The basics of line moiré patterns and optical speedup

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1 The asics of line oié attens and otical seedu Ein Gaielyan Tale of contents The asics of line oié attens and otical seedu Tale of contents Astact Intoduction Sile oié attens Sueosition of layes with eiodically eeating aallel lines Seedu of oveents with oié Sueosition of layes with inclined lines Couting oié lines inclination as function of the inclination of layes lines Deducing the known foulas fo ou equations The evealing lines inclination as a function of the sueosition iage s lines inclination Sueosition of eiodic cicula attens Sueosition of cicula eiodic attens with adial lines Sueosition of cicula attens with adial cuves Conclusions Refeences Tale of figues Links Astact We ae addessing the otical seedu of oveents of layes in oié attens. We intoduce a set of equations fo couting cuved attens, whee the foulas of otical seedu and oié eiods ae ket in thei silest fo. We conside linea oveents and otations. In the esented notation, all eiods ae elative to the axis of oveents of layes and oié ands. 3. Intoduction Moié attens aea when sueosing two tansaent layes containing coelated oaque attens. The case when laye attens coise staight o cuved lines is called line oié. This docuent esents the asics of line oié attens. We esent nueous exales and we focus also on the otical seedu of oié shaes when oving laye attens. Nueous exales ae esent. Dynaic exales deonstating the oveents of layes ae esented y GIF files. Page 1 of 25

2 We develo hee the ost iotant foulas fo couting the eiods of sueosition attens, the inclination angles and the velocities of otical shaes when oving one of the layes. In section 4, we deonstate the henoenon on the exales with hoizontal aallel lines, which ae futhe extended to cases with inclined and cuved lines. In section 5 we esent cicula exales with staight adial lines, which ae analogously extended. 4. Sile oié attens 4.1. Sueosition of layes with eiodically eeating aallel lines Sile oié attens can e oseved when sueosing two tansaent layes coising eiodically eeating oaque aallel lines as shown in Figue 1. The lines of one laye ae aallel to the lines of the second laye. Figue 1. Sueosition of two layes consisting of aallel lines, whee the lines of the evealing laye ae aallel to the lines of the ase laye [es], [tif], [ng] The sueosition iage does not change if tansaent layes with thei oaque attens ae inveted. We denote one of the layes as the ase laye and the othe one as the evealing laye. When consideing inted sales, we assue that the evealing laye is inted on a tansaency and is sueosed on to of the ase laye, which can e inted eithe on a tansaency o on an oaque ae. The eiods of the two laye attens, i.e. the sace etween the axes of aallel lines, ae close. We denote the eiod of the ase laye as and the eiod of the evealing laye as. In Figue 1, the eiod of lines of the ase laye is equal to 6 units, and the eiod of lines of the evealing laye is equal to 5.5 units. The sueosition iage of Figue 1 outlines eiodically eeating dak aallel ands, called oié lines. Sacing etween the oié lines is uch lage than the eiodicity of lines in the layes. Light aeas of the sueosition iage coesond to the zones whee the lines of oth layes ovela. The dak aeas of the sueosition iage foing the oié Page 2 of 25

3 lines coesond to the zones whee the lines of the two layes inteleave, hiding the white ackgound. The laels of Figue 2 show the assages fo light zones with ovelaing laye lines to dak zones with inteleaving laye lines. The light and dak zones ae eiodically intechanging. Figue 2. Sueosition of two layes consisting of hoizontal aallel lines [es], [tif], [ng] Figue 3 shows a detailed diaga of the sueosition iage etween two light zones, whee the lines of the evealing and ase layes ovela [Sciaaella62a,. 584] Figue 3. Couting the eiod of oié lines in a sueosition iage as a function of the eiods of lines of the evealing and ase layes Page 3 of 25

4 The eiod of oié lines is the distance fo one oint whee the lines of oth layes ovela (at the otto of the figue) to the next such oint (at the to). Let us count the laye lines, stating fo the oint whee they ovela. Since in ou case <, fo the sae nue of counted lines, the ase laye lines with a long eiod advance faste than the evealing laye lines with a shot eiod. At the halfway of the distance, the ase laye lines ae ahead the evealing laye lines y a half a eiod ( / 2) of the evealing laye lines, due to which the lines ae inteleaving, foing a dak zone. At the full distance, the ase laye lines ae ahead of the evealing laye lines y a full eiod, so the lines of the layes again ovela. The ase laye lines gain the distance with as any lines ( / ) as the nue of the evealing laye lines ( / ) fo the sae distance inus one: = +1 Fo equation (4.1) we otain the well known foula fo the eiod the sueosition iage [Aido00a,.20]: (4.1) = (4.2) The sueosition of two layes coising aallel lines fos an otical iage coising aallel oié lines with a agnified eiod. Accoding to equation (4.2), the close the eiods of the two layes, the stonge the agnification facto is. If the nues / and / ae integes, then if at soe oié light zone the lines of oth layes efectly ovela, as shown in Figue 3, the laye lines will also efectly ovela also at the centes of each othe light zone. If / and / ae not integes, then the centes of white oié zones do not necessaily atch with the centes of laye lines. In any case, equation (4.2) eains valid. Fo the case when the evealing laye eiod is longe than the ase laye eiod, the sace etween oié lines of the sueosition atten is the asolute value of foula of (4.2). The thicknesses of laye lines affect the oveall dakness of the sueosition iage and the thickness of the oié lines, ut the eiod does not deend on the laye lines thickness. In ou exales the ase laye lines thickness is equal to / 2, and the evealing laye lines thickness is equal to / Seedu of oveents with oié The oié ands of Figue 1 will ove if we dislace the evealing laye. When the evealing laye oves eendiculaly to laye lines, the oié ands ove along the sae axis, ut seveal ties faste than the oveent of the evealing laye. The thee iages of Figue 4 show the sueosition iage fo diffeent ositions of the evealing laye. In the second iage () of Figue 4, coaed to the fist iage (a), the evealing laye is shifted u y one thid of the evealing laye eiod ( / 3 ). In the thid iage (c), coaed to the fist iage (a), the evealing laye is shifted u of Page 4 of 25

5 y two thid of the evealing laye eiod ( 2 / 3). The iages show that the oié lines of the sueosition iage ove u at a seed, uch faste than the seed of oveent of the evealing laye. Figue 4. Sueosition of two layes with aallel hoizontal lines, whee the evealing laye oves vetically at a slow seed [es (a)], [ng (a)], [es () ], [ng ()], [es (c)], [ng (c)] When the evealing laye is shifted u eendiculaly to the laye lines y one full eiod ( ) of its atten, the sueosition otical iage ust e the sae as the initial one. It eans that the oié lines tavese a distance equal to the eiod of the sueosition iage while the evealing laye taveses the distance equal to its eiod. Assuing that the ase laye is ioile ( v = 0 ), the following equation holds fo the atio of the otical iage s seed to the evealing laye s seed: Accoding to equation (4.2) we have: v = (4.3) v v v = (4.4) In case the eiod of the evealing laye is longe than the eiod of the ase laye, the otical iage oves in the oosite diection. The negative value of the Page 5 of 25

6 atio couted accoding to equation (4.4) signifies the oveent in the evese diection. The GIF aniation of the sueosition iage coesonding to a slow oveent of the evealing laye is ovided in Figue 5. The GIF file eeatedly aniates a eendicula oveent of the evealing laye acoss a distance equal to. Figue 5. GIF aniation of the slow vetical oveent of the evealing laye [s], [gif], [tif] 4.3. Sueosition of layes with inclined lines In this section we develo equations fo attens with inclined lines. Since ost of all we ae inteested in otical seedu, instead of using the well known equations, we eesent the case of inclined attens such that the equations (4.2), (4.3), and (4.4) eain valid in thei cuent sile fo. The values of eiods,, and fo the exales of Figue 4 coesond to the distances etween the lines along the vetical axis coesonding to the axis of oveents. When the laye lines ae hoizontal (o eendicula to the oveent axis) the eiods () ae equal to the distances (denoted as T) etween the lines (as in Figue 1, Figue 3, and Figue 4). If the lines ae inclined the eiods () along the vetical axis does not coesond anyoe to the distances (T) etween the lines. Accoding to ou notations, the eiods do not eesent the distances T etween the lines, ut the distances etween the lines along the axis of oveents. By adoting the new notation, equations (4.2), (4.3), and (4.4) ae valid all the tie. Equations fo inclination angles fo such notation of eiods () ae esented in this section. Fo otational oveents values eesent the eiods along cicufeence, i.e. the angula eiods Couting oié lines inclination as function of the inclination of layes lines The sueosition of two layes with identically inclined lines fos oié lines inclined at the sae angle. Figue 6 is otained fo Figue 1 with a vetical sheaing. In Figue 6 the laye lines and the oié lines ae inclined y 10 degees. Inclination is not a otation. Duing the inclination the distance etween the laye lines along the vetical axis () is conseved, ut the tue distance T etween the lines Page 6 of 25

7 (along an axis eendicula to these lines) changes. The diaga of Figue 10 shows the diffeence etween the vetical eiods and, and the distances T and T. Figue 6. Sueosition of layes consisting of inclined aallel lines whee the lines of the ase and evealing layes ae inclined at the sae angle [es], [ng] The inclination degee of laye lines ay change along the hoizontal axis foing cuves. The sueosition of two layes with identical inclination atten fos oié cuves with the sae inclination atten. In Figue 7 the inclination degee of laye lines gadually changes accoding the following sequence of degees (+30, 30, +30, 30, +30), eaning that the cuve is divided along the hoizontal axis into fou equal intevals and in each such inteval the cuve s inclination degee linealy changes fo one degee to the next accoding to the sequence of five degees. Laye eiods and eesent the distances etween the cuves along the vetical axis. In Figue 6 and Figue 7, is equal to 6 units and is 5.5. units. Figue 7 can e otained fo Figue 1 y inteolating the iage along the hoizontal axis into vetical ands and y alying a coesonding vetical sheaing and shifting to each of these ands. Equation (4.2) is valid fo couting the sacing etween the oié cuves along the vetical axis and equation (4.4) fo couting the otical seedu atio when the evealing laye oves along the vetical axis. Page 7 of 25

8 Figue 7. Two layes consisting of cuves with identical inclination attens, and the sueosition iage of these layes [es], [ng] Moe inteesting is the case when the inclination degees of laye lines ae not the sae fo the ase and evealing layes. Figue 8 shows fou sueosition iages whee the inclination degee of ase laye lines is the sae fo all iages (10 degees), ut the inclination of the evealing laye lines is diffeent fo iages (a), (), (c), and (d) and is equal to 7, 9, 11, and 13 degees coesondingly. The eiods of layes along the vetical axis and (6 and 5.5 units coesondingly) ae the sae fo all iages. Coesondingly, the eiod also the sae fo all iages. couted with equation (4.2) is Page 8 of 25

9 Figue 8. Sueosition of layes consisting of inclined aallel lines, whee the ase laye lines inclination is 10 degees and the evealing laye lines inclination is 7, 9, 11, and 13 degees [es (a)], [ng (a)], [es ()], [ng ()], [es (c)], [ng (c)], [es (d)], [ng (d)] The GIF aniation of Figue 9 shows the sueosition iage when the evealing laye s inclination oscillates etween 5 and 15 degees: Page 9 of 25

10 Figue 9. GIF aniation, whee the inclination of aallel lines of the evealing laye oscillates etween 5 and 15 degees [s], [gif], [tif] Figue 10 hels to coute the inclination degee of oié otical lines as a function of the inclination of the evealing and the ase laye lines. We daw the laye lines scheatically without showing thei tue thicknesses. The old lines of the diaga inclined y degees ae the ase laye lines. The old lines inclined y degees ae the evealing laye lines. The ase laye lines ae vetically saced y a distance equal to, and the evealing laye lines ae vetically saced y a distance equal to. The distances T etween the ase laye lines and the distance etween the evealing laye lines ae not used fo the develoent of the next equations. The intesections of the lines of the ase and the evealing layes (aked in the figue y two aows) lie on a cental axis of a light oié and etween dak oié lines. The dashed line of Figue 10 coesonds to the axis of the light oié and etween two oié lines. The inclination degee of oié lines is theefoe the inclination of the dashed line. T Page 10 of 25

11 l T + l tan( ) T Figue 10. Couting the inclination angle of oié lines as a function of inclination angles of the ase laye and evealing laye lines Fo Figue 10 we deduce the following two equations: + l tan tan = l + l tan tan = l (4.5) Fo these equations we deduce the equation fo couting the inclination of oié lines as a function of the inclinations of the ase laye and the evealing laye lines: tan tan tan = (4.6) Tale 1 shows the oié lines inclinations fo seveal degees of the evealing laye inclination, fo the ase laye inclination fixed to 30 degee, with a ase laye eiod equal to 6 units, and with a evealing laye eiod equal to 5.5 units. The tale shows that when the inclination of the lines of the evealing laye is the sae as the inclination of the lines of the ase laye, the inclination of oié lines is also identical to the laye lines inclination. Page 11 of 25

12 Tale 1. Inclinations of oié lines of the sueosition iage fo the ase laye lines inclination equal to 30 degees and fo the evealing laye lines inclination fo 25 to 35 degees: Fo the sae set of aaetes, the old cuve of Figue 11 eesents the oié line inclination degee as a function of the evealing laye line inclination. The two othe cuves coesond to cases, when the ase laye inclination is equal to 20 and 40 degees coesondingly. The cicle aks coesond to the oints whee oth layes lines inclinations ae equal, and the oié lines inclination also ecoe the sae oié lines inclination evealing lines inclination Figue 11. Moié lines inclination as a function of the evealing laye lines inclination fo the ase laye lines inclination equal to 30 degees [xls] Deducing the known foulas fo ou equations The eiods T, T, and T used in the liteatue ae couted as follows (see Figue 10): Page 12 of 25

13 T cos =, T = cos, T = cos (4.7) Fo hee, using equation (4.6) we deduce the well known foula fo the angle of oié lines [Aido00a]: T sin T sin = actan (4.8) T cos T cos Recall fo tigonoety the following sile foulas: cos( cos = 1 1+ tan 2 1 2) = cos 1 cos 2 + sin1 sin Fo equations (4.8) and (4.9) we have: cos = T 2 = + T 2 T 2 T cos T cos 2 T T T cos T + T 2 2 T Fo equations (4.2) and (4.7) we have: T T T ( cos cos + sin sin ) cos T cos( ) 2 (4.9) (4.10) = cos (4.11) T cos T cos Fo equations (4.10) and (4.11) we deduce the second well known foula fo the eiod of oié lines: T T T T = 2 2 (4.12) T + T 2 T T cos( ) Recall fo tigonoety that: 1 cos sin = (4.13) 2 2 In the aticula case when T = T, taking in account equation (4.13), equation (4.12) is futhe educed into well known foula: T = T (4.14) sin 2 2 Still fo the case when T = T, we can teoaily assue that all angles ae elative to the ase laye lines and ewite equation (4.8) as follows: Recall fo tigonoety that: sin = actan (4.15) cos 1 Page 13 of 25

14 zeo: 1 cos tan = 2 sin 1 tan( 90 + ) = tan Theefoe fo equations (4.15) and (4.16): (4.16) = 90 + (4.17) 2 Now fo the case when the evealing laye lines do not eesent the angle = (4.18) 2 We otain the well known foula [Aido00a]: + = 90 + (4.19) 2 Equations (4.8) and (4.12) ae the geneal case foulas known in the liteatue, and equations (4.14) and (4.19) ae the foulas fo otation of identical attens with aallel lines (i.e. the case when T = T ) [Aido00a], [Nishijia64a], [Oste63a], [Mose61a]. Assuing in the well known equation (4.8) that = 0, Figue 12 shows the chats of the oié lines degee as a function of the evealing laye s otation degee fo diffeent values of T / T / = T T 1.1 T T { 0.9, 0.925, K1.075, 1.1} oié lines's angle / = T T 1 / = T T 0.9 = actan sin T cos T evealing lines' angle Figue 12. Moié lines inclination as a function of the otation degee of the evealing laye [xls] Page 14 of 25

15 Only fo the case when T = T (the old cuve) the otation of oié lines is linea with esect to the otation of the evealing laye. Coaisons of Figue 12 and Figue 11 show the significant diffeence etween sheaing (i.e. inclination of lines) and otation of the evealing laye atten The evealing lines inclination as a function of the sueosition iage s lines inclination Fo equation (4.6) we can deduce the equation fo couting the evealing laye line inclination fo a given ase laye line inclination, and a desied oié line inclination : tan = tan + 1 tan (4.20) The inceent of the tangent of evealing lines angle ( tan( ) tan( ) ) elatively to the tangent of the ase laye lines angle can e exessed, as follows: tan ( tan tan ) tan = 1 (4.21) Accoding to equation (4.4), facto, and theefoe equation (4.21) can e ewitten as follows: tan tan v = tan tan v 1 is the invese of the otical acceleation (4.22) Equation (4.22) shows that elative to the tangent of the ase laye lines angle, the inceent of the tangent of the evealing laye lines angle needs to e salle than the inceent of the tangent of the oié lines angle, y the sae facto as the otical seedu. Fo any given ase laye line inclination, equation (4.20) eits us to otain a desied oié line inclination y oely choosing the evealing laye inclination. In Figue 7 we showed an exale whee the cuves of layes follow an identical inclination atten foing a sueosition iage with the sae inclination atten. The inclination degees of the layes and oié lines change along the hoizontal axis accoding the following sequence of altenating degee values (+30, 30, +30, 30, +30). In Figue 13 we otained the sae sueosition atten as in Figue 7, ut the ase laye consists of staight lines inclined y 10 degees. The evealing atten of Figue 13 is couted y inteolating the cuves into connected staight lines, whee fo each osition along the hoizontal axis, the evealing line s inclination angle is couted as a function of and, accoding to equation (4.20). Figue 13 deonstates what is aleady exessed y equation (4.22): the diffeence etween the inclination attens of the evealing laye and the ase laye ae seveal ties salle than the diffeence etween the inclination attens of oié lines and the ase laye lines. Page 15 of 25

16 Figue 13. The ase laye with inclined staight lines, the evealing laye couted so as to fo the desied sueosition iage [es], [ng] Anothe exale foing the sae sueosition attens as in Figue 7 and Figue 13 is shown in Figue 14. Note that in Figue 14 the desied inclination atten (+30, 30, +30, 30, +30) is otained using a ase laye with an inveted inclination atten ( 30, +30, 30, +30, 30). Figue 14. A sueosition iage, whee the ase laye and oié cuves ae ioed elatively to the hoizontal axis [es], [ng] The GIF aniation of Figue 15 shows a sueosition iage with a constant inclination atten of oié lines (+30, 30, +30, 30, +30) fo odifying ais of ase and evealing layes. The ase laye inclination atten gadually changes and the evealing laye inclination atten coesondingly adats such that the sueosition iage s inclination atten eains the sae. Page 16 of 25

17 Figue 15. The evealing, ase, and sueosition iages, whee the ase laye inclination atten gadually changes, and the evealing laye coesondingly adats such that the sueosition iage s inclination atten eains the sae [s], [tif], [gif] 5. Sueosition of eiodic cicula attens 5.1. Sueosition of cicula eiodic attens with adial lines Siilaly to laye and oié attens coising aallel lines (see Figue 1, Figue 2, and Figue 3), concentic sueosition of dense eiodic laye attens coising adial lines fos agnified eiodic oié attens also with adial lines. Figue 16 is the counteat of Figue 1, whee the hoizontal axis is elaced y the adius and the vetical axis y the angle. Full cicufeences of laye attens ae equally divided y intege nues of adial lines. The nue of adial lines of the ase laye is denoted as and the nue of adial lines of the evealing laye is n denoted as. n Page 17 of 25

18 Figue 16. Sueosition of two layes with egulaly saced adial segents (a otion of the evealing laye is cut out to show a at of the ase laye in the ackgound) [es], [ng] The eiods and denote the angles etween the cental adial axes of adjacent lines. Theefoe: = 360 n = 360 (5.1) n Accoding to equations (5.1), equation (4.2) can e ewitten as follows: Theefoe the nue of oié adial lines etween the nues of laye lines: n = 360 (5.2) n n n coesonds to the diffeence = n n (5.3) If in the laye attens, the full cicufeences ae divided y intege nues of laye lines, the cicufeence of the sueosition iage is also divided y an intege nue of oe lines. Radial lines in Figue 16 have constant angula thickness, giving the the fos of segents, thick at thei oute ends and thin at thei inne ends. The thickness of adial lines affects the oveall dakness of the sueosition iage and the width of oié ands, ut thee is no iact on othe factos, such as eiod of sueosition atten (i.e. values of and n ). In ou exales the angula thicknesses of laye Page 18 of 25

19 lines ae equal to the laye s half-eiod, i.e. the thickness of the ase laye lines is equal to / 2 and the thickness of the evealing laye lines is / 2. The otical seedu facto of equation (4.4) can e ewitten y elacing the eiods and y thei exessions fo equations (5.1): v v = (5.4) n n n The values v and v eesent the angula seeds. The negative seedu signifies a otation of the sueosition iage in a diection invese to the otation of the evealing laye. Consideing (5.3), the asolute value of the otical seedu facto is: v v n n = (5.5) In Figue 16, the nue of adial lines of the evealing laye is equal to 180, and the nue of adial lines of the ase laye is 174. Theefoe, accoding to equations (5.4) and (5.3), the otical seedu is equal to 30, confied y the two iages (a) and () of Figue 17, and the nue of oié lines is equal to 6, confied y the iage of Figue 16. Figue 17. Rotation of the evealing laye y 1 degee in the clockwise diection otates the otical iage y 30 degees in the sae diection [es (a)], [ng (a)], [es ()], [ng ()] Figue 18 shows a GIF aniation of the sueosition iage of Figue 16, whee the evealing laye slowly otates in the clockwise diection. Page 19 of 25

20 Figue 18. GIF aniation of the sueosition iage of the layes with eiodic adial lines, whee the evealing laye slowly otates clockwise [s], [tif], [gif] 5.2. Sueosition of cicula attens with adial cuves In cicula eiodic attens cuved adial lines can e constucted using the sae efeence sequences of inclination degees as used in section 4.3 fo cuves of Figue 7. The inclination angle at any oint of the adial cuve coesonds to the angle etween the cuve and the axis of the adius assing though the cuent oint. Thus inclination angle 0 coesonds to staight adial lines as shown in Figue 16. With the esent notion of inclination angles fo,, and, equations (4.6) and (4.20) ae alicale fo cicula attens without odifications. Cuves can e constucted inceentally with a constant adial inceent equal to. Figue 19 shows a segent of a cuve, aked y a thick line, which has an inclination angle equal to. β tan() Figue 19. Constucting a cuve in a ola coodinate syste with a desied inclination While constucting the cuve, the cuent angula inceent β ust e couted so as to esect the inclination angle : tan( ) 180 β = actan tan( ) + π (5.6) Page 20 of 25

21 Figue 20 shows a sueosition of layes with cuved adial lines. The inclination of cuves of oth layes follows an identical atten coesonding to the following sequence of degees (+30, 30, +30, 30, +30). Laye cuves ae iteatively constucted with inceent ais (, β ) couted accoding to equation (5.6). Since the inclination attens of oth layes of Figue 20 ae identical, the oié cuves also follow the sae atten. Figue 20. Sueosition of layes in a ola coodinate syste with identical inclination attens of cuves coesonding to (+30, 30, +30, 30, +30); a otion of the evealing laye is cut away exosing the ase laye in the ackgound [es], [ng], ulti-age [tif], [gif] Siilaly to exales of Figue 7, Figue 13, and Figue 14, whee the sae oié atten is otained y sueosing diffeent ais of laye attens, the cicula oié atten of Figue 20 can e analogously otained y sueosing othe ais of cicula laye attens. Taking into account equations (5.1), equations (4.6) and (4.20) can e ewitten as follows: n tan n tan tan = (5.7) n n tan n tan 1 n tan = + n n (5.8) Thanks to equation (5.8), othe ais of laye attens can e ceated (see Figue 21) which oduce the sae sueosition iage as in Figue 20. In the fist iage (a) of Figue 21, the ase laye lines ae staight. In the second iage (), the ase laye lines inclination atten is evesed with esect to the oié lines. Page 21 of 25

22 Figue 21. Sueosition iages with identical inclination atten (+45, 45, +45, 45, +45) of oié cuves, whee in one case the ase laye coise staight adial segents, and in the second case the ase laye coise cuves which ae the ioed counteats of the esulting oié cuves [es (a)], [ng (a)], [es ()], [ng ()] Figue 22 shows an aniation, whee the oié cuves of the sueosition iage ae always the sae, ut the inclination atten of the ase laye cuves gadually altenates etween the following two io attens (+45, 45, +45, 45, +45), and ( 45, +45, 45, +45, 45). Fo each instance of the aniation, the evealing laye lines ae couted accoding to equation (5.8) in ode to constantly aintain the sae oié atten. Figue 22. Aniation of the sueosition iage, whee the ase laye lines gadually change thei inclination atten fo (+45, 45, +45, 45, +45) to ( 45, +45, 45, +45, 45) [es], [tif], [gif] Equations (5.4) and (5.3) eain valid fo attens with cuved adial lines. In Figue 20 thee ae 180 cuves in the evealing laye and 171 cuves in the ase laye. Theefoe otical seedu facto accoding to equation (5.4) is equal to 20, and the nue of oié cuves accoding to equation (5.3) is equal to 9, as seen in the sueosition iage of Figue 20. Page 22 of 25

23 6. Conclusions We edeveloed the ost iotant foulas fo couting the eiods, inclination angles of oié attens, and the velocities of otical shaes. Instead of using the well known equations, we eesent the case of inclined attens such that equations (4.2), (4.3), and (4.4) fo linea attens and thei counteats (5.3), (5.5), and (5.4) fo cicula attes, eain valid in thei sile fos. In ou equations, the values eesent the eiods along the axis of the oveent of the evealing laye. In section we coaed ou foulas with the foulas known in the liteatue. 7. Refeences [Aido00a] [Aido03a] [Nishijia64a] [Oste63a] [Sciaaella62a] [Mose61a] 8. Tale of figues Isaac Aido, The Theoy of the Moié Phenoenon, Kluwe Acadeic Pulishe, 2000 [CH], [US] Isaac Aido, Glass attens in the sueosition of ando line gatings, Jounal of Otics A: Pue and Alied Otics, 28 Mach 2003, [CH], [US] Y. Nishijia and G. Oste, Moié attens: thei alication to efactive index and efactive index gadient easueents, Jounal of the Otical Society of Aeica, Vol. 54, No. 1, Januay 1964,. 1-5 [CH], [US] G. Oste and Y. Nishijia, Moié attens, Scientific Aeican, Vol. 208, May 1963, C. A. Sciaaella and A. J. Duelli, Moié finges as a eans of analyzing stains, Aeican Society of Civil Enginees, Vol. 127, Pat I, 1962, [CH], [US] Stanley Mose, August J. Duelli, and Cesa A. Sciaaella, Geoety of oié finges in stain analysis, Aeican Society of Civil Enginees, Vol. 126, Pat I, 1961, [CH], [US] Figue 1. Sueosition of two layes consisting of aallel lines, whee the lines of the evealing laye ae aallel to the lines of the ase laye [es], [tif], [ng]... 2 Figue 2. Sueosition of two layes consisting of hoizontal aallel lines [es], [tif], [ng]... 3 Figue 3. Couting the eiod of oié lines in a sueosition iage as a function of the eiods of lines of the evealing and ase layes... 3 Figue 4. Sueosition of two layes with aallel hoizontal lines, whee the evealing laye oves vetically at a slow seed [es (a)], [ng (a)], [es () ], [ng ()], [es (c)], [ng (c)]... 5 Figue 5. GIF aniation of the slow vetical oveent of the evealing laye [s], [gif], [tif]... 6 Page 23 of 25

24 Figue 6. Sueosition of layes consisting of inclined aallel lines whee the lines of the ase and evealing layes ae inclined at the sae angle [es], [ng]... 7 Figue 7. Two layes consisting of cuves with identical inclination attens, and the sueosition iage of these layes [es], [ng]... 8 Figue 8. Sueosition of layes consisting of inclined aallel lines, whee the ase laye lines inclination is 10 degees and the evealing laye lines inclination is 7, 9, 11, and 13 degees [es (a)], [ng (a)], [es ()], [ng ()], [es (c)], [ng (c)], [es (d)], [ng (d)]... 9 Figue 9. GIF aniation, whee the inclination of aallel lines of the evealing laye oscillates etween 5 and 15 degees [s], [gif], [tif] Figue 10. Couting the inclination angle of oié lines as a function of inclination angles of the ase laye and evealing laye lines Figue 11. Moié lines inclination as a function of the evealing laye lines inclination fo the ase laye lines inclination equal to 30 degees [xls] Figue 12. Moié lines inclination as a function of the otation degee of the evealing laye [xls] Figue 13. The ase laye with inclined staight lines, the evealing laye couted so as to fo the desied sueosition iage [es], [ng] Figue 14. A sueosition iage, whee the ase laye and oié cuves ae ioed elatively to the hoizontal axis [es], [ng] Figue 15. The evealing, ase, and sueosition iages, whee the ase laye inclination atten gadually changes, and the evealing laye coesondingly adats such that the sueosition iage s inclination atten eains the sae [s], [tif], [gif] Figue 16. Sueosition of two layes with egulaly saced adial segents (a otion of the evealing laye is cut out to show a at of the ase laye in the ackgound) [es], [ng] Figue 17. Rotation of the evealing laye y 1 degee in the clockwise diection otates the otical iage y 30 degees in the sae diection [es (a)], [ng (a)], [es ()], [ng ()] Figue 18. GIF aniation of the sueosition iage of the layes with eiodic adial lines, whee the evealing laye slowly otates clockwise [s], [tif], [gif] Figue 19. Constucting a cuve in a ola coodinate syste with a desied inclination Figue 20. Sueosition of layes in a ola coodinate syste with identical inclination attens of cuves coesonding to (+30, 30, +30, 30, +30); a otion of the evealing laye is cut away exosing the ase laye in the ackgound [es], [ng], ulti-age [tif], [gif] Figue 21. Sueosition iages with identical inclination atten (+45, 45, +45, 45, +45) of oié cuves, whee in one case the ase laye coise staight adial segents, and in the second case the ase laye coise cuves which ae the ioed counteats of the esulting oié cuves [es (a)], [ng (a)], [es ()], [ng ()] Figue 22. Aniation of the sueosition iage, whee the ase laye lines gadually change thei inclination atten fo (+45, 45, +45, 45, +45) to ( 45, +45, 45, +45, 45) [es], [tif], [gif] Page 24 of 25

25 9. Links (070212) Rando oié [CH], [US] (070227) Rando line oié [CH], [US] (070306) This we site: eiodic line oié attens and otical seedu [CH], [US] Foat [doc], [df], [ht], [ht (s)] * * * Page 25 of 25