Content 1. Experimental Setup... 2

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Kenneth Watkins
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1 Cotet 1. Exerimetal Setu.... Theory Reflectio Selliu law Parallel hift Total iteral reflectio Beam roagatio through rim Prim dierio Gratig dierio Birefrigece ad otical activity... 13

$1. Exerimetal Setu Uig the etu how i FIGURE 1 variou otical heomea like refractio, reflectio, ad total iteral reflectio will be ivetigated.$

2 1. Exerimetal Setu Uig the etu how i FIGURE 1 variou otical heomea like refractio, reflectio, ad total iteral reflectio will be ivetigated. With the laer module, mouted o the movig rail, the laer light ca be directed o differet amle. The et of amle iclude a lae-arallel late, tramiio cavity, rim, ad a movig table with mall rim ad lee. The trace of the laer beam ca be ee very well i the et u. The beam will be rojected o a cree ad mall beam termiator. The differet heomea will be ivetigated uig ecial temlate reared for each tet, o which all beam trajectorie have to be draw. FIGURE 1: Setu for traarecy tet.

3 . Theory Light exit a log a time, howig image from the begiig of the uivere. The imortace of light became eve higher with the ivetio of the laer: from thi time o light wa ot oly iformatio carrier, but became a bae of key techologie i 1 t cetury. Emloymet of laer accelerated idutrial roceig i weldig, cuttig, drillig, ad rofilig ad at the ame time icreaed the reciio of work. I medicie the laer techologie are ued ractically i all dicilie every day, for examle, i edocoy baed o a total iteral reflectio heomea. The modulated otical igal tramit iformatio with the velocity of light through tramiio ytem ad waveguide. Further develomet of laer ytem led to a creatio of mall module coverig wide ectrum rage, icludig wavelegth viible for huma eye..1. Reflectio The traitio betwee two media lit the beam ito two art the reflected ad tramitted beam. I cae of gla, the reflected beam ha 4%, while the tramitted oe ha 96% of the ower relative to the icomig beam ower. The trajectory of the reflected light i decribed by the reflectio law, ad the trajectory of the tramitted light i decribed by a Selliu law ' '' ' (1.1) It defie the relatio betwee the icidet agle ad agle of reflectio (FIGURE ). 1. iterface air Medium. iterface gla air FIGURE : Tramitted ad reflected beam trajectorie.

4 .. Selliu law Selliu law decribe the relatio betwee the icomig ad refracted agle uig α ad β, which are the refractive idice of the media i (1.) We coider beam trajectorie how i FIGURE. The α-beam emitted by a laer i lit at the iterface of urface 1 ito a reflected ad a tramitted beam. The β-beam i tramitted through the medium (here: lexigla). O the iterface of urface the β-beam i lit agai ito a tramitted beam αtra, which roagate further i medium 1 (here: air) ad a reflected beam β. The beam β i lit agai o the iterface of urface 1, but oly beam α i obervable. The iteity of α i almot equal to the iteity of α. The ecod beam α ca be trogly cattered by otical elemet. To miimize thi effect, otical comoet uually have atireflectio coatig. The refractive idice i have ubcrit (α, β, γ etc.), which deote the refractive idex relative to vacuum, which i the miimum oible refractive idex for ay traaret media. Due to the extremely mall differece betwee refractive idexe of vacuum ad air, i literature, they are ofte coidered a idetical vacuum air (1.3) A accuracy of agle meauremet i the lab i about 0.1. Becaue of that, the refractive idex of air i alo give oly with the accuracy of two digit: α = Table 1: Variou, relative refractive idice. Otical medium Refractive idex relative to vacuum Refractive idex ued i thi lab air air = air = 1.00 crow gla K13 K13 = K13 = 1.5 flit gla F F = F = 1.6 heavy flitgla SF3 SF3 = SF3 = 1.74 diamod diamod = diamod =.4

5 The value of the refractive idex at a iterface determie the trajectory at the urface. The refractive idex i alway greater the 1, i 1, but it relative couterart ca have ay value. Table : Differet relative refractive idice ad beam trajectorie. Relative refractive idex at iterface 1 Beam trajectory chage refractive beam get cloer to eredicular Examle air to gla 1 o refractio ame media 1 refractive beam move away from eredicular gla to air.3. Parallel hift We coider agai beam roagatio eterig ad exitig a lae arallel late (FIGURE ). The beam α, icidet o a lae arallel late at the agle 0 α, roagate through thi late with a width d ad agle β to a eredicular. O the iterface of urface, the beam exit the late at the agle αtra. There i o chage i roagatio directio, but rather a hift i comario to a trajectory without the lae arallel late. Thi arallel hift i eaily calculated. We firt calculate the agle β i the medium ad tramitted agle αtra ito air uig Selliu law twice. Calculatio of i gla Calculatio of tra i air 0 (1.4) i (1.5) tra i (1.6)

6 (1.5) i (1.6): tra (1.7) yield ad the arallel hift b i give by the followig equatio tra, (1.8) b d. (1.9) co.4. Total iteral reflectio We ivetigate thi effect at the traitio betwee water ad air (FIGURE 3). We deote air a medium 1 ad water a a medium 3. A beam γ i lit ito a refracted beam α ad reflected oe γ at the iterface betwee water ad air. We icreae the agle γ ad oberve the chage of the tramitted agle α. At ome articular agle γ the refracted beam α vaihe ad, oly the reflected beam γ i left. medium 1: air medium : water FIGURE 3: Traitio of light from dee to le dee medium. The roduct 1 (1.10) with (1.11)

7 ha o olutio, becaue the fuctio y = i(x) i ot defied for y 1. Therefore, we do ot oberve the beam aymore, which leave the water o the to urface (FIGURE 3). Thi agle i deoted a the agle of total iteral reflectio, ad the heomeo i called total iteral reflectio (TIR). The agle of total iteral reflectio i give by arci (1.1) TIR The total iteral reflectio heomeo i ued i medical edocoe ad otical waveguide..5. Beam roagatio through rim I cae of the roagatio of a beam through a rim, the icomig beam will be refracted twice toward the directio of bai of the rim. The total refractio δ i calculated uig the followig equatio (1.13) 1 1 i (1.14) 1 1 (1.15) 1 1 (1.16) 1 1 (1.17) 1 1 i (1.18) If the agle 1 i le tha 1, total iteral reflectio take lace o the other ide of the rim.

$FIGURE 4: Refractio ad reflectio i a rim. Agle of miimum deviatio betwee the icidet ad tramitted beam aear whe the beam roagate arallel to the bae iide of the rim 1.$

8 FIGURE 4: Refractio ad reflectio i a rim. Agle of miimum deviatio betwee the icidet ad tramitted beam aear whe the beam roagate arallel to the bae iide of the rim 1. I thi cae, the refractive idex of the rim ca be calculated accordig to the exreio mi, (1.19) i where γ i the rim agle..6. Prim dierio Dierio of a rim exhibit itelf a a wavelegth deedece of the refractio agle of the beam. For examle, two beam with differet wavelegth (red ad gree) are refracted by a differet agle (FIGURE 5).

9 Dierio of a rim i characterized by: FIGURE 5: Diffractio i a rim. red Dierio red gree gree (1.0) It ca be how that the dierio ha maximum value for the beam roagatig arallel to the bae of the rim, which correod to the agle of miimum icliatio, e.g. mi, dev 1 (FIGURE 5) betwee the cotiuatio (dahed lie) of the icidet ad refracted beam..7. Gratig dierio If a lae wave i diffracted o a gratig (FIGURE 6), each oit of the gratig create ecodary wave, which iterfere with differet hae hift reultig i differet diffractio order. A gratig coit of may thi lit with eriod d. A gratig i characterized by a umber of lit N er mm, havig tyical value of 100mm -1, 300 mm -1 or 600 mm -1. Each wavelegth i diffracted at the agle Θ, which i calculated by: k i, k 0, 1,,... (1.1) d Thi equatio how that the loger wavelegth (red) i diffracted by a larger agle comared to the horter oe (gree). Due to cotructive iterferece, two ymmetric et of differet order aear o the cree.

$diffractio d order left beam gratig 1 t order left 0 order 1 t order right cree d order right FIGURE 6: Gratig geerate differet diffractio order, each of them cotaiig a diffracted ectrum. 1.1. Light a electromagetic wave We coider a electromagetic wave.$ lae of icidece lae of icidece FIGURE 7: A electromagetic wave E B x (left); electromagetic wave olarized i icidet lae E z ad eredicular to the icidet lae E y (right).

lae of icidece lae of icidece FIGURE 7: A electromagetic wave E B x (left); electromagetic wave olarized i icidet lae E z ad eredicular to the icidet lae E y (right).

10 diffractio d order left beam gratig 1 t order left 0 order 1 t order right cree d order right FIGURE 6: Gratig geerate differet diffractio order, each of them cotaiig a diffracted ectrum Light a electromagetic wave We coider a electromagetic wave. Thi wave coit of electric E ad magetic H field, orthogoal to each other ad roagatig alog the x - axi. The fudametal roertie of the wave roagatio are decribed by the Maxwell equatio. lae of icidece lae of icidece FIGURE 7: A electromagetic wave E B x (left); electromagetic wave olarized i icidet lae E z ad eredicular to the icidet lae E y (right). Whe olarizatio effect are dicued, oly the electric field i coidered. Here we coider the roagatio i x - directio ad rojectio of the field olarizatio o the icidet lae xz ad eredicular to the icidet lae. Brewter agle (David Brewter, )

11 The olarizatio roertie of a electromagetic wave ca be how at reflectio uder Brewter agle, where oe of the olarizatio exeriece o reflectio. The electric field coit of arallel E ad eredicular E comoet the icidet e e lae field. Oe ca calculate reflectio ad tramiio coefficiet for the both wave (tramiio i ot coidered hereafter). r E r (1.) Ee ad r E r (1.3) E e ad calculate iteity reflectio for both olarizatio R r (1.4) R r (1.5) Uder the Brewter agle, oly oe olarizatio E i reflected ( R 0 ), while the other oe i comletely tramitted ( R 0 ). lae of icidece eredicular iterface 1 FIGURE 8: Reflectio of a icidet electromagetic wave for arallel E ad erediculare to the icidet lae olarizatio uder Brewter agle: oly oe olarizatio E i reflected ( R 0 ), while the other oe i fully tramitted ( R 0 ). e

12 The Brewter agle i a ecial olutio of the Freel equatio, decribig Br reflectio ad tramiio at boudarie. Reflectio for olarizatio eredicular to the icidet lae: R r i i (1.6) Reflectio for olarizatio arallel to the icidet lae: R r ta ta (1.7) The Brewter agle i give by the coditio R 0 : Br g 90, (1.8) whe the reflected ad refracted beam are eredicular to each other. The tramiio of the wave with the olarizatio arallel to the icidet lae for thi agle i 100%. Brewter agle calculatio Alyig Selliu law, oe ca eaily calculate the Brewter agle Br i 90 Br (1.9) Br co (1.30) The, the o-called exteral Brewter agle from le dee to dee media i I cae of traitio from air: Br ta Br (1.31) ta Br (1.3)

13 The agle deedece of the iteity reflectio coefficiet R ad olarizatio are how i FIGURE 9. R for both FIGURE 9: Iteity reflectio coefficiet for the both olarizatio i cae of traitio from air to lexigla..8. Birefrigece ad otical activity Birefriget crytal have differet refractive idice for differet roagatio directio. Here, we deal oly with the igle axi birefriget crytal, which have to refractive idice, amely o (ordiary) ad eo (extraordiary). otical axi o eo differet directio o eo FIGURE 10: Otical axi of a birefriget crytal. Tramiio i birefriget crytal cut arallel to otical axi A lae wave, travellig through a birefriget crytal which i cut arallel to a otical axi, how a rotatio of the olarizatio. Thi effect ca be detected uig two croed olarizer. Suoe, the icomig, uolarized wave frot coit of two wave frot with two mutually eredicular olarizatio tate E ad E. At the olarizer P e (FIGURE 11) the wave frot with the olarizatio tate E will be aborbed. The wave e e

frot after the olarizer P i liearly olarized. I cae of the ecod olarizer i laced eredicular to the firt oe, the ecod wave frot E, ad hece the tramiio, e i comletely blocked.

After the firt olarizer, the liearly olarized wave frot roagate through the crytal. The olarizatio lae i rotated, ad the ecod olarizer doe ot block the light comletely.

14 frot after the olarizer P i liearly olarized. I cae of the ecod olarizer i laced eredicular to the firt oe, the ecod wave frot E, ad hece the tramiio, e i comletely blocked. cacellatio of E e cacellatio of E e FIGURE 11: Tramiio i blocked by two eredicular aliged olarizer. The birefriget crytal i laced betwee the olarizer with otical axe arallel to the roagatio directio. After the firt olarizer, the liearly olarized wave frot roagate through the crytal. The olarizatio lae i rotated, ad the ecod olarizer doe ot block the light comletely. After agle adjutmet of the ecod olarizer the rotatio agle ca be meaured. The rotatio agle deed o the wavelegth ad thicke of the crytal. The otical activity i how here with SiO; right- ad left rotatig crytal exit, deedig o the crytal tructure. oberver otical axi See of rotatio for olarizer FIGURE 1: Liearly olarized wave roagate arallel to otical axi of a birefriget crytal. The olarizatio lae rotate durig the roagatio through the crytal.

INF-GEO Solutions, Geometrical Optics, Part 1

INF-GEO Solutions, Geometrical Optics, Part 1 INF-GEO430 20 Solutios, Geometrical Optics, Part Reflectio by a symmetric triagular prism Let be the agle betwee the two faces of a symmetric triagular prism. Let the edge A where the two faces meet be