plots. In Sec. II, I list the conditions necessary for making scratches that are reproducible. 1-3 The purpose of this analysis is to simulate the intensity of the radiation pattern produced by existing secondary standards over a selected range of angles and to use that knowledge to construct new standards which can be used to evaluate the quality of optical surfaces. Since I wish only to demonstrate ideas, the mathe- matical derivations in Sec. III are made brief; they are summarized by key formulas so that an interested party, who has worked with integral equations, Hilbert space, and mode theory, can derive the formulas.4- 6 For the reader interested only in a general overview, I include a brief explanation of the physics of each step. To be sure we are all thinking about the same thing, I first give a brief overview of scattering by a defect. My prime purpose is to design a series of artifacts that will cause diffraction of light so that there is a display of the necessary range of scattering power in the far field that would be present in the current artifacts. Because we are observing only the far-field radiation under a se- lected range of angles, it is possible to use well-defined structures to simulate a scratch; microscopically, these structures do not have to look like scratches seen during the manufacture of high-quality optics. This means that the dimensions of the grooves used for the new artifacts have no necessary correspondence to the di- mensions of actual scratches. The key point of interest is to design a technique which allows a range of radiation intensity, which will be reproducible in the manufacture of the simulated scratches, and which will require a minimum of fabrication procedures. 4056 APPLIED OPTICS / Vol. 22, No. 24 / 15 December 1983
Transcript
Simulating the scratch standards for optical surfaces:theory
Eric G. Johnson, Jr.
I show how to simulate the scattering generated by a scratch on the surface of high-quality optics and theirelements. This is accomplished by first describing how the present cosmetic scratch standards tend to beused in the optics industry. Second, I derive from first principles, using the scalar model for electromagneticradiation, the first-order scattering coefficients for the far-field radiation due to a particular scratch pattern.There are approximations made to get these coefficients. The results allow construction of a set of secon-dary scratch standards. These are a pattern of rectangular grooves that can be made precisely reproducibleduring the manufacturing phase. Appropriate selection from this set can provide the same range of scatter-ing power and character as is present in the current scratch standards, which are not easily reproducible.Because the method for construction of these new secondary standards is nonrandom, to guarantee the re-producible construction between these standards it is necessary to restrict the observation range 5-10° fromthe direct beam.
1. IntroductionThe scratch standard is the most common descriptor
of optical surface quality. It is used by making a visualcomparison between a secondary standard and the op-tics under evaluation. This activity is done where theoptics are being manufactured as well as at other sites;therefore there is a need for a large number of secondarystandards. The current method of manufacture ofsecondary standards has significant problems of re-producibility. This theoretical paper shows that anetched pattern of rectangular grooves can be used tosimulate the far-field radiation intensity of an arbitraryscratch. The primary advantage of such grooves is thatit is then possible to generate a scattering pattern thatwill be exactly the same for all the secondary standards.The primary standards can then be used to define thescratch number that is assigned to each secondarystandard. The etched pattern suggested here is a seriesof rectangular grooves with a width -1.0 ,um and adepth of either 0.55 Aum or <0.1 gzm. The form and in-tensity of the radiation pattern are controlled by thenumber and depth of these grooves and by their relativespacing. This paper goes from first principles to derivethe required formulas necessary to simulate the scat-tering pattern.
This paper has three additional sections: (Sec. II)the definition of the problem, (Sec. III) the summaryderivation of a simplified equation showing the keyfactors creating the conditions for radiation scattered
The author is with U.S. National Bureau of Standards, Electro-magnetic Technology Division, Boulder, Colorado 80303.
Received 7 March 1983.
or diffracted from a scratch, and (Sec. IV) a discussionof results of the scattering model using plots.
In Sec. II, I list the conditions necessary for makingscratches that are reproducible. 1-3 The purpose of thisanalysis is to simulate the intensity of the radiationpattern produced by existing secondary standards overa selected range of angles and to use that knowledge toconstruct new standards which can be used to evaluatethe quality of optical surfaces.
Since I wish only to demonstrate ideas, the mathe-matical derivations in Sec. III are made brief; they aresummarized by key formulas so that an interested party,who has worked with integral equations, Hilbert space,and mode theory, can derive the formulas.4-6 For thereader interested only in a general overview, I includea brief explanation of the physics of each step.
To be sure we are all thinking about the same thing,I first give a brief overview of scattering by a defect. Myprime purpose is to design a series of artifacts that willcause diffraction of light so that there is a display of thenecessary range of scattering power in the far field thatwould be present in the current artifacts. Because weare observing only the far-field radiation under a se-lected range of angles, it is possible to use well-definedstructures to simulate a scratch; microscopically, thesestructures do not have to look like scratches seen duringthe manufacture of high-quality optics. This meansthat the dimensions of the grooves used for the newartifacts have no necessary correspondence to the di-mensions of actual scratches. The key point of interestis to design a technique which allows a range of radiationintensity, which will be reproducible in the manufactureof the simulated scratches, and which will require aminimum of fabrication procedures.
Actual scratches are complex single dip structureswith rough edges and bottom. The radiation intensityis determined approximately by the depth and widthof the scratch. The roughness and roundness of thescratch bottom act to fill in interference maxima in thefar-field pattern. The actual pattern from suchscratches will be slightly different for each scratch. Ina similar manner the artifacts would also be variable.We need them to be less variable than actual scratchesand to have variability that is insignificant between eachartifact. We eliminate variability in the manufactureof the artifacts by using rectangular grooves which aresmall enough to make their structural details not criticalto the intensity of the scattering pattern. We then geta range of intensity by using a selected number of thesegrooves placed at specified locations so that there areno interference dips seen in the far field over the ob-servation range. There are two distinct choices for thedepth of the grooves, namely, about the wavelength ofthe visible light or substantially less than a wavelength.A discussion of the details now follows.
II. Definition of the ProblemFigure 1 shows the nominal observation and mea-
surement conditions along the x and z coordinates. Inpractice, the following are true during the visual ob-servation of scattered radiation from a scratch:
(1) The illumination source is a lamp with wave-lengths ranging from the infrared to the ultraviolet.
(2) The observation angle, , for observing the scat-tered radiation is presumed to be confined between 50and 100 of the direct beam.
(3) The beam divergence of the incoherent illumi-nation is 2°.
(4) The polarization of the beam is unspecified andis usually random.
(5) The method of observation is to compare therelative intensity of the scattered light from eachstandard scratch with that from the scratch to be clas-sified.
(6) The transverse dimension of a scratch of interestless than the resolution limit for that observer (100 umor so at a viewing distance of 25 cm). This fact meanswe can construct a series of grooves as long as the overallwidth is less than this 100-,4m resolution dimension.
(7) The secondary standard has a scratch of 25-mmlength or more. This fact allows us to treat the radia-tion pattern as 1-D in the x coordinate and independentof the y coordinate.
(8) The index of refraction is near 1.5 since the arti-facts are manufactured on optical glass.
(9) The illumination of the scratch or artifact isusually from the opposite side to the scratch. The ob-servation is on the side closer to the scratch.
(10) The direction of illumination is nearly normalto the glass surface in the neighborhood of the defect.
(11) There is no color observed from the scratch.Please note that this list differs slightly from the
current MIL specification. Modifications in themethod of observation will be necessary to allow con-struction of reproducible secondary standards.
Glass
EyeDirect /Beam 9 Observed
I Scatteredv Radiation
Scratch -
x
hZ hi
/ White-- I / Light
Beam/ SourceLimitingAperture
Fig. 1. Nominal observation conditions for scattered radiation froma scratch.
Ill. Deriving the Model Equation for the ScatteringProcess
I start from the scalar theory to represent the elec-tromagnetic field. The vector theory is unnecessarybecause the scattering is at small angles to the normalto the glass surface. In addition, the accuracy of thismodel is relative rather than absolute. We are tryingto get a well-controlled means for manufacturing benchmarks of scattering intensity, we are not trying to du-plicate the actual scratches. This fact allows us to ig-nore details that could affect the absolute scatteringstrength. Our effort is to find a way to get reproducibleresults. The polarization affects only the absolutescattering pattern. I further assume there is no y de-pendence in the scattering process; hence I drop all ydependence in the equations. As seen in Table I, thereare three regions denoted by the subscript a = 1, 2, or3. The first surface of the glass contains the scratch andis represented by z = h (x); the second surface, which ispresumed flat here, is represented by z = z 1.
Table 1. Region Specification
z-CoordinateRegion range
a = 1 (air) z < h(x)a = 2 (glass) h(x) < z < z1 (1)a = 3 (air) z1 <.z
I assume that h(x) is a single-valued function of xwhich is nonzero over an 50-Mm range. The scalarwave equation is written as
[a 1 + aZ + k2 ]E(x,z) = 0 (2a)
in the air, and
[a2 + 92 + h2n
2]E(x,z) = 0 (2b)
in the glass, where E is the electric field and k is thewave number 27r/X. In air, k = k3 = (k 2
- s 2) 1/2 , andin glass k2 = (k 2n 2
- 2 )1 /2.Here s is an index for each plane wave and corre-
sponds to k sin(0) when the angle 0 is real. (There areother terms where this angle needs to be purely imagi-nary since s can go to ±c.) Equation (3) shows theform. n is the index of refraction of the glass and X isthe wavelength of the coherent component of the illu-mination source.
The complete solution using the plane wave for eachregion is
Ea(Xz) = ' ds exp(isx)[CaU(a,z) + DaV(a,z)], (3)
where Ca (s) and Da (s) are the amplitudes of each planewave,
U(a,z) - exp(ikaz)
are the right traveling waves,
V(a,z) - exp(-ikaz)
and are the left traveling waves. We fix the amplitudesby requiring continuity of the electric field and itsnormal derivative at each glass-air interface. Thenormal derivative at the defect is the more complexexpression. Equations (4a)-(4d) show the four rela-tionships:
(a) The electric field continuity condition at the firstsurface implies
[E1 -E21 lz=h(x) = 0. (4a)
(b) The normal derivative of the electric field con-tinuity condition at the first surface implies
% -d 8[h(x)]8x9[El-E2]1z=h(x) = 0. (4b)
(c) The electric field continuity condition at thesecond surface implies
[E2 -E3] Iz=zE = 0. (4c)
(d) Finally, the normal derivative of the electric fieldcontinuity condition at the second surface implies
az [E2 - Ed 1I=z1 = 0. (4d)
These algebraic and differential equations are con-verted by an inverse Fourier transform into algebraicand integral equations in the Fourier space as desig-nated by s. The results are listed in Eqs. (5a)-(5d).The necessary definition follows in Eq. (6). Equations(5a) and (5b) are the constraints on C2 and D2 due to thecontinuity conditions at the second surface. Equations(5c) and (5d) are the constraints on C2 and D2 due to thefirst surface of the glass. Both constraints must besatisfied simultaneously.
the defect at the first surface. Diffraction or scatteringcorresponds to a change in the direction of propagationof the light from the plane wave labeled s to the planewave labeled s1. The Mr in Eqs. (5c) and (5d) gives thestrength of this scattering process [see definition (4)below].
(2)gl = (1 + kl/k 2 )/2, g2 = (1 - kl/k2)/2,
G _ (k2 - sis)/[kl(s)k], G2 (k2n2-S S ,
Ul a U(2,zl), U2- U(3,zl),
U3 V(2,zl), and U4 V(3,zl).
(6)
(3) When there is no defect on the surface, the re-flected and transmitted beams can be derived from Eqs.(5a)-(5d) by noting that the scattering process due tothe defect is described on the right-hand side of Eqs.(5c) and (5d). If these terms are set to zero, we canderive the usual Fresnel equations. In the notation ofthis paper the Fresnel equations are given below.
The reflected electric field is
(7a)R - DU4 gig2 (U3 - Ul)/[U2 (g U -
and the transmitted electric field is
T - DUlU3 U4(g2 - gl)/(g2U - g2U3).
We have set D3 D which is the amplitude of the inci-dent electric field from the right-hand side of the glassas shown in Fig. 1, and C1 =_ 0 since there is no source onthe left-hand side of the glass.
(4) The Mr terms define the scattering due to thedefect. There are three such terms; r is the index forthese terms:
Mr(Sis) - 3' 2 exp[i(s - sl)x][Fr - 1],
where
F1 - V[1,h(x)],
F2 - U[2,h(x)], and F3 - V[2,h(x)].
Briefly, we have scattering due to a mismatch of theplane wave from a flat surface to that due to the de-fect.
The definitions used in the above equations are:(1) k 1(s 1) = ki(s), where the plane wave index s is
a new label for the plane wave index label in kI(s). Inlike manner, we have a new label for the s, in the k2, D1,C2, and b2. The caret indicates the new label sl. Thiss 1 label is needed because there is mixing between twodifferent pairs of plane waves in the two integrals on theright-hand side of Eqs. (5c) and (5d). This mixingcorresponds to the diffraction (scattering) induced by
To get the equation which shows the key features ofthe scattering process without unnecessary complica-tions, we make a series of approximations which elimi-nates details that affect the final results by errors of theorder of 10%. We are interested only in the single-orderscattering within a small angular region at almost nor-mal incidence with a reasonably collimated incoherentbeam.
Approximation 1Because the scattering is weak and because the ob-
servation conditions are restricted, the ranges of s ands1 which have any important contribution are <10% ofk; therefore we need not consider the s or s1 dependencein many of the terms used in Eqs. (5a)-(5d). This isespecially true when we have expressions that containa ratio of s or s to . Those terms can be neglectedcompared with the terms that are constant. Thismeans the preceding expressions reduce to
kl=/ 1l=k, k2 = 2 = kn, g= (1 + n)/2n, (8a)
92 = (n - 1)/2n, G = 1, G2 = n, F = exp[-ikh(x)], (8b)
Approximation 2We separate the amplitude of the traveling waves Ca
and Da into a part which is unmodified to first order bythe defect and a part that is the scattering due to thedefect. Because we are interested only in the first-orderprocess, which is weak and nonresonant, we can assumethat the scattered part can be neglected inside the in-tegrals in Eqs. (5c) and (5d). We define, for conve-nience in writing these equations,
a r' ds[M2 2 + M1 1 - MI] T (9a)
22 a J ds[G2(g2M2 - g1M3) + G1M1]T, (9b)
where the unscattered part of C2D2 is given as C2 = g2 Tand D2 = gT here. We further define
Y' = (g + 01g2) a2, (l0a)
2 = ( 2C71 - 0 1 )k2a2 /k. (lOb)
Remember that the caret means that the index labelissi for the plane waves. The two integral Eqs. (5c) and(5d) become with these notational steps
-1 = 2 1, (lla)
-41 S3/k + 293 = 22, (llb)
where 91 is the scattered part in the air on the left-handside of the glass and 93 is the scattered part in the air onthe right-hand side of the glass. We are ultimately in-terested only in the S1 expression due to the observationconditions in Fig. 1, so we eliminate 3 from Eqs. (la)and (lIb) to get
[ Y2 - ,/k = 221 + 122- (12)
We now insert the simplifications defined in Eq. (8)to get
Here the s is a third label index for the plane waves.Its purpose is to allow us to relate the coherent waveformulation we have used so far to the incoherent formthat will finally be used in the plots. B (so) is a statis-tical amplitude which is defined below in Eqs. (17a)-(17c). Its purpose is to allow incoherence. TheGaussian function in Eq. (14) represents the finite widthfor a coherent beam. The term p gives a convenientestimate of that beamwidth in the Fourier transform orwave number space. There is a second width that willcome from the incoherent part, which is defined in Eq.(17c). These widths determine the critical form of theresulting intensity pattern for the far field.Approximation 3
The first term in Eq. (13) containing g1 comes fromthe double reflection process. It is 4% of the secondterm; therefore we drop it as unimportant. In Eq. (14)we have assumed that the source term can be writtenas
D = Do J dso exp[-(s - so)2/p2]B(so), (16)
where Do scales the net amplitude of all equivalentsources. Here we assume that each radiating atom actsas a coherent source which exhibits a Gaussian radiationprofile of width p for its plane wave radiation pattern.This is not strictly true for the atoms but is adequate tocharacterize the features of interest for white-light ra-diation processes. The coherent beamwidth p = 10-4ko which corresponds to a far-field radiation conditionof each atom located 30 cm from the glass defect andwith an effective beam size of 3 mm at the glass. Weassume that the incoherent radiation will have abeamwidth with pf = 0.05 ko which corresponds to abeam divergence of 2, ko = 27r/Xo, where Xo = 0.55 m.We now define the statistical amplitude B (so) with thefollowing statistics:
(a) The average is
(B(so)) = 0, (17a)
which implies total incoherence of the source.(b) The variance has uncorrelated behavior between
each plane wave of the source, thus
(B(so)B*(so)) = 3(so - o)f (so).
(c) The power spectrum is
f (so) = exp[-s/pf'j,
(17b)
(17c)
in which the pf gives the width of the incoherent beam.Here the width corresponds to the allowed beam di-vergence for the incoherent part of 2°. We have as-sumed that the source is completely incoherent and thatit is centered about the plane wave direction so = 0,which is normal to the surface of the glass.
The critical approximations are now done. We canproceed to do the indicated integrations and reduce theformulas in view of these approximations.
Given the small p, we can integrate directly the svariable in Eq. (13). We define the first-order scat-tering kernel in that equation as
and the intensity of the directly transmitted beam isgiven as
T(O) = To f dsoB(so) exp[-s /p2] (19b)
These expressions contain all the relevant details forunderstanding first-order scattering by incoherentsources.
We now calculate the intensities of the electric fieldat the angle defined by s1 and at the direct beam centerwhere s, = 0. In addition, we perform the statisticalaveraging indicated in Eqs. (17a)-(17c). The scatteredintensity is
F(sj) = X dsoWW*TOTO exp[-s /p ]
The directly transmitted intensity is
G(O) = ToTowx/7r,
(20)
X coordinate XbI
Width, C=1.065 Lm
cl-
=Jfl~flJ7Jthth. Z 2 = 0.55 rm
X X < or 0.1 Lmb b+1
Fig. 2. Example of groove pattern for inducing scattering in anartifact.
-1/2, 1/2, etc. when N is half-integer such as 5/2, where-N • b • N and where o and :3 are unspecified at thistime. The number N1 is equal to (N - 1)/2 where N isthe number of grooves. This number can be varied torealize a different scattering intensity for each desiredscratch standard.
The W term in Eq. (18) becomes
W = WWbWcs (23a)
where
(21a) WQ 2 r 2 g [exp(-k0Z2n) - exp(-ikoz 2 )1, (23b)
where the convolved width, w, of the beam is givenexplicitly by
1/w2 = 1/pp + 2/p2. (21b)
We can define the relative intensity between the directbeam and the scattered beam as
R(sj) - F(sO)/G(o) = f dso exp[ s/p ]WW*(wxv/;) (21c)
There is dependence on both the coherence width, p,and on the incoherence width, pf, of the source. Thisis one of the reasons for the need to use a comparisonmethod for evaluation of a scratch strength. Absoluteevaluations require strong control of the source so thatthe scattering pattern is kept the same for the secondarystandard and for the scratch of interest. The controlis unnecessary in relative comparisons when the changebetween neighboring secondary standards is at least afactor of 3.
To progress further we need to define the structureof the artifact. Figure 2 shows the suggested patternfor these secondary standards. Here we assume thatthere are small rectangular grooves with dimensionssuch that the far-field features are insensitive to thosedimensions within the observation range of 5-10°. Wechose the width in this discussion to be c = 1.065 um.(The actual fabrication dimension need only be in theneighborhood of this value.) This is a dimension thatcan be realized reproducibly by etching glass and theinterference maximum is well outside the observationrange. We chose the depth of this structure to be z 2 =0.55 Am or <0.1 ,m. The former choice will cause amaximum intensity in scattering with minimum de-pendence on the wavelength over the visible light range.The latter causes the intensity in scattering to be about13 times less than the maximum intensity case. Wefurther chose the location of the rectangular grooves tobe Xb. Thus,
Xb = ob(a + fb), (22)
where b = -1, 0, +1, etc. when N is an integer, and b =
(23c)Wb = F exp[i(so - l)xbl,b
and finally
W, = {sin[(so- s)c/2 . (23d)
Equation (23d) shows the interference maximum whenWc = 0. This is implied by the width of a single groovewhen we let so = 0 and select the direction s1 = kosin(31°) to get Wc = 0 with c = 1.065 which is equal toXo/sin(31'). The wavelength X has been chosen to be0.55 ,um. We use the symbol X = Xo to remind us of thatassumption in the remaining discussion. This as-sumption is useful because there is no significantwavelength dependence over the observation range.
We can now complete the integration and set the ratioin Eq. (21c). We neglect the s0 in Eq. (23d) because theconvolution effect on it is small since the incoherentbeamwidth pf is small relative to the interference widthdefined by the width c of the groove.
First, we define a normalized multiple groove strengthfunction as
SN(Sl) = * E CoS[SI(Xb - Xb 1)]Y b,bi
X expi(xb - Xbl)pf/2]2}. (24a)
Here the Y is the normalization factor necessary tomake SN(sl) = 1.0 at some angle between 5° and 100.
Next, we define a scale factor which gives the inten-sity of the scattered radiation relative to the directbeam. This is a dimensionless number,
Sf a W w Y(Pf/wx) (24b)
In this notation, the relative intensity defined in Eq.(21c) is given as
R(si) = Sf * SN(S1). (24c)
We are now ready to discuss the values ae and 3. Theproper values for a and / will depend on the ability of
the manufacturer to make reproducible lines and on thedesired far-field pattern. The value a = 7.5 or = 0.75is suggested as one where it is possible to get a separa-tion between lines and yet get etched lines that are re-producible. The purpose of the a term is to minimizethe interference effects as the number of grooves is in-creased. (This discussion assumes the etched lines arerectangular. We can expect the actual lines to have anapproximately trapezoidal cross section.) This dif-ference is of no importance as long as the pattern is thesame in the main details from artifact to artifact.
We have completed the derivation of the mathe-matical model for representing the scattering from adefect. The next section shows plots of SN(s1) over therange of interest, namely, 5-10° in 0.10 increments.Several values of a and are shown as well as otherparameter changes.
IV. Discussion of the GraphsHere we look at Figs. 3-9, each of which contains four
plots of the function shown in Eq. (24a). These plotsshow how flat is SN(S1), and hence is R(sD), for the an-gular range of 5-10°. The data beside each plot give thevalue of the parameters used in that plot. Not all pa-rameters are indicated here. Only those that may bechanged are indicated. SCALE is the value of thefunction S/ in Eq. (24b). ALPHA is a and BETA is A.INCOHPF is thepf defined in Eq. (17c). It has unitsof inverse micrometers. The GROOVE COUNT is N.Z2 is the depth, Z2, of the groove in micrometers. C isthe width, c, of the groove in micrometers. INDEX isthe index of refraction, n, of the glass. Finally,WAVELENGTH is X in micrometers.
Figure 3 shows how much the pattern changes whenthe wavelength is changed from 0.5 to 0.6 ,um. Thereare few important changes expected for the parameterschosen here. The lower right plot shows what happensif the incoherence angle is reduced by a factor of 4. Thedip shown in the top right plot becomes much morepronounced. This implies it is necessary to have somelevel of incoherence in the illumination beam so thatsuch interference effects are unimportant.
Figure 4 shows four plots. The top pair shows whathappens when the groove width is doubled. The bot-tom pair shows what happens when the alpha, beta pairare changed. In both sets there is a shift of the peak anda corresponding change in the range of the strengthwithin the 5-10° region.
Figure 5 shows for N = 1, 2, 3, and 4 grooves that it ispossible to have a scatter pattern that is flat within afactor of 2. This figure set could be used to scale the lowrange of scattering strength from 4.6 X 10-6 to 2.3 X10-5.
Figure 6 shows for N = 5, 7, 10, and 20 grooves thatit is possible to have a scatter pattern that is flat withina factor of 2. This figure set could be used to scale themidrange of scattering strength. The choice of groovenumber would depend on how this midrange would bescaled. This scattering strength ranges from 2.4 X 10-5to 9.8 X 10-5. The approximate width of the 20-groovepattern is 154 im.
Figure 7 shows what happens when the depth ischanged to 0.55 from 0.1 gim. The patterns are thesame as those in Figs. 5 and 6. The strength haschanged by a factor of 12.6. This means the scatteringstrength can range from 5.8 X 10-5 to 1.2 X 10-3.
Figure 8 shows what happens when the spacing im-plied by alpha and beta is reduced by a factor of 3.75.The range of strength goes from 4.6 X 10-6 to 2.5 X 10-4.The width of the groove pattern is -43 gim. It is pos-sible to increase the strength by a factor of 12.6 by usingthe depth of 0.55 Aim. These patterns have a verysmooth scattering feature. The choice between thoseshown in Fig. 8 and those in Figs. 5-7 depends on howdesirable it is to have the flat pattern within the ob-servation range. I suggest that the flat pattern is morelikely to have agreement between the various observers.The pattern of Fig. 8 may be closer to the usualscratches on high-quality glass, but it is more likely tohave variability in classification of a scatter number.
Figure 9 shows two cases where the incoherence fac-tor, pf, is reduced by a factor of 10 from the previousvalues of 20. The top pair shows a case where beta isthe value used in a test sample we have on hand. Thebottom pair shows the case which is suggested as thebest choice for alpha and beta. There is substantialinterference effect and corresponding modification inthe pattern. This means that the checking of thescattering pattern with a laser will require careful sig-nature analysis. For the intended use of these scratchstandards with incoherent sources this interferencepattern is of no importance.
We conclude that a procedure has been demonstratedwhich will produce standard scratches that are repro-ducible. A subsequent paper by Matt Young will showactual patterns which were produced from scratchsamples made by several manufacturers.
Partial funding for this activity was provided by theFire Control Division, U.S. Army Armament Researchand Development Command, and the Combined Cali-bration Group for the Department of Defense.
I wish to thank Matt Young for his extensive editorialand technical help with this paper.
References1. Military specification MIL-0-13830A, 1963. For the use of present
standards.2. J. M. Elson, H. E. Bennett, and J. M. Bennett, "Scattering from
Optical Surfaces," in Applied Optics and Optical Engineering,R. R. Shannon and J. C. Wyant, Eds. (Academic, New York, 1979),Vol. 7, Chap. 7.
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