.EF _ CTION OF LGHT -4 FOiI _ _ . IMUTI- T YLR FILT by ivionarch L. Cutler Submitted inr P3artial Fulfillment of the Requirements for the Degree of BACHELOR OF SCENCE from the Massachusetts Institute of Technology 1939 Signatu're of Author . . . -.. Department of Physics, MIy 1, 1939 Signature of Professor in Charge of Research . - Signature of Head of Department i1 I: . .. -. . PREPFACE This thesis is the result of work leading to the degree o achelor o Science in Physics. Recently Dr. Hawley C. Cartwright ad Dr. Arthur F. Turner proposed the placing of thin films of varying indices on lass to roduce color filters through the interference of light by these films. This paper is a theoretical study of the reflection and transmission characteristics of these proposed filters. In Part I, equations are derived to enable the calculation of the reflection and transmission curves of multi-layer films. In Part II, these equclations are applied to the special case of the color filters. In the Appendix are several tables and charts that were conmputed in the early stages of this work. They ere not found to be as useful as expected and are included here only because it is believed that similar material is not available at present in other sources. Grateful acknowledment is ext ended to Professor Arthur C. Hardy, my tesis advisor, to Dr. Cartwright and Dr. Turner for information concerning their work in the production of he color filters, Fey3mmn connection with Part I. for his aid in and to Richard P. i~onarch L. Cutler Camibridge, hivass., Mlay, 1939. Sk.'M .. ~G TABLE OF CONTETS Page Part I: General Solution for the Determination of Reflection from Multi-la er Films. The Vector Method of Analysis . ... . Reflection and Transmission from Two Surfaces. ....... .... . . Case I: . The Single Film .. .5. .... Normalization of the Equations . . m + 1 Films Case II: Case III: . 6 8 . . . . . . . m + 1 + n Films Use of the Eauations 3 . . . . 9 10 .. Part II: Conditions Necessarcf to Produce Color Filters From mult-layer FlmsP .s . Conditions for High Reflectivity Conditions for a Sharp . . Special Situation . . . . . Maximum Reflection . . . . . . . . . . "Latitude'" of the Maximum ...... .. Secondary Maxima Varying nd cos r . Summary . . .. . ..... 14 . . . . . . . . .... Appendix. Charts and Tables of Angles of Refraction and Reflection Factors for Various Indices. II""'"I"'""~-~L- --PIII------------ ------- -- 21 21 ........ . 18 19 . . . 12 14aximmn 14 . . . . 22 77 7': ;i if ~PART GE1IERAI SOLJTIO I FOGR THE DTP'B'lThATTIOJ OF R FL CTI ON FROM " MUT-LAYER F'TMS Relection from a thin film is basically a problem in the interference of light. Essentially, the amplitude of the incident vibrating light is divided into two sections by a combination of reflection and refraction at the first surface of the film. These two sections are later reunited, this time by reflection from the lower surface, to produce interference. THE VECTOR iETHOD OF AITALYSIS If a plane wave is incident on the surface 1-2 of the film in figure 1, the wave reflected at this surface will interfere with the one reflected at surface 34 in the plane ab (or ab'). The difference in the phase of these two waves is 2. 2 where: = 4nd cos r/noAo, (1) d = thickness of film, r = angle of refraction, no= index of first miedium, n = index of reflecting medium, A/= wavelength of the light in the first Id j1 --- -- ---- ---- i· c E P1IPY·-·IP-r--UI ~us*wr~-n--i~o~i~ ~ )_ e__by vA L. Z __ (_ l--_Yi-jL*iiilY%-I-1Yi.i_ si·---r~l·.-~---I~-- .I·-C :· ------ ... 1; Figaure 1. .... . .. . - .1 I . . I .. ,. . .. I -- I. ~~~~~~ ~ ~ ~ I -. . .. .. .. .. axi . ~ ~ ~ . ~ .I . . ~ ----, - ~ ~ ---~~~?s' m~~u-·rr-nurrn-r of imainmriess I 1 . axi s of re als .. \ .---- F igu re 2. i(lOll--------- -·I-- - I- _ ___ A: J -2- Since the light waves under consideration are amlitbud.es, fi gre 2, they mayr be treated as vectors. A represents the incident vector of unit amplitude incident on a single surface. t is In An amount transmitted and travels throu-gh one thickness of the f ilm, thus producing a phase change, . This phase change is represented on the vector diagram as a rotation of t through an angle is the complex quantity teJi. ; the vector notation The reflected vector, r, encounters a phase change of re jv, and its notation is which is simply -r; on the vector diagram, it therefore points in the negative direction. If 'te transmitted light is desgnated by T t and the reflected light by R', where both are complex quantities, then to obtain the intensity of each, T' and R' mrst be multiplied by their respective complex conjugates (notated by double priming), so that It = TT" and Ir = Rt1" If the example in figure 2 is used, j It = (tei) I Since I r + I t .^ ______1__1_ (teie) = (rej w ) (re'J) 2 2 = 1, r +t =1. r _I _1_1 t 2 = r2 . __ RE=F 'CTIONA3, In TRAT-',irSSTO FROuI T0 STRFACES. determining the reflection and transmission from two surfaces, sidered. the interreflections must be con- Figure 3 shows the values of the amplitudes of several interreflected plane waves. The arrow indicates the direction in which the waves are traveling, while the value of the amplitudes are taken at the points market with a small circle. The solution evolved is for any thickness of film and for any combination of indices, therefore it is assumaed that there is no phase change on reflection. This assumption is justified since in the actual use of the final equations, a + or - sign is placed before the reflection factor r or s, depending upon which of the mediums used is the denser. For notation in this discussion and that which follows, r or R' indicates that the reflection is to the left, while s or S' indicates that the reflection is to the right; u or U is taken for transmission to the left and t or T', to the right. The small letters are used for reflection factors from a single boundary surface, while the capitals are the comple: reflection factors for one or more films. __·_1_ 111 1 The light changes ______ / i surfpe :p I -·r-·-; - *- .. 16 . rIo t .... - · -·--P· - /< .... e6 '. --- tI l I 2) 1 '· . t I 10, I I ` r i I. 2 . . · 2 uI2 - - . - . t TI1 /r surface I1 tlte ., I: n -1.1 t. I - - T, . i L Is c I t t ituul rS2r1' .2 .S I 3 .1 i i t 2 t tzr t 2 rs. t~ ir s'I 1 '~ 12if 1 2 3 ' 2 Q' -- --° 12K- et : eera tc Light Incident from Right _- ..z. .. _ . t.3 . .................................................... surIf .. e , , .>. .s. .+#v^ . 1/1 --... . surface i ~~~~~~~~ 2 b- - A !: 1 .t-·v -7,-U------ V o SI Y I r r If,,rc ; L-_;rs 4 I-- . - Ir e :k sl·_ l__r__ S .. . iL2 , et cetera Li z 2r2rH J tgu2si ht Incident from Left Figure 3. ----- - - -- i -4- in passing through one thick- phase by the angle ness of the film. For convenience, eje is represented by . The total transmitted amuplitude is obtained by 1 igure 3. 22 + tlt 2r s 2 2e 2 2 1 + + r2 s2- "-). adding the vectors on the right in T' = tt + 2 =tlt 2 L tt 12r2 (1 + r2s 1 ..... Since the expression in parenthesis is a simple power series, it can be written as t t T' tlt2eJ 1 - (2) slr 2 e J The reflected amplitude is obtained in the same manner * = rl+tlulr 2 2 R' (lr 2s l J2 + r2 2 1 2 + ..... ) 2 tlulr2 eji 1 - r 2 ls ei 2 e (3) If the amplitude, A = 1, is incident from the right in figure 3, rather than from the left, the total transmitted wave is j' r:j~~~~~~- = Uluu 2e ( r2sleJ and the reflected amplitude is "_~~~~r·~1 S' = s2 + t2u2sleJ2e r 1 (5) I 1I : ) :· : LI II I _'k- ··· single film ri, Il1 w_ SI tl Sl, , .. .- . , . . . t. . . ", nd cos r ' r 1 =-S 1 &Clj-LLII U1 r 2 , 'Ua =' t 1 (a) R . I . e_ m- singl.e film l' , ii mT m nd cO r - i_ SZ t - film.J films fnM .I s inile Case I: i·iU -,· uZ' t2 Ii ..... II r ( b) Case I: m filmrns R,' U'r I i I S i i ,, tz ilms. n fi'lms i T' & nd cos r ir I iI i i 'Figure 4: m + 1 sLingle fillm i· (c) , uz Sz IR', Case r1.1 U n S irn, T' Tf n m + 1 + n films. Cases to which the multi-layer film equations apply, "11 I . I. I .~ 1..... - .1 I - . , , .. - % - . -1 -_~ 1 I , -.. I.. .1 ~ - .- I . i .1 I .1 -5Equations (2), (3), (4), and (5) are the general solutions for any two surfaces. In the case of a single thin film, the quantities r, s, t, and u are simple owever, when multiple films are used, reflectAfactors. Figure 4 these values become complex themraselves. illustrates the three possible cases under which the above eau.ations may be applied. TkT CAS1' I: S TM GLE FILI. The reflection and transmission of a single film is readily determined by setting s = -r and u = t, which facts ,rue in such a case, so that by using are necessarily 2 2 equations (2), (3), (4), and (5) become = 1 + t r R' = r + r2 ej 2 e l+rlr 2 e j 2e ei j(6b) r _? l+rrl T' = U' = (sa) e2 3 j tl t 2 .e (6c) ltrlr2eJ 2 As has been stated previously, these vectors must be multiplied by their complex conjugates, R't , S", and T", respectively, to obtain the intensity, so that t1 2 2 It = T'T" = (lr12r _- _ ---~~_~~~~~_ 2 2 ) +2rlr 2 cos 23 , (7a) -6 - Ir = R"= (312+ 2 )+2r es CO (7b) (l+rl2r2 2 ) +2rlr 2 cos 2e Tr = IR = R r s S 'S". If equations (7) are added it will be seen that their sum is unity. NONI'.R }IZATION OF THEEO'- J ATIOTS . Equations () and their conjugates may be cumber- some to handle when the action of several films is Since they are complex numbers, they may considered. be expressed by the form Ke ix The normalized eauations are R' = Re j u and R" = Re-ju; (8a) S' = sejV, and S" = Se - jv; (8b) T' = TelW, and T" = Te-JW; (8c) Since, I t = T R 4jr -S = , T " = T = t. Likewise, , so that the only difference between the reflection factors R' and S' is in the normalization angles, v and w. If a vector diagram, similar to figure 1, is set u-p, it will be seen that u, v, and w are merely the phases of the vectors of magnitudes R', S, and T', respectively. In order to determine tdhe value of u, ecuations (8a) are expanded into ,? i !I2 .. -7- = R (cos u + j sin u) R R" = R (cos u - and j sin u); adding the above: R' + R" = 2R cos u; (9) subtracting: (0) R' - R" = 2Rj sin u. If equation (10) is divided by ecuation (9), the result is tan u = R' - R" . j (ila) Rt + Rt" By doing the same operations with (b) and (8c) the following are obtained: j tan v =S' - S j tan wT' T' T" T11 (11b) (lc) . laTen the equations for R t, S', and T and their conjugates are placed in equations (11), the t's cancel out and the real angles u, v, and w may be determined that by remembering e i and + e ix eJx _- -x = 2 cos x = 2j sin x. In the case of a single film, equations (6) are placed in equations (11) and the results tlhus obtained are __ .-1.1-_1_-^.-·11 111·-·1·11··-·--- 1 -8- tlulr 2 sin 28 tan u r(i+r2 (12a) )+r2 (l+r )cos e t 2 u 2 rlsin 2e tan v = (12b) r 2 (l 2 tan w = ( 1 l + r l 2 ) +r l 1 l-rlr 2 +r2 2 ) 2 cos 28 tan 8 (12c) l+rlr 2 CASE II: m + 1 FILMSf. If the reflection and trsnsmission for m films have been determined and it is desirous to find them for one more film, or m + 1 films, the process is analagous to that just performed for a single film. (3), (4), and (5) can be used if , Equations (2), sl, t, and u are replaced by the complex reflection and transmission factors (R'm, S't, m~ T'm ~ and~ U'm) of m films. m~ When these operations are completed, the results With reference to iLure 4b are R'm+l R m + T U'mr2 e j2 = s 2+ j2e t2u2-S 'rme2 j 1-S'mr2e (a) e 1-Sf mr2e S m+ e 2e ( + T I (13d) = U'Jm+ 1 ^ T2 TTT · A 'sJL.~ .'l..L l. ±T * e e2 3 'mr2 !-S i (13c) 2J t I ' gl .LJT-L TTkitc LL LLJ In some cases it may be desirous to calculate the reflection and transmission of m films and of n films and then determine these characteristics if the two sets are put together with a single film between them (i. e. m + 1 + n films) as shown in fugure 4c. The forms of equations (2), (3), (4), and (5) in this case are R' :R' m m+l+n + e T' U m 2 1-S lne mR m+l+n = n,, ~ TnU' S nI (14b) S'mRnej2 e 1-S U'm+l+n = m Sm nn e j2e U'U e j2e 1-S 'mRt e, ^----lc-r*··)-m·nn_. ____ _ (14a) j · ., ~ ~~. ~I 1 1.5.... .I · .. i : : , :s : i:: : ;z:!{ ~? i ::o !~' ~ivfb : ! !i: :0 :' . }-;' .; .-H~ . .-4 -C4 - .~ .,~- 'C ; , -1e . - _. .-i .1-11 p~~~ ulS C) r-4' , .r -4 · 40 I . ,o 4 , ,,4 - u~~~~ t W %:%~~~~~~~~ / C .X s_~~~~~. Y IJ t ~~n ~ , -- . . 4 sC . . < s, - .-..................... - ' m-0£ 0 H- /i 0;2 . . . ---. ,, . _ - 1 I ;4 w ~ /~~ W . -- I ' ~~~ I· to d 5' 0 - -- c .: · po, U) pq~~~~~~~~~~~~~~~~~~~~~~~~~~~~: . t M.l' ¢ . :iiwS i'. < :s - if t {; : . 0 H~ - . .~ . SC)~~~It ~ +> X &, -Q ~ r| ..,t ~tQ '. . 11~~~~ ~ ~ ~ ~~11 (1;as ; -~~~~~~~~~~~~+ II VBcC ,,, :~~-.*H 4 v- -~i C)b ' t - "- 44 --. I-C3 SIQ k ~ cr;~ FC9 ~U dikJ>:;~~~ . --- . .r . - 1, ,CQ i , ItC 19 U W : 4 +r - . . r ........ p.j . k -k .W -10- '1 ~ - I Wdhen an attempt I -- - - I -1 _ I - _ _ s mace to proceea Lurizer, in the case of a single . as ilrm, with the generalized forms of equations (13) and (14), the results become rather complicated. A For this reason it is : best to con- tinue from here to whatever special case may be under consideration; a special case often has enough idiosyncrasies to help reduce the formulae. USE OF THE EQUJATIONS. If the equations are studied it will be seen that : -[ it is far easier to calculate the transmission than the reflection. throughout.- This procedure is followed Since, as will be seen later, the calculations are quite tedious, it is desirous to use these formulae in such a manner that the least amount of computation need be done. The ost straight forward method' - ' of determining the transmission wold be to calculate it for a single film by euations (6) face and use ecq4,uations and then eep a.ding oe (13) thereafter. Such a proceduro is illustrated diagrisnatically in figure 5a. method is is illustrated in that if all the h-h figure 5b. the H films Another The advant age here- indices and all the are respectively the same, more sur- 1ow indices the values calculated for ould be the samie and thus alleviate uch unnecessar-y work re(qired in usinLg the previous ly 'F MaL r-· i"' '"~~~~~~~~~~~~~~~~~~~~ ,; 1 I i I II* -------"-- -1-·c- ----------- ·-----·~`"p"P*'`'LJ.U; le·iiL-·LUU·Y-DiCirr;b-;· ; : - ::: I3 I i I IoI .Y-.l Sw LOICLi·III1_ ar<.-:_ffie .. I ~~a f t if s .1i I I z I . ' ' Ii. I_. _wE;i>-S_- ·· ., K ,_. W I ! i- '§leo<~,p~r. .sF-~--yrod w rc- -. _r ,.11- 1- -'*r-O t-. wrw -wa -. ~--· i i 7-- --- -i i i il ¥Ik I I t i I v' 1. Yg ,e~ 'igre 61 A ew ways in which the m-alt -layer film eqtUtins Oay be applied to a series of sevehn fiims. -I- mentioned successive surface nmeth.od. T'h-is -m!ethod may be called the alter-nate l.ayer method. and entails the use of equa.tions (14), h.ich are for rm + 1 + n films. There are a large number of combine these two mrethods. It is ossible ways to that with es tiate. seven films (eight bloundarry surf ces) there are w-ell over fifty possible methods of approach to the final solution of the reflection or trr.nlsml ssio n few of these _ ___II________II__I___I__ curve- A re show.n diagramatically in fi~gre 6. -C----·-- ·-. I- -12; ._~,." ?:PT II I PiOL .. JLTI-LYWR FT¥h,}S .- Tlhe ideal color filter would be one thai lar g e reflection (or transmission) l ength band and a ssha,.rp ;wvelengths of te gave a for a narrow wave- rop to around zero sectr-ai. or other The conditions necessary to -.roduce such a response from a series o may be stAdied with in Part I. fi ters CO-I>TITOPS tlil t.lin films aid of the equations This discussion deals with 4e e.ol ved reflection OR 13 ICR FI LiSCTTTy7. If we have a film, as in fiLure 7, with an inex higher than Ltht of lass and of infinite thinnes3, then the reflection factors r inter-reflectionls) and r 2 (neglecting from surfaces ab and cd, respectively, will point in opposite directions since rl is negative and r 2 is positive. The ilmA f m,-st theremore be O that the vector r 2, on passing fromr; -A;[i l rad-ans; the 2 . of euaton (1) euals ¶r a 1iickness ab to cd and back a ain, will be rotated through or so that This condition, it can be s een, occurs when nd cos r = kA0 ,/4,where k is 'an odd integer. }?/i.[i sch 'i ··· ; '' ';' : ''': ': ",;' ~' l' :)~~~~~~'i ~-.--, -.-..' ~|:--. .-'-''':' .,,r ' ~': 0:''''','0'',08 ..... ' ' '' .'~~~~~~~~~~~~~~~~~~~~~. . .' ,: '=' ·-;-:-- -- -lbhW1I : .: --7-1--l-r Z_ ':- bX ": r· . . '·: · ··):?:. ·. :·; .''·.L ·;L.' :-: I· 'Wi;;lL:;-:l-s.,.:i-,s_,i.-.;;;.r :::::;-:i ___ _______________ ,, . .1 I ., a __ . -;·Ue~jt~;l t%0: air / b - .-.. .- :-' '· i ~.w E,>tDVjt 'd ) .. , ..- . . ''.. . . . ..... ' .:.=-,. (b) d ia-n. ., -4 I h i gh' -nd ex l r a'r -et 3t teeet or- d Iarm 7. . - . - - --.-1- . -I-ri1 r ror r 4 r5 r6 ht i hi .r rl ;·· . r ·f71 c SS (a) Schematic dia ram i (b) Vector diaram: Filt,~ infiniely thin Fifu re 8 m_ _ _ >_ _ _Ki; -x' '^' ;IUIW--O"PUYCWLWW;_·YY··*·i;UMhii* I . -. ... ·iiiLL\ei_ii:-i-iLSliLieil)llRL ` _1_1_4 -3-- If several filams are used, it is desir.ble that actors rl, the reflection ,r r3, etc. e high. condition can be ffilled by alternt'nl so tha This the films t e upper one has a high index, the next a low-; index, then high., and so on throuGhut the f ilter, as in figure . The reflection factors may also be increased by (designing the filter for use at hi -,h angles of incid.ence. If each. ilm is of infinite thinness, the vectors point in the directions shov. To make r, r4, and r 6 pomnt in the sme rd'rectionas vectors, it is necessary to make nd cos r eal od --quarter wavelength for each film. rr radians; rotated the other r, 2£Fradians; r, to an Then r 3T 2 is radisns and so on, until all the vectors point in the same -di.rcction as rl, which of course cannot be changed since it does ot pass throug-h the films. WUith these vectors all direction, att -i nable they ad ointin2 in the se up to give the reflection b- Just vary ing the thicirkness of the fil:ls. For other wavelengths nd cos r does not eccal oneone-quarter wavelenot'lh and. the vectors point in the sam.e direction ill not all and thus the refelection ~~~~~~~~~~~~~~~~~~ I ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~- I· '· · .` ru · Ci-·----i--·i ..I - . . i -· . 'v-- l., i 1 ,:5> = I., . --·--- i-:·_---; - n =- -·..;:-· -;;.Ur-:-- ,· r L .J :---·;i--- ·· · . L . .C) 7J7 "0 - ; - -I - 4l - - ''' 4 0 ·- · I --. . . - 2 42 . _ - -r~ 2 7 rC angle of incider.ce nd cos r (a) in 450 fAm/4 for each film escripton o f te tic Cv.te m from .Liruire.Ot10 c co -Out e d. vir 7 1.e 55 >2 trnlch the curves air ! r Js I/ l-al (b) ivgrrrmetic i:llustration of toe rmulti-layr film equt'-tiz'xr pute th~e eiarvcsin igd{re 10. " - . ...- ~~ I --1,~~~ .···--. -- -LI--'----- I~.1. . . . }siamrcr inr ;hici 'crc: t'ic-ed to com- ihe, -~- - I I - I . I - 1-.- I --I ------ ------- --------I-- a Pi gure 9- -9 --. -111193111·1*·111411ILslllrr(·------· · .·_ __ _ I-I _·· . · _ *- - - br;-·l ': -1 ,L ' i I· Other L-mailma a.r,-,arat on each side. if cuarter-wave C T"iTrIT'iTS FOR A s'fH-R. films to make the as sharp ..a;imml rom sevr;al lrayers of thin case is a special Rath.er than build.ing several films and deteimning teir S etc., To obtain an idea of the sharpness of films, a quan titative aalysis o phot;roeter, 3Ao, S5A, T K ITlT:J the maxium obtainable necesssry. values are used throu-llhcul. roble-m is The ne xt as possible. and cdecrcasingn a:imnuM at , curve will show a chara,-ct eristics on a spectro- the eua.tions evolved in Prt I m-3ay be C. · SPPOiiCT. S.TUl ~ ,ON. The series of£ fil S that will be discu ssed is illustrated in figsre 9 with the values of the indices and the reflection and transmission factors. lr.Ls ae r-anged to act asa The reflection filter with t an angle of 45 degrees. the light to be incident The light must, of course, be parallel to have the filter The reflection factors finction as desi-red. of the bouncary surf. ce are calculted from Presnel's f orrnml a R = - sin (i - r) , I: .jl L 1 _i ----ir-r lll--IllC5 -- -._ - ·- ''-==---- --- I 1· - % 1"1 1 - E. _ .- 1:i ' -- Reflectdion curves, rrAr iB and-I for the system c in figuum 4-I i'T' S) .1 I H i.is.t.1-il -Li w@ --- la.-sis · r ,,, - . w:;· .r '"r- I T-lwvml ~I ~ . .t1-' t 9 __ i t ------ .T- : r7 Itl.-T ' I1 I 1WHH ¢flJX rt771-i I j7T 4-' -t'I tH "I t -!h-4 11'f '-t- I I at-4 I + 1-' "lit·t-: ThT h-~1- . . 4,4_,- .1 , ' 4j-j J lj2dH2 _- 40 I_ij , iFI~ ',4_ I.S, f iJi I f * .,I q441i ~.~J' ; ¥~ ~IZ I -Hj+ _! . ! { III', f -:?4- H-H .-". t- T I 1'q44 -. ;·.· ij i kt4-! ¶4 1+ - i 1 _ Ie -H' ;· ..:..: I 1:1- _ :·- I; ' !- , z "; " " '` 78.if- , H- f ee s, . i:i'+ /i 's.r I l l I ,t.4+1 I: _1.i ff k Hih i7 t -i 4- ;'11-111±1 , 4- _D t 1_ A., 14111TV -- t -rc--tn* m f 1Lt4P iFi-Hr t . It1_:: 7H_ f-i' .zI- ihii _, j -i H4 i"i E .j +i ,." -. I,{I t5!%_:- I- i.. !':-iY! mt g TI ti i!:.r4;1 2_ ~;j ,4,1-4 ,l- .4 i-- t--t4~ I; I l I I ' '; (rI:( I tI:_I f LE 14T - _P~t±+_gj ffIi2I li' .!i_"JT! · -L - '.v . Li i; A - 't1, 'I ' t-l +P`+H_ -+ _4 1 d:,L I :J.L'_, 2t .i Ii] 414· -H.-lf YT4 f.91'' II i i 4 Vf+' .LF-f£+M 1 -, - _iH rt' 'ti r~-- T-"I1tT. 3 . L--t+.. JuI g Tlle i1 - 1 ?74 TTT IE!- IM T-j- i I. ?ft .[ZT-F-'~'. Huff~t-t ;. .-. IT 11'W111-11: 4T44TRIP-11tW. tt tH'14: T l iTT. -: T l4J1-44l -I a I,'1'Al ig _i-il] p- -+ I-s' Sleti+' :!t fil ms i 'l ust rated ,. . W-4X TX i - 4 'v" ' .. :~~~~" " i%· . _L.4- ?^ pTIL--, I . i- I'"~_ 3() ii ·i· ,· ;; Pq 20 10 TTTTTT1, _I U. .-1 7- I _.4t 1 ;7Ft- '_;"J't -J--1 t· I4 ;SL,__-T . T-, Pt-t 4 'tE,~- .;r4 :i - t' ! i {: _114- ' 4 I. ' - :- i.-i- -i l , ar ..... L . , . S 0Q 2.0 1 t.9 Z 1.8 _44_ -1 . i ..... , 1.9 I-r . 1+'I I ] T r . 1.6 1.5 4 1-a- : 17;n ,,.. _ .7 .5 .3 .4 I- L:,,J. i ; L:;. 'i k -4.L'4I-A. _`_ IL-iI I 'Y" .' renU;-i· _ , FRMi 3 T ,.A-M ; U _L tL ',-- - -,T Fhfw-.H 1.4 'l . .. _.-L.~ L-~I Al LI½LAL !d ,4- ,t114:;t T- -t-· j-.IMIT .i I LL 1-r-T-f-T : , ,6 7 r liltI _ ": .. . 1, t . -, i·IIIi ii 1: III f . t-i- i . . .9 1.1 I I0 . ., I .. . A/Ar I 10 Figure 10. ......... ~},' t', ":' . . . ,. ' , .' :' . .. I /j i~ so that ani additional recuirement is be olarized in a that the light I i i lane parallel to its trace upon i i I the plane of incidence. The value of nd cos r fillm, xwhere \ is s m/4 for each t.ken is the wavelength have the Baximumm. reflection. (in Since 2 air) that is to = 4rndcs r/ rT/AA , the curves can be plotted a, ainst the X = Am/ raction I I II i I i i I! i1 1! I =7r , or 2 Figure 10 shows the curves calculat>sd to illustrate the effect of building up a series of films. is the reflection from the first fil , um.nder the cnditions shown in f shows the result whener the first Curve A of index 1.92, ure 10; curve B, two films are con- sidered; and curve C, for the frst three films. The forula used to determine curve A is deduced from elquation (7a): 2 1 A (! -r ! 2 2 2 2 r 2 ) -2rr The only variable here is 2 2 cos28 = XTr, so that this cl- culation is relatively simple and. can be reduced to .765 ItA 1 after the values .109 cos X r iven in filure 9 have been inserted. I t;i : I· I -- ·----- -------------- -16- Before determining curve B, one more value Ylmst be found: tan v t 2 rsin29 = A r2 (l+r i - 2 )+r,(l+r 2 )cos2Q 2 .392 sin 20 _m - I - .574 Sin 2@ ItB is obtained from equation (13c) by multiplying 7 = complex congugate and remembering that TT" it by its I and S' t Se j3v and S-R, It see pge 6). t22 0 ItB B (l+r2 IrA) -2r 2 RACOS (20+vA) ii In a similar manner 2 ii It illii i Itc :li t2 (l+rz2IrB)+2r2 RBcos (2Q+vB) Ir; where . S t tan VB - iin·, -St B IB - B B ii which reduce to t2RA sin (20+v A ) tan v B '!··if. i I; i. 3 I W1 """' _s T| r -r IrAcos tt(2@+vA). r22 Ai- RAoSA) ttIr A * )-.-r ? .l~~ ~ ~,-17, To calculate tan v a rather td for around twenty values would be ious ,.ask for an indi-,_vi>il since there four variables in re he- ecuation For thi s reason it is bst to revert from the successive surface met;od of using the forulae to the alternatte ayer et:!od:, shon sm d agrmat.it:ca1ly in f iure 8b. Such a system would require the detiermination of ItH but it would also eliminate the calculation of ItB and of tan v . Actusnlly the curve or IB is not required and is shown here only or the purpose of illustrating the effect of building up a series of films. I e t2 ....b i ItH( .903 = (1+r 2 4 )-2r2 cos24 Me (14c), . 1-.096cos 2 ormula for ItC -maybe resolved from equation -whichis for m + + n films. ItAItH 1+I AIr - 2R AR * c u o s ( 2 + v ·IrA +f ;s ) where tan v ,...,-· W -. 3592 sin 2e A - - - ~ 1- . 574cos2( J -·------------- ~~~~~~~~---- . w.i I .i W · · · · . -· l I · I I, I · · '!.: E-I! -,4. -. '1-4 ' FiHi 4-4t4H4l1M-l RFj--E;4+4+ti4-t4-Hi lt1t t!~ itt~·~~~~·.t-- t-4_1j:q1 -i--q i4.i, FtHi i-i -, ~ :If ~ -~YX!~: -.. : '--r itti-l +!-H!-I -J iri :7 1 :i -TTTI IFg: -, 14 I~-rI' -1,,~'!I! L1J--L-" 4 --, ff I~~~~-q · 4:-t' -i'.~, !-H- ;.:+--. hi ii4 .i fi -'--;-''--'t',:'"-.-!-:-,-,,'7-:,,-f';:,?'_t~ '?-'' TLI ixIt_ d i.- _- _tIltt-N __,*4-t4H-Hi itnUfl .I1'== ;0; ~-t 7: + -- '5x-t~-$ ''qm-u?~? ~i '*k ~, :i~~ ~*. .i ~-S;:T iS~.t:'~~' i-Hltt'J~J~., f i I-e lEt ,';~.:-r _?i rrr .riE'l¥ , J 4 -4., ;-4.',4+Wq~i- 44---,.- 4;-'.4 . ;q ,I-~-~ ~.~,*~~~~~~"~ a:..-¢~t :..-~,. i...q~ ·. 1 00~~~~~~~~~~ 44+ S: , -s n-i -_.- - ii~~" t.20 J'2-;-;-u'l_Yii_i 0 .': ,. I-. I] ,I t _ rq--.-H,-t l. .'.L. r -' , k. e444--?-'/f H ,I;.-k- ; t r -l~~~~~~~~~~~~~I : .I I, tIJ i ! ! +i t ti-i." 4 -vM"'+ I 1,144 iI :1 14 fill] 2'lijjjL EI L i;4"i7-TCFf ttt't¢=4irII'M a-,- 4;- .'j ti+ t'- - -t-i ti -- - i i . 1. - l !-, . I r -k-',4f 1.11- .1- . ',' 'i~·i--:;-.,:~i?;:_t :,r:-:--4::1~~~,-:_ ' _ ~ 60~~L t4~-~'.. 4 ~ - --t- 'i--i '~.,,r.;i.,; - :'-?Mq -4-.4-.- *,-"; '.~ 2~t .~.-I ~, ~ :~'Tt,-,,~_L '"'~!T llJ.'T!-,L O T ---. -: -.- &i ~ ,_ , i ] i ,' ii!f''.' ]S i Z.T--''[t!.~q] q ::-,- .-- "-.r .: i-'-T '--. ~ F~ t -4--. tS.-C- : - i$-.-F---i 5A0 P,40 I W :" ~'- !~i,,t' -F:Z..+--i ' i :--4'' i i- i" ; !q4-. ~~,j r-' '-: R:1-'$. . . HI . 4qH--iI ;~~~~~~~~~~~~~~~~~~~~~~~7 _Tlk-' - -"+q' ;; '~j- + ' i.L:~~~~~~~~~~~~~~~] ;A ...... r !--H -F: :· -: F~~Tfi 44 I- .i-. .. -''f''i-.N .. H yt -1... TiI 4 ;iS: i.~---~-'.,4.-' ~:-- . '+-. i4-. . ... - ,. ,~..i--41H...,.. , ,,, LI Number Of films on Al-ass 1'-- ~ t .I--,! -' . ~ . i.~-I: --:-~-- -";t' -,-+.-.,.~~-,--~.~~ , .. I : ,' ,!· ~ ]~~.-t ... 4- H-!i--:~,.. ,, t. -? ~4i+-f-!, -+- -~" ~''~,"xi.''N-.-hr r,[' ,i--1--,i.',k~ '-lf -Ht--.'---l.~-H I.' i''t~" t'? -Tc-'-',v'''-rt ;10,Li-'q· ~~--4'-.;-.- ; ;n: n1' .. Z ~ iF e,--.';i,;!-;~ L :N m.T"":q1 t':: {," q4F ;!' !! r :/!;i! :: ,i+ , 'dS 4f~~'rti---,' l-~-~';-i-ih, on.1-F~.,eE ~ltumber of films Figare 11. -- ·--- -- I ·, - - -- ILt -18- as previously deterrmined, and t 2 2 r 2 sin 2@ tan .r =. -0.0D sin 2e r 2 (l+r 2 2 )(1-cs 2, It 1 - cos cn be readily seen that cor^utation is reduced by eli-'inatins ,!IIM.UivU 29 appreciabl. t n v - REFLECTION. It is evident from figure 10 that as more films are added, th6 intensity of A increases. The nxt step is to determine the nixnber of films necessary to produce a relatively high reflection atA . were made with one to five Calculations ilms on glass and with the high index varying from 2.0 to 2.8; the low index was kept at 1.35. The results are reproduced in figure 11. From examining the chart it should be obvious that five layers are sufficient to roduce a satisfactory high reflection (between 85 and 100 percent), even with the high index at 2.0. The multi-layer film equations reduce to a very simple form for reflection at Am. The general reflection equation becomes R tn+r fn + 1 + R nr n where only the mgnitudes of R'n and r are placed in -- -b· I"-"ls- 70 T +I .I i I Hi ; !·i 4 .... i- - ' 'T , --*.. T41 t : .... ,.. 50 t~~~~~~~~ ~~~~ ~' .. ... '- ' cL -141* t - '. ~-, "'' · .'-~r ·- '- 3...· - - ' ' 9 - . -.. ... -l'~-F.kr .. f{~p-,'~ pl.t- .. ~ .. ~ : ... I .... .[L i~I'"l!:-' 40 ---i, ll----- ?- ~: ... :~----~. ri -?. ' - _..+4. ' " - . f +i i~ ~ - , ' ~.c& II~~~~~~~~~~I .. ,' -~r- .-~ .,,, ,_,.T;: - -'i . . 1 i! . I ' .. [~'--" ~~" "',t'~" .~...I : ~ - .- ~ ' ~ t '-C " v -tIir-'~t~'':t~t-7 '- ~ .r:/- ~i' Li.'1i , ' ,'4--:.'~ ~ I;t':-i" 'Z ,.L'".: .11.._:.:~-+[4..?. : ~.;r',,"r· _,'~1 r~7 f-t :.iJ-i . ..... ........... .. [ t1, ,,..-:2 ..........E ~-w~-,1 .... t ....... ......................... ~* ..........+. - -~...-''t- -,+1 -:, -¢ H . ,'4"d_ r"'.h.-~ . 4 ;.', ~' r .~-~ + T l.'''-4+q--T T:.L =~.~.. -t-TX "r' H 4 '-'-' i'. - '~- -:.-.- ~_.L -4 i -' i-i~~~~~~~~~~~~-4ti-~-~-1~lPi~rY~l-I~-~Ii.. l, :'--t -!' -: -+'~---.+h-.--; -. ¥-~!'},~-% P ; ".-~ !-i- ~: -+il-zN'?' 7t.· i L 'H i -'i'i ~-i4,lq!) [-.-i -n -'1~ L -- 'r ,~'7't:-Hii-:T, ii !&' 'L 1--'ff' ._ ~ ' i'?Ml_'~-?t ~ 'r tri -~i t ... J .. .. , t . r rr-...j i-,,,4 Tl--ij -IT * i i I FR-~T l' 14 H 44-k0.. : __.. . i i-~-~ . ~-t-:{.., . . - :~t;--t ;2.. 4-,'-~ i t"' -~1i'-~'-i - '." .%1 : .t~,_ . ,r 1 , ['~' t· ~ " , t ~' + ' .. _~ ~ .... h , ,...r-,' ~ -- '.~ ,-... 'T~ .. ~~J ' ': _c..+ ~.,_,' t L -'-!...~:+- e ! .. 14h .H .- J ':- ~~ ~ ,~T-!4 iH'r ri "-- + +'.. +'q-Lt+-44 --t i~'-I:;·-l! ,~~ IL.~-~ ,l~-~. --- I . ;-4:.-~ .. : i, -&- ~i CO 0 P4 +iL 4j I ~...... 1· -~ -: - -;-~ rC ,::~7,,, ,u : - , . ..... ~ ,;_ ...i **:H;-~ ~~~~~~~~~~~~~~~~~~~~~~~~~~:-, 20 4- , 10 . 2 3 .4. 5 6 1.8 1 7 1.6 1. 5 1e4 .1 0 2.0 1.9 -u _l " -ro- , .A im Figure 12.- · · · ~~~~~~~~~~~~~~~~~~~~~I Il110 7 8 -1 .3 1v2 .9 1.0 1.1 1.0 ii ·i -19the equation since the + or - signs of these The above have already been taken into account. applies to the special case of an actors rrangement of alternately high and low index film-s (figure wfhich nd cos r of each film is length. a), in an odd quarter wave- film, R'n is merely the For the first reflection factor of the air-film boundary surface. LA TiTU" ITHEr OF -FMAXIOiIH. . The latitude of the maximaum is defined as one minus the value of A 7A at the last point of zero slope, before the curve begins to rise toward the maximum at Am . _ This latitude may also be termed the half-breadth of the maximum. In figure 10, the latitude for curve IC - IrA, .52 = .48; for curve IrB 1 - .37r = .3; is for curve it is unity, since the only low turning point on the curve is at ero. A small latitude is desired in order that a filter have a sharp maximum. latitude decreases. As more films are added, the However, the rate of decrease is not constant, but decreases as the latitude decreases. These facts may be seen from figure 10. It has been estimated fromn somre rough calculations (not indicated here) that an addition of two more layers to the three already considered will reduce the latitude to no less than between . -~-- -I-" and .40; a further increase in -n-'------------ ----------.-III·llrrr--·---------·-- the - - ------ - : A8 I 70 -!i'tC '-~ -jE_,_ML I;- , - t,.-. . . ------L-( I-- 1t_ __4 : :',.1 Fi I I e gI ____ 1.; 1 60 il , ; r.l . , , I i .I; t! 4jSt [ , 5O , ',1- I..L k l '. if _ f _!.. I -'- , nt .T . ' 1 I -,- t , = _,i , :,t - - --1 . r, , , , i__ I , -- ::i :_ : : q -_ ` i, IJ -.-.,- - 4 - !' - [?iiq 'i- . -: i- I , L4 , !F-' j ..-H t;-,..; ^^-I b ~;ti I i :- ,. ' ; I ,! ' · t tTit r -:-.>i ^ t: ?.- '_': t %:..: -- -I ;. ; . I . It H__:. I L . ii W., j., j. :: t ;-t?-,- :-t .ii :. .r . , 1;-i'i '#i-- t t- i* ? ..[ r. -i6 - .F - 4... i;',L L I _. I , 7: . -- :1 , _ J_L_4 . i : r ,ij f l- ::- r I.; iI ; ._,!.T r--r-'. Q) i ' ' i rj f4 3 ]_ a i: 'X' i 'i1 7, i -m,-, ;. *1 1, , ; I. t- i [ it , ;_,_. : 4 " ,-,F i~i ,i i, t i I :i; o iti,} .X_ _ -, 1- I, ,i! -., , ' ' i, , .v_ I I - , it i i- ' II.._ . I I. .I i i! i . :..] -.? ?, . . + ?i; .j . -r.: . 1 i- ' ''1 . '| ' 'j ,: i, : ,, ?, " 1 . :i . ',+t , t ' . i- + . .ii . ii i i' -- - ` i; i, 'i '.-L!: ILl . I .; , : -, i . L, I ' ,:i;.7 . I. Ji "7 -1 i I ... 7 I .:.J I- J : i It - pt, 4Jl L _ .. I ti' :1 - Q) ; ? , -- : ? o 4 .. _ _ .i- t- - . I ,i i r.i g r; '- t 'tFtZ L! li·- 7' ' ' .°4 40 t -1:, _ . + w- 44j ,1._, 4 .- ,j T t-"T TL; , , ... I; . .- 0 -. . E . . I7,,: _1; .I ::- . i-, r;"L_ ,:'.! I I1_ - I .. .i -1. ill ; -- ._, I,4 -''1- .t'..IsI IT I ...... +: ' i. - .,-:Z: 4 ,- ,-4 -I .. -- i , ", , ; ! -.- I . -, , , ,-- I _ltt---t .It, ''it Ii .: '' . II ..il4 I c -- .1I L_ I I ; -'-i . , t. 1 --:';'1 . . : , : , _ I '' i' . ___ t--f--- -,.: i:., - . , I +, I'. . ,: ' 1L_] ,9.; I . . : , :- __, : 't 4L'_ I i ~~~~ FF tgl,- T~~~~ '_~~~~~~ L i_ ,* 1 LEt f~i i':vsr~ . __ ti-L - !. I,- !i' i T ' i, .> 10 3 I.I ZoO 1 9 1 e8 1 7 4 e . 1 6 1 5 6 1 4 e7 1 .3 a Ais/A Figurfe 13. - __ ·I_ -- I 1.i 9 1.e2 I ....___ 1 1 iC -20the number of films causes a very small reduction. I-i A very good filter .01. Such a condition will give a spread of the peak in te Am should have a latitude of about reflection curve from is 475 mn-. if 70 mai to 480 m An arrangement of films to this sharpness has not been found b rocduce the theoretical study made in this paper. However, if the films are increased in thickness, a reasonably small half-breadth may be obtained. It vill be retmembered that the nd cos r of each film must be an odd quarter-wavelength to produc;e a maximum reflection atA considered . . if, in figsure 9, increased to 3 12 results; i occurs. So far only a value of m/4 has been nd cos r of each film is m/4, the reflection curve for I 5 /4 is used, the curve of figure In the former the latitude is while by using 5Am/4, it ; reduced to .16, can be made as low as .096. The half-breadth for k m/4 divided by k. in f iglre ,/4 is smply that for rom a purely theoretical stand- point, the value of k may be made large and thus produce a very sharp filter. II In doing so, the films are made *1 so thick that serious effects caused by scatterinc and absorption occur. Also, it is evident from figures 10, 12, and 13 that as the thickness of the ilms increases, the number of II Iz? 70 it y'. i%: ;i -l .' 6o -ji;1i t~i ~t"iI ~ i' 1 7 '- " , 4. , t i 15O i TF ~~~.: ~,, :-,=~-:.::~ :.~ ~ .::_~_I :~?-'ta ~!:- , i . if"!~ -t-- i ''r'. ; -I 04 o1 -L. i 1 : ,f r : f I :?:~-d ~.:~:;~ t f~ :~.:. i -/ '~.:~ J::,-..l ""I ~L .=-:-:N:. C) o-a , 4 - ...;...i+~ ..l ';!:~:J ~ ?-"' : t p~t Ml~ 20 ' ~_I : IH; 'i-1 -: i !.T:'i-.1 "I-1ii;-IZ;;!..1-I ILI!LLY t -f' I 'T_ i-: I ---. i -I -1 '- I I ;I , 1.j j !'; II" . I 7 -I- L'' ' ' -7T 1 ' r I: - 7 .! I t i 1i, .... F ~ ~ ~ ~~~~~_.. ,-!' -" - -':* :'i :-i ti-1 13------- ~ :i i .1" - ;- I --I I ~ -~ii+' ; _. f ; .-. i -Ali!, ;- 1_ Tilt-l' :'.'1_7 10 "" t'_' I 2' . I _i :II '~_ ; 1i.!] : i i- 1' I-i f~~~~~~~~~I I . _1__i I I I- i. . i "' i: -.-i- -.-.--,-,7: : i _`I :1I 'I - "I . I 11 i 1'o!, f 4 .; -,i I "_r I z l-LL.'- II j I il . .. . I ; . I IIt I ''LI"-"I"- .~ ,:: --· , . -1171-71 IT ' 71 ,-·rL---LLU-i-LL Thl'i' I , 4; -I .__- -1 . i_ -7 c---ii;-· f iijt_4111:'- - 1:;i- -·.- ; I .-; --. I . -. I .-1 '" .L"'.-'·l:i.')' . .1 I ' lil I' +t -I_11 - C--i·:·· ^' f. ' i., ' F 0f~I 2; I3 0 .1 ,2 ,3 .4 .5 . .7 .$6 .9 1. 2~0 1.9 1. 8 1,.7 1. 6 1 .5 1,4 t. -3 1,g2 1. I t.0 Pigu.re 14,- -II -- - -21- airimu', reflecti on peaks of ras. if a filter for the visible region (400 mp to 700 m) is desired, the 5j\ m/4 flms can be used only whenAm is between 470 m. and 530 mi, otherwise two peaks of maximum reflection or the 3m/4 films will occur in the visible region. the 1imits are from 335 m two to 610 m. eaks occur in the visible Above 610 m rn art of the sectrm.uv A7:-lA' -::y~(i ,....A~. MAXI',A (DARY SEC It is believed that the small secondary mrraxima in '; the reflection curves will decrease in intensity as the number of films increase, but no special calculations have been made to prove this point. 7TARVuINTG nd cos r. Another interesting in a series of films. ossibility is varying nd cos r In the previous discussion all of a sirngle series have been 1/4 or 3/4 or 5/4 wavelength films. It is possible to makce a filter with the nd cos r of the first on. film, 3 m/4,' of second film,)m/4, and so Figures 14 and 15 represent the reflection curves of Ir for the system sketched in figure 9, but with the films of different thicknesses. first and third films are 3 3 m/ In figsre 14, the m/4, while second iim is ; in figure 15, the first and third filrms are X4, while the second is a Xm/4 film. m~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ . .~~~~~~~ ,. - : - - ! 70 t...... :<i., | -L- t - S i 60 ! 4-~~~~~~~~~~~~~~. !'iL~. .~! I~i ; -!1 ,~; i -:| . '. " ' i~' j ; 1 T.,i;!4|.....0Xt 1" it * ,i - - "' -W-4-: '' ' j . m r, 0 :L-!'i":'~-.,' :'-':! N; ' ! i m.-:..L < + l ' ! < : ....::'- 7 ::: 1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ :-itX i*' ''{t . i , .-; 7! ir : --r-.... :t-:'i--: ; r': ~t....*- 0 0 *f .. ,.. .,.:; i L t ~g . , , :_.-:; X ': |_'-',4 ~ ~~~ ~ 30 is i0 ,- __, <l ti:'riti; . ~'1 :t ' ;-f 4 -. : ~.~'~~~~~~~~........ ~ i;'~'~"':"~~~~~~~~~~ i:-;l.: :-{i: ... i' {t,:i; e t:'-t4.i:., ' i 't " ·. . , I , ' -::' ¢ ' ::L 'i J':'''<i'j'* 1'-!t' i' -X~~::,~_>:.: .... i:i_iI:_ ~~4i' _, __L_,_-, . ._,_ . --: _. ........... '-- i'.': :4~~~~~~~~. . ... j | .' ' .\ 1 [ ..j: .-. *; ~:T '- ! wt~~~~~~~~~~~ t, [~_'[i --.. ~' ~ t I ~ ;~'~ ~:'" -r ' W-Wj~~~ I :... :: ...... .j4;,-...Ii ', !.[~~~~~~~~~~~~~ :;jf-;W;s tF .---~~*---~-~-J.....L-~-f--:: .. t L : .~i~,: "'' i: ' ~' ' ~ ; :' i I '~ J::,~~~~~~~~~~~~~~~ :.:i !77 j;~~~~~~~~~~~~~~~~~~~~~~~~~~ ;, 10 [~~~~~~~~1 0 .1 2.0 1.9 .2 e3 1.8 1.7 c4 e5 o6 1.6 1t5 1.4 s7 Figure 15. I-----* _ -1.3 e8 ii9 1c 1.2_ 1.1 1.0 -2 ) The seconSry raxim.s arc ,uite cases and it Is. probable that the- icsh in these two will be d-ifficult to reduce with the addition of more films. it Consequentl, probably undesirable to make the filters with is varying values of nd cos r. STJ- mARY . From the foregoing discussion the following limitations in d esignin a filter by using thin films have been men- tioned: A Factors determining the number of films. 1. The intensity of m. 2. Scattering and absorption. 3· The size of the latitude (sharpness of the filter). B actors determining the thictmess of the films. 1. The size of the latitude (sharpness of the filter). 2. The number of peaks of maxirmum reflection occuring 3. Sc~4ttering and absorption. 4. The intensity and numiber of the secondary maxima nd cos r of the films in a filter wh en ried. is v -- .. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~--- .. 10(--. 11II-I.. I...I ;I;l"IiY.. ,_:..1, ! .1I:.-I` ...I .__ .-.. 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I.,-;.1.7 ., 1 .--- - - ' , ,- --------..Itr;7,:-.-,,-,, -,-:I---.,,."-...-L1 , ,:tL-___ j,-__T., TE II.-,.I _T3 : -__t. .-.-..: ! -I:-,-- .11.. - -,., .!-t4 . .I -."I ... ., ! _'.,. . -:1.T., -1--I,_.:, ,41.-_I.. .. --.-.I L1, ... -.. ...; ,_ --. , i-- -, -- : It ----.. - .-' ,:-. I I.. -'. -'-- .- ..I.,.,.,.-I :... .t_ -I.4--t-.-1I _-.__-...;_,_--_.-I I -:, I..1 7 :w_., 1. ._. - - - ,-, II.I, ... )-..I;II..1,-_.. 1.:I_.___ -. !IA,,; ,.,-,,-.-1. -, 2C 10 400 450 BOO 560 600 Wavelength in millimicrons 1; f _ 6, .~~~~~~~~, 700 Fisuire 16 shows how a filter react with, iin the blue region. o three films would The reflection curve was detenine. from figure 13. i ------- -- i--·-·--n--*p·--·a----p------------·rrr-·--·^··cl-ac·--c·-----·· This appendix cons~sts of tables and charts of angles of refraction and reflection factors for various indices and at angles of incidence for 0 to 80 ° . ::; ill ~_ ~ ~ _~~~~ ~~~~~~~~ ~ ~ ~ __ _~~~~~~~~~~~~1_~~~~~_I----- T, 3L:E I ,::-GL 4: __ i= of anl,e n = ilie.e -E :;aFR 'i TI i:c- de.nc e. r si = sin i n 1.1 tn 1.4 1.3 1.2 1.5 1.6 . 666 .625 i _o .833 8.,03 . 71 4- .79 1 .909 5 4.55 4.17 3.8~6 10 9.08 8,32- -7.67 .57 3.35 3.14 7.12 3-j 6 III 6.23 10.56 9.93 9.31 _. 15 1 .1 !2l:3.46 12 t4 l.49 20 18.12 16 56 5.° 155 v . 2 ~ 5 14.14 13 .13 12.35 1^ 25 22 5 0.2 18 . 97 17.57 16.36 15.31 30 27 . 03 1 22.58 20.93 !c.4-7 18.21 ,4. _- : 35 31.43 28.56 26.18 24.19 '22. :8 -21.01 40 35.76 32.39 29 .64 21,7 33 25.37 23.75 45 40.00 32.95 30.34 2 12 2.0. 2 2 50 4 .'f 39.66 36 .10 33.17 30.63 28.61 55 48.13 1305 Z . Q5 {:o 39.06 35.81 3 3 . !o lbo 33. 30.8! 3 51. 93 zi6 ·.19 41. 77 38.21 35.26 32.77 60 W2' 1 1 : il . 65 55.47 4-9 .05 44.20 11-0 . 4 37 .17 58. 51.54X 46.28 412.16 38.9 LL.50 ii' O70 8 35.96 75 61.41 5S.60 47.98 43.62 40.08 V7.13 80 6 . 5 4- 55.15 4-9.29 14. t70 41.03 37.99 85 64.90 o .1 50 . 02 45.36 4-1.6 38.50 00 65.33 6,4 50.28 45.58 41 81 3.68 ---- -cl--``-1-1------ - i i i i z TArBLE II RTEFL]CTION PICTOB'S Rs = - sin (i- - : r) tan (i ip '~ $ O r) rY z.1 *, i} - tan? (i 1.2 ' R, "s 1.3 R 1476 +1476 - 909 i 477 476 911 5 485 463 10 477 15 Rs ±1909 o1304 +°1304 909 1304 1304 914 903 1317 1297 453 939 886 1336 1273 506 445 961 855 1377 1234 20 532 419 °1006 S01 1433 1175 25 570 384 1071 749 1515 1095 30 618 336 1149 663 1616 989 35 680 272 1244 562 1752 854 40 713 189 1387 414 1939 679 45 875 78 1568 246 2 2133 456 95 2408 171 2 50 O 1047 - 79 1788 55 1232 1 ,1 281 2083 + 300 2747 -1207 60 1518 573 2483 714 3190 696 35 1917 989 3002 01277 3760 9326 70 2501 01605 3722 2051 4483 2172 75 3409 2535 4669 3134 5413 3317 8o0 4774 4002 5954 4659 6597 4864 -`^I-"I-'-----~----- - TABLE II (cont.) REFLECTION FACTORS \n ------------- - R IiIp x,~-, 1.4 1.5 Rs 16 Rp RS Rp 0 -°1667 +01667 -02000 +°2000 -02303 +02303 1 1668 1666 2003 1999 2307 2300 5 1674 1657 2011 1990 2318 2296 10 1707 1630 2041 1959 2355 2263 15 1748 1582 2096 1906 2408 2205 20 1821 1514 2168 1829 2490 2122 25 1909 1420 2261 1725 2601 2014 30 2033 1297' 2396 L589 2743 1852 35 2179 1139 2566 1478 2917 1680 40 2382 1939 2773 1196 3135 1447 45 2618 685 3038 999 3410 1155 50 2910 364 3367 574 3722 1790 55 3289 50 3733 133 4110 329 60 3752 1572. 4197 -1430 4578 65 4329 -0o1267 4776 01138 5145 984 70 5037 2154 5472 2064 5821 o 1934 75 5934 3337 6317 3274 6630 3170 80 7030 4806 7341 4870 7578 4770 -- __.__.__.I:_________ -262 I \ \ i~~~~~~; -~-:': : ; '~1:: _ .,....:' :-' ~7r -__ Z '4-----(,· i-'1-. :? :i; ...... .~..,. ;-' ;/i '.: 4 ,.,..I ¥_z.si·· ; i a k 1 f a 1 ... ··-r )-i ·· c f T I--· I--· ii-i-i-i ·;;*. ii-i-i-i ·- ;;*. ··1.·lii..: i ,-··· ··--!- ·---;·:--·:·; a·- i "qII." ··.9 ·t .t F IiI I ·1;-i ii! ·1;-i ·I·I..--.i;i .-. 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