Optics Formulas. is the wave impedance of vacuum, and η is the wave impedance of a medium with refractive index n. Wave Quantity Relationship.

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1 Optics 57 Light Right-Had Rule Light is a trasverse electromagetic wave. The electric ad magetic M fields are perpedicular to each other ad to the propagatio vector k, as show below. Power desity is give by Poytig s vector, P, the vector product of ad H. You ca easily remember the directios if you curl ito H with the figers of the right had: your thumb poits i the directio of propagatio. Light Itesity H P k P = x H The light itesity, I is measured i Watts/m, i Volts/m, ad H i Amperes/m. The equatios relatig I to ad H are quite aalogous to OHMS LAW. or peak values these equatios are: = ηh, H=, η = η H H ηh I =, I =, I = η I = ηi, H= η η = 377 ohms ( Ω η = η Optics ormulas The quatity η is the wave impedace of vacuum, ad η is the wave impedace of a medium with refractive idex. Wave Quatity Relatioship π π k = = λ λ πν ω = = c c c c ν = = λ λ kc ω = = π π c λ λ = = ν π πc = = k ω k: wave vector [radias/m] ν: frequecy [Hertz] ω: agular frequecy [radias/sec] λ: wavelegth [m] λ : wavelegth i vacuum [m] : refractive idex ergy Coversios Wave Number ( ν[ cm ] = λ µm lectro volts ev per photo =. λ µm Phoe: -8-- ax: mail: sales@ewport.com Web: ewport.com ( [ ] [ ] Itesity Nomogram The omogram below relates, H, ad the light itesity I i vacuum. You may also use it for other area uits, for example, [V/mm], [A/mm] ad [W/mm ]. If you chage the electrical uits, remember to chage the uits of I by the product of the uits of ad H: for example [V/m], [ma/m], [mw/m ] or [kv/m], [ka/m], [MW/m ]. [V/m] 5 5 I [W/m ] H [A/m] WINOWS BAMSPLITTRS ILTRS & ATTNUATORS POLARIZATION OPTICS ULTRAAST LASR OPTICS ACCSSORIS TCHNICAL RRNC

2 58 O ptics BAMSPLITTRS WINOWS ILTRS & ATTNUATORS Wavelegth Coversios m = Agstroms(Å = 9 m = 7 cm = 3 µm Plae Polarized Light or plae polarized light the ad H fields remai i perpedicular plaes parallel to the propagatio vector k as show below. H λ k x The omogram relates waveumber, photo eergy ad wavelegth. /λ [cm ] [µm] [m] 5.. SOT X RAY VACCUM UV 3 NAR UV VIOLT R 3 NAR IR hv [ev] Sell s Law Sell s Law describes how a light ray behaves whe it passes from a medium with idex of refractio, to a medium with a differet idex of refractio,. I geeral the light will eter the iterface betwee the two medii at a agle. This agle is called the agle of icidece. It is the agle measured betwee the ormal to the surface (iterface ad the icomig light beam (see figure. I the case that is smaller tha, the light is bet towards the ormal. If is greater tha, the light is bet away from the ormal (see figure below. Sell s Law is expressed as siθ = siθ θ TCHNICAL RRNC ACCSSORIS ULTRAAST LASR OPTICS POLARIZATION OPTICS Both ad H oscillate i time ad space as: si (ωt-kx AR IR 5 Phoe: -8-- ax: mail: sales@ewport.com Web: ewport.com. > > θ θ θ

3 Optics 59 Beam isplacemet A flat piece of glass ca be used to displace a light ray laterally without chagig its directio. The displacemet varies with the agle of icidece; it is zero at ormal icidece ad equals the thickess h of the flat at grazig icidece. (Grazig icidece: light icidet at almost or close to 9 to the ormal of the surface. h The relatioship betwee the tilt agle of the flat ad the two differet refractive idices is show i the graph below. θ d= hsi θ ISPLACMNT/THICKNSS (d/h..5 N = N cos θ si.5 d θ 3 9 TILT ANGL (GRS Beam eviatio Both displacemet ad deviatio occur if the media o the two sides of the tilted flat are differet for example, a tilted widow i a fish tak. The displacemet is the same, but the agular deviatio δ is give by the formula. Note that δ is idepedet of the idex of the flat; it is the same as if a sigle boudary existed betwee media ad 3. xample: The refractive idex of air at STP is about.3. The deviatio of a light ray passig through a glass Brewster s agle widow o a HeNe laser is the: δ= ( 3 - ta θ At Brewster s agle, ta θ= δ= (.3 x.5 =.5 mrad At, ft. altitude, air pressure is /3 that at sea level; the deviatio is.3 mrad. This chage may misalig the laser if its two widows are symmetrical rather tha parallel. Phoe: -8-- ax: mail: sales@ewport.com Web: ewport.com θ δ = θ si si θ 3 ta θ, if ( δ Agular eviatio of a Prism Agular deviatio of a prism depeds o the prism agle α, the refractive idex,, ad the agle of icidece θ i. Miimum deviatio occurs whe the ray withi the prism is ormal to the bisector of the prism agle. or small prism agles (optical wedges, the deviatio is costat over a fairly wide rage of agles aroud ormal icidece. or such wedges the deviatio is: δ ( - α VIATION (GRS 9 3 θ i 3 9 α 5 3 INCINT ANGL (GRS α δ WINOWS BAMSPLITTRS ILTRS & ATTNUATORS POLARIZATION OPTICS ULTRAAST LASR OPTICS ACCSSORIS TCHNICAL RRNC

4 57 O ptics ILTRS & ATTNUATORS BAMSPLITTRS WINOWS TCHNICAL RRNC ACCSSORIS ULTRAAST LASR OPTICS POLARIZATION OPTICS Prism Total Iteral Reflectio (TIR TIR depeds o a clea glass-air iterface. Reflective surfaces must be free of foreig materials. TIR may also be defeated by decreasig the icidece agle beyod a critical value. or a right agle prism of idex, rays should eter the prism face at a agle θ: θ < arcsi ((( - / -/ I the visible rage, θ = 5.8 for BK 7 ( =.57 ad. for fused silica ( =.. ially, prisms icrease the optical path. Although effects are miimal i laser applicatios, focus shift ad chromatic effects i diverget beams should be cosidered. resel quatios: i - icidet medium t - trasmitted medium use Sell s law to fid θ t Normal Icidece: r = ( i - t /( i + t t = i /( i + t Brewster's Agle: θ β = arcta ( t / i Oly s-polarized light reflected. Total Iteral Reflectio (TIR: θ TIR > arcsi ( t / i t < i is required for TIR ield Reflectio ad Trasmissio Coefficiets: The field reflectio ad trasmissio coefficiets are give by: r = r / i t = t / i No-ormal Icidece: r s = ( i - t /( i + t r p = ( t cos θ i - i / t + i t s = i /( i + t t p = i /( t + i Power Reflectio: The power reflectio ad trasmissio coefficiets are deoted by capital letters: R = r T = t ( t /( i The refractive idices accout for the differet light velocities i the two media; the cosie ratio corrects for the differet cross sectioal areas of the beams o the two sides of the boudary. The itesities (watts/area must also be corrected by this geometric obliquity factor: I t = T x I i ( / Coservatio of ergy: R + T = This relatio holds for p ad s compoets idividually ad for total power. Polarizatio To simplify reflectio ad trasmissio calculatios, the icidet electric field is broke ito two plae polarized compoets. The plae of icidece is deoted by the wheel i the pictures below. The ormal to the surface ad all propagatio vectors (k i, k r, k t lie i this plae. parallel to the plae of icidece; p-polarized. PLAN O INCINC NORMAL TO SURAC SURAC Phoe: -8-- ax: mail: sales@ewport.com Web: ewport.com i H i θ i k i ormal to the plae of icidece; s-polarized. PLAN O INCINC i H i θ i k i θ r t θ t r θ r θ t r H r K t K t t H t H t k r H r k r SURAC

5 Optics 57 Power Reflectio Coefficiets Power reflectio coefficiets ad are plotted liearly ad logarithmically for light travelig from air ( i = ito BK 7 glass ( t =.573. Brewster s agle = 5.. RACTION RLCT LOG RACTION RLCT RSNL RLCTION OR BK 7 N BRWSTR S ANGL INCINT ANGL (GRS RSNL RLCTION OR BK 7 N BRWSTR S ANGL INCINT ANGL (GRS The correspodig reflectio coefficiets are show below for light travelig from BK 7 glass ito air Brewster s agle = 33.. Critical agle (TIR agle =.5. RACTION RLCT LOG RACTION RLCT RSNL RLCTION OR BK 7 BRWSTR S ANGL 33. N CRITICAL ANGL INCINT ANGL (GRS RSNL RLCTION OR BK 7 N CRITICAL ANGL.5 BRWSTR S ANGL INCINT ANGL (GRS Thi Les quatios If a les ca be characterized by a sigle plae the the les is thi. Various relatios hold amog the quatities show i the figure. Gaussia: Newtoia: x x = - X + = s s Magificatio: Phoe: -8-- ax: mail: sales@ewport.com Web: ewport.com Y X Sig Covetios for Images ad Leses Quatity + - s real virtual s real virtual covex les cocave les Les Types for Miimum Aberratio S s /s Best les <. plao-covex/cocave >5 plao-covex/cocave >. or <5 bi-covex/cocave X S BL P V H TC H V Trasverse: M = y T y M T <, image iverted Logitudial: M L = s s x = = M x M L <, o frot to back iversio S X Thick Leses Y X A thick les caot be characterized by a sigle focal legth measured from a sigle plae. A sigle focal legth may be retaied if it is measured from two plaes, H, H, at distaces P, P from the vertices of the les, V, V. The two back focal legths, BL ad BL, are measured from the vertices. The thi les equatios may be used, provided all quatities are measured from the pricipal plaes. BL P S X T WINOWS BAMSPLITTRS ILTRS & ATTNUATORS POLARIZATION OPTICS ULTRAAST LASR OPTICS ACCSSORIS TCHNICAL RRNC

6 57 O ptics Les Nomogram: BAMSPLITTRS WINOWS 8 This omogram solves the Gaussia Les equatio. The dotted lie shows the example S =, ter ay two of S, S or, ad draw a straight lie to fid the third. Two differet scales are 3 =, S = or the example S =, =.33, S =. You may multiply by powers of ad give, oe o the clockwise sides of the 8 use ay uits. axes ad oe o the couterclockwise 5 sides. 3 8 S 3 8 S This omogram is simply three uiform scales itersectig at. You may make your ow i ay size. This omogram may also be used to fid the impedace of parallel resistors, parallel iductors or series capacitors, or their duals. 5 8 TCHNICAL RRNC ACCSSORIS ULTRAAST LASR OPTICS POLARIZATION OPTICS ILTRS & ATTNUATORS The Lesmaker s quatio Covex surfaces facig left have positive radii. Below, R >, R <. Pricipal plae offsets, P, are positive to the right. As illustrated, P >, P <. The thi les focal legth is give whe T c =. P T C P R V H H V = Tc P = R Tc P = R R ( ( + T c R R RR ( ( Numerical Aperture φ MAX is the full agle of the coe of light rays that ca pass through the system (below. MAX NA = si φ or small φ: SYSTM f /# = NA Both f-umber ad NA refer to the system ad ot the exit les. Costats ad Prefixes Speed of light i vacuum c = m/s Plack s cost. h =.5 x -3 Js Boltzma s cost. k =.38 x -3 J/K Stefa-Boltzma σ = 5.7 x -8 W/m K electro volt ev =. x -9 J exa ( 8 peta (P 5 tera (T giga (G 9 mega (M kilo (k 3 milli (m -3 micro (µ - ao ( -9 pico (p - femto (f -5 atto (a -8 Phoe: -8-- ax: mail: sales@ewport.com Web: ewport.com

7 Optics 573 Wavelegths of Commo Lasers Source (m Ar 93 Kr 8 Nd:YAG( XeCl 38 HeCd 35,. N 337., 7 Xe 35 Nd:YAG( Ar 88, 5.5, 35., 33.8 Cu 5., 578. Nd:YAG( 53 HeNe 3.8, 53.5, 59.,.9, 53, 53 Kr 7., 7. Ruby 9.3 Nd:Glass Nd:YAG, 39 Ho:YAG r:yag 9 Gaussia Itesity istributio The Gaussia itesity distributio: I(r = I( exp(-r /ω is show below. RLATIV INTNSITY e - e NORMALIZ RAIUS (r/ω The right had ordiate gives the fractio of the total power ecircled at radius r: [ ( ] Pr ( = P( exp r / ω The total beam power, P( [watts], ad the o-axis itesity I( [watts/area] are related by: P( =( πω / I( I( = ( / πω P( iffractio The figure below compares the farfield itesity distributios of a uiformly illumiated slit, a circular hole, ad Gaussia distributios with /e diameters of ad. (99% of a. Gaussia will pass through a aperture of diameter. The poit of observatio is Y off axis at a distace X>Y from the source. LOG INTNSITY Y/ λx Phoe: -8-- ax: mail: sales@ewport.com Web: ewport.com. ocusig a Collimated Gaussia Beam I the figure below the /e radius, ω(x, ad the wavefrot curvature, R(x, chage with x through a beam waist at x =. The goverig equatios are: ω ( x ω λx/ ω Rx ( x πω / λx = +( = +( ω is the waist diameter at the /e itesity poits. The wavefrots are plaar at the waist [R( = ]. At the waist, the distace from the les will be approximately the focal legth: s. = collimated beam diameter or diameter illumiated o les. S f-umber f/# = ω X = R(X ω(x θ X epth of ocus (O O = (8λ/π(f/# Oly if O <, the: New Waist iameter ω = f λ = λ (/# π π Beam Spread θ= (/# f WINOWS BAMSPLITTRS ILTRS & ATTNUATORS POLARIZATION OPTICS ULTRAAST LASR OPTICS ACCSSORIS TCHNICAL RRNC

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