Paraxial Beam, Gaussian Basics

Transcription

1 Paraxia Beam, Gaussian Basics ECE 5368/6358 han q e - copyrighted Use soey for students registered for UH ECE 6358/5368 during courses - DO NOT distributed (copyrighted materias). Introduction. Paraxia approximation for Gaussian beam (can skip). Important properties of Gaussian beam. Beam description From the above (we drop the subscript from k for simpicity):

2 LectSet - Gaussian beam basic 5358_p.nb E z, r = p wk  Q z  k z+ r Q z (..a) where: Q z = z - ; = kw = p w (..b) Note: we have E[z,r] in the unit of so that the normaized profie: ength p E z, r r r q= is unitess. Rea E has rea physica unit, of course. The formua is ony for mathematica convenience. We can aso drop the normaization factor for convenience. It doesn t have any specia meaning. We can aso write: where: p E z, r = wk Abs Q z  -k r Im Q z  k z+ r Re - Arg Q z Q z (..a) Q z = z - ; = kw Re = z-â = p w z ; Im = z + z-â and: Arg Q z =-ArcTan z ; z + What are the meaning of a these formuas? What can we infer?. Beam intensity.. Transverse profie For beam intensity: I E = Abs I w k Abs Q z It is a Gaussian profie: with a beam intensity radius: Substitute: And with We obtain: W z = wk Abs Q z  -k r Im Q z  k z+ r Re k Im - k r Im Q z = w k Abs Q z - ==> Im Q z W z = z + = k k = kw W z = w + z w k Abs Q z - r W z Q z = kw z = (..b) (..c) Q z - Arg Q z (..) r W z (..) p W z (..3) + z (..4) = W + z (..5) The transverse profie of a Gaussian beam is a Gaussian profie everywhere, with the radius changing as W diameter). + z. We aso refer to beam spot size as times the beam radius (beam Gaussian beam does not originate from a point: there is no such thing as a point source of ight, but there is a pace where the Gaussian beam has the smaest profie. The smaest beam radius

3 LectSet - Gaussian beam basic 5358_p.nb 3 is caed beam waist: W[]. (see note above aso that shows that the point source of a Gaussian beam is a compex number) HW: pot beam intensity Manipuate k = p ; =.5 k w ;W= w + z ; Show Pot -w + v, w + v, v, -5, 5, AspectRatio Æ.5, PotRange Æ -5, 5, Fiing Æ Æ, Hue.4,,,.7, Graphics Red, Thickness.5, Line z, -5, z, 5, ImageSize Æ 5, 75, PotRange Æ -5, 5, -5, 5, DensityPot w k z + - x +y W, x, -5, 5, y, -5, 5, Mesh Æ Fase, w k CoorFunction Æ GrayLeve, PotRange Æ,, PotPoints Æ 5, 5 z +, z, -5, 5, w,,.5, 4 z w,.. Beam divergence (focusing) Obviousy, at arge z, the beam radius diverges as: W + z Thus, one can define the divergence ange q with: Tan q ~ Sin q = w z z ~ wz (..6) = w (..7a) When we dea with sma ange: Tan q ~ Sin q ~q Aso, with -> p w Tan q ~ Sin q = = = (..7b). p w p w p D where D is the diameter of the beam. As it turns out, this formua with a sight modification is amost generic

4 4 LectSet - Gaussian beam basic 5358_p.nb for any beam: Tan[Divergence ange] = Tan q = m (..7c) D where m is some number that is a characteristic of a specific beam profie, and which is for the case of p Gaussian. It is. for exampe for a top hat beam. (there is no hard definition for the divergence ange either, so the number m is not something fundamenta). Manipuate z w ; Pot w z z,w z, z,,, AspectRatio.5, z PotRange 8, 8, Fiing, Hue.4,,,.7, w,.5, 3 For sma ange: Often, we tak about beam fu width: qº p D D qº 4 p (..8) An exampe for non-gaussian beam: for top hat beam, instead of 4 ~.7, it is.44. p Let s think: sma waist -> arge divergence arge waist -> sma divergence As seen above: waist x divergence = constant. This gives rise to the concept of etendue and reated to the definition of brightness. A simiar formua can be used for focusing (see more in section 4)...3 Longitudina profie Again, for the beam intensity above: I E = Abs I w k Abs Q z wk Abs Q z  -k r Im Q z  k z+ r Re - k r Im Q z = But now, we are interested in variation aong z. At r = the center of the beam w k = w k = w k Abs Q z z + w k Abs Q z - Q z - Arg Q z (..) r W z (..) = I + z z As z->, the intensity drops as as expected for any spherica wave z + z (..9) As the transverse profie of a Gaussian beam gets arger (or smaer), the intensity drops (or increases) as (think Lorentzian) by virtue of power conservation. + z

5 LectSet - Gaussian beam basic 5358_p.nb 5 Pot, z, -,, PotRange Æ A + z Power conservation We now can express the beam intensity I - + z z r W z (..) where I[] is the center of the beam intensity at its waist. (the highest intensity or peak intensity of the beam), Power is obtained by integrating over a area: Int + z - r W z pr r ConditionaExpression p Int W z,re z + W z > FuSimpify p Int W z p Int w z +. W z -> w + z Power is: P = I p w = I A (..a) where: A =pw (..b) is the beam "area". We may ask why the factor? It is because I[] is the peak intensity at the center of the beam waist: I = P p w (..) The power is an integration over a intensity of the whoe beam, not just the peak. Hence, we can think of (..a) as: P = I ave A (..3) with average intensity I ave = I. Think ike the area of a triange. The area is / of the height times the base. Sometime we write the beam intensity as:

6 6 LectSet - Gaussian beam basic 5358_p.nb I r, z = P p w - + z r W z or P p W z - r W z (..4).3 Beam phase.3. Phase front Let's ook at the beam again: E z, r = wk Abs Q z  -k r Im Q z  k z+ r Re - Arg Q z Q z (..a) = p w where: Q z = z - ; = kw z Re = ; Im = z-â z + z-â and: Arg Q z =-ArcTan ; z The phase term is: k z +r Re The phase front is defined as: k z +r Re Q z Q z z + - Arg Q z (.3.) - Arg Q z =constant. (.3.) This surface is a function of z and r. What are the meanings of each term? kz is the pane wave term. k r Re (..b) (..c) Q z the wavefront. Arg Q z =-ArcTan is indepent of r, is just a shift that is dependent on z. z ceary describes the curvature of If we have a pane wave traveing aong a Gaussian wave of the same k, do they traveing in step on the axis?.3. Guoy effect Not quite. Because for pane wave, it is k z ; for Gaussian, it is k z + ArcTan. So as z goes from - to, z The Gaussian is retarded by a p phase shift. This is caed Guoy's effect..3.3 Phase front curvature Phase front term: k z +r Re Q z - Arg Q z (.3.) At arge z, where z>>r, the variation of the term is ~ z +r Re Q z But this is the equation for a parabooida surface: r = constant (.3.3) z + = constant (.3.4) R where R is the radius of curvature. Therefore, we can define: Re = (.3.5a) Q z R z where R z is the radius of curvature of the phase front.

7 LectSet - Gaussian beam basic 5358_p.nb 7 From (..b) Re Q z = Re z-â = R z = z + Now we find out what the meaning of Re Q z = R z ; z : Q z and Im Q z = kw z = Thus, we can write: Q z z z + (.3.5b) (.3.5a) p W z (.3.6) = +Â R z p W z (.3.7) (..b) Remember the comparizon of paraxia spherica wave vs Gaussian: in spherica becomes in Gaussian. Since z Q z in spherica is the radius of curvature, it makes sense that the Re part of is aso the radius of curvature. z Q z Remember: is NOT the Rayeigh range of the beam. We reserve that term ony for, where W[] is p W z p W the beam waist. z =.; k = Pi; w = ; z = kw ; x z phasefrontpot = ContourPot Mod k z + z + z - Arg z - z,*pi == Pi, z,,, x, -5, 5, ImageSize ->,, PotPoints Æ 5, ContourStye Æ Red ; profiepot = Pot -w + z z, w + z, z,,, ImageSize ->,, z PotRange -> -5, 5, PotStye Æ RGBCoor.,., ; Show phasefrontpot, profiepot Ü Graphics Ü The term, where Q z = z -Â z Q z, tes us the beam radius of curvature in the rea part, and the beam transverse profie radius in the imaginary part.

8 8 LectSet - Gaussian beam basic 5358_p.nb 3. Practica techniques to measure Gaussian (or non- Gaussian) beam Discuss in cass ony, and for those who reay dea with aser beam in the ab. 3. Intensity profiing: beam scan method How do we measure the profie of a beam? Why don't we use things ike scanning sit? Scanning of a Gaussian beam W u x y W x - y W y - p W p W p W - erf u W Pot - erf u W. W ->, u, -.5,.5, GridLines -> Automatic, Frame -> True, AxesOrigin ->,, PotRange -> -.5,.5,,

9 LectSet - Gaussian beam basic 5358_p.nb 9 N - erf u W. u -> W N - erf u W. u -> -W N - erf u W. u -> -W N - erf u W. u -> W FindRoot - erf x ==.5, x,.5 x Ø.843 x. x Æ ` 3. Beam divergence (not at thin ens foca pane) How do we determine the divergence of a beam? Not a aser beams are Gaussian (remember that). But first, what is divergence? What is far fied. The distribution of a beam intensity as a function of ange at infinity is defined as FF. We define a function: f[q,f]. Sometime, it is written: f[w], where W={q,f}. If we ook at a Gaussian beam, the resuts we saw above is that the beam profie is - r W + z - r W + z We define. We need to convert this into ange {q,f} as zø. Ø - r W z = - W = Sin q r W z = - z Remember that Sin q = W = W p W Sin q W Sin q (aso use Tan q ). Thus: - Sin q : this is the far fied profie function = p W. So now we have a practica question: how do we determine a beam far-fied? Go out very far, measure the spatia profie, and divide by distance z? how far? cm? m? km? The key condition is z >> : Rayeigh range. Rayeigh range is the range where the beam diverges very itte and the wavefront is neary panar: it is where the pane wave approximation is good. Suppose we have a Gaussian beam with -cm diameter (that is, beam waist=.5 cm), what is the beam divergence? What is its Rayeigh range?

10 LectSet - Gaussian beam basic 5358_p.nb We need to know the waveength aso. Let's =.638 mm; the beam Rayeigh range is:.5 cm ^.638 m. m ^ 4 cm 4. cm Suppose we want to measure at a ocation ~ times the Rayeigh range, it' be: 4 m or. km! How big is the beam there? w + z.5 cm. w ->.5 cm. z -> To measure the far fied of any beam, we can just focus the beam with a ens, then measure the profie at a reasonabe distance (do not confuse this with measuring the foca pane of a thin ens that we wi earn ater on). But then, the divergence we get is just the divergence of the focused beam, not the origina beam! How do we convert? We can convert for a Gaussian beam! But first, et's ook at a concept: etendue. For a beam or an optica system, the product of accepting area and accepting soid ange is caed beam etendue. We can have -D etendue or -D etendue. Earier, we ask a question about Gaussian. Let's see: Beam divergence: Tan q = or q p W º ; p W Beam waist: W ; Thus, -D etendue = W = 4 : constant! p W p (What is its -D etendue?) So now, if a ens transforms a Gaussian beam into another Gaussian beam (just of different beam waist), then if we measure the etendue of one, we knows the etendue of the other! Appy this to the aboratory exercise. 3.3 Laboratory exercise: M and number of times of diffraction-imited Ever wonder when you focus a beam ike aser beam, what is at the focus spot? NOT a beams wi give the same spot size even with the same focusing convergence ange! Perform a Gaussian beam measurement in the ab and report your measurement: ocate and measure beam waist, measure divergence ange. Focus the beam with a ens and measure. 4. Gaussian beam transformation through enses (and parabooida/spherica mirrors) In practice, rarey any beam is used without any transformation through optics.

11 LectSet - Gaussian beam basic 5358_p.nb Except for integrated optics, which use waveguide to conduct a beam, and the beam profie is determined by the waveguide modes, many systems sti use free-space optics. Even at a very macroscopic eve... The beam profie properties are essentia. 4. Review of thin ens and spherica or parabooida mirror

12 LectSet - Gaussian beam basic 5358_p.nb Thin ens Lighty curved mirror Thin ens and ow-curvature spherica or parabooida mirror has an interesting property: they convert input r - k beam into an output beam with an approximatey parabooida wavefront: f for thin ens where f is the ens foca point and  k r R M where R M is the mirror radius of curvature. (derivation comes from paraxia approximation). Convention: f> for a positive ens, and R M < for concave, R M > for convex. Derivation for this comes from ray approximation in which the phase at ocation r is proportiona to r. (Linked to Simpe Optica Eements). Thus, if a ens is ocated at ocation z, and the input wave is U x, y, z, then the output wave at the same r - k ocation is: U x, y, z f. Whatever happens to the wave afterward? at arbitrary z? this is precisey the interesting property of these optics. In particuar, we wi see that Gaussian beam behaves quite nicey with this type of optics and that is the reason we study it here. 4. Gaussian beam after a thin ens. (derivation) A Gaussian beam remains a Gaussian through a thin ens (or equivaent paraboic mirror) but with a different beam waist ocation, vaue, or direction of propagation (the case of mirror). Thin ens What becomes of the beam here? z Let's set z= at the waist of the input beam. Then, et s write the input Gaussian profie at z just before the ens. From: to: E z, r = wk Abs Q z  -k r Im Q z  k z+ r Re - Arg Q z Q z (..a) E z, r -k r Im Q z  k z +r Re - Arg Q z Q z (4..)

13 LectSet - Gaussian beam basic 5358_p.nb 3 w k (We drop the term which is just an overa ampitude factor without any effect on profie and phase Abs Q z  for simpicity). Remember that: Im = = (4..a) Q z kw z p W z and: Re = (4..b) Q z R z Then, by substitute (4..), (4..) becomes: - r W z The output beam just after the ens is: - r W z  k z + r R - Arg Q z  k z + r - Arg Q z R - k r f (4..3) (4..4) = - r W z  kz  k r R - f - Arg Q z If we define: = -, (4..5) R' R f then we can write: - r W z  kz  k r R' - Arg Q z (4..6) Let's ook at the expression. What does it te us? Obviousy, it has a Gaussian ampitude profie: - r W z ; and a parabooida phase term:  kz  k r R'. This is a typica Gaussian profie! (note: The phase term - Arg Q z wk, ike the ampitude term that we drop, is Abs Q z  ONLY a constant phase term without any r-dependency hence, just a factor). The ony difference between this output Gaussian and the input Gaussian is that the phase front has a radius of curvature R' instead of R. They both have the same beam spot radius at z : W z. Thus, this is aso a soution of the Hemhotz equation. The output of a Gaussian beam through a an on-axis thin ens is a Gaussian beam. BUT this Gaussian beam is NOT the same as the input (inspite of the fact that BOTH have the same beam spot radius at z : they have difference phase front at z ). So, to find out the property of this output Gaussian from the ens, we must find a Gaussian beam expression that gives us exacty the beam spot radius: W z and radius of curvature R' at ocation z. Let's write the expression again: - r W z  kz  k r R' - Arg Q z =  k r + R' kw z  kz - Arg Q z Now we reca that in genera, + = R' kw z P z (4..7) where P z is the Q-function (virtua point) for some Gaussian beam. How do we do that? Basicay we must find the beam waist ocation z and beam waist W '. We write the genera expression for a Gaussian beam: - r r  k z+ W z R' z -Arg z-â' (4..8) This is expression for a beam with waist at z=. We shift the coordinate to z : - r W' z-z  k z-z + r R' z-z -Arg z-z - ' (4..9)

14 4 LectSet - Gaussian beam basic 5358_p.nb At z = z : r W' z -z  k z -z + r R' z -z -Arg z -z - ' (4..) Now, this beam must match with the output Gaussian beam from the ens at z, right? So, we compare terms by terms: - Ampitude profie: - r W' z -z = - r W z (4..) which yieds:w ' z - z = W z ; or: W ' + z -z = W ' + z W ' + z -z p W = W ' + z 4 W ' + z -z = W p W ' + z p W p W 4 p W ' + z -z = p W p W ' + z p W ' + z -z ' - Phase profie: r = + z (4..)  k z -z + -Arg z R' z -z -z - '  kz  k r - R z f - Arg z - (4..3) Again, separate the r-dependent term from the others:  k r R' z -z =  k r - R z f f = - (4..4) R' z -z R z f which is precisey the wavefront reation we mention earier. The other term is NOT important here since it is just a constant phase shift (no r or z-dependence). Recaing that = z, we have the equation: R z z + z -z = z - (4..5) z -z + ' z + f Soving the equations Let's rewrite the Eqs. again: ' + z -z =z ' + z (4..) z -z z z -z + ' z + f (4..5) Changing z - z -> -d ' + d =z ' + z (4..6a) d z - d + ' f z + (4..6b) So we have eqs. with unknowns.

15 LectSet - Gaussian beam basic 5358_p.nb 5 FuSimpify Sove zp + d zp == z + z z d, == d + zp f - z, d, zp z + z Genera::spe : Possibe speing error: new symbo name "z" is simiar to existing symbo "zp". d Ø z - f f + f, zp Ø f - z f +z + z f z f - z f +z + z 4.3 Resut discussion 4.3. Reations between the beams The first resut: ' = f z - f + or (4.3.) W ' = W f z - f + = W M (4.3.) where we define: M = f z - f + = f z - f + z - f (4.3.3) Next: d Ø z - f f + f f d - f = z f f - z f +z - f +z z - f +z d = f + z - f M (4.3.4) The quantity M is interesting: it is the ratio between the beam waists. Thus we define M to be the "magnification" factor (as if we use the ens to magnify or focus the beam). We see these reations are somewhat simiar to ray-optics of ens that we are famiiar. In fact we have other reations invoving M: f Rayeigh range: ' =z = M z z - f (4.3.5) + Beam divergence: q' = = = q. (4.3.6) p W ' M p W M Write a code cacuate output beam waist and ocation as a function of input beam parameters 4.3. Meaning of the "magnification" (or reduction) term

16 6 LectSet - Gaussian beam basic 5358_p.nb Let's ook at the term: f M = (4.3.3) z - f + If the beam is point-ike: it has a sma beam waist and short Rayeigh range, then: M = f z - f + = f z - f º + z - f f z - f (4.3.7) z If z - f >> so that º. z - f But this is the typica ray-optics ens reation: z d z-f f Image magnification: W ' = W M In fact, we ook at other reation: d - f = z - f M or: d - f = z - f d - f = f z - f f z - f d - f z - f = f dz - f d + z + f = f d+z dz = f f z + d = f (4.3.8) This is the ens imaging reation in geometrica (ray) optics. Depth of focus: ' = M (4.3.9) A of these reations are compatibe with ray optics. When is this concept not reevant? obviousy, when the beam waist is arge: the beam is more "pane-wave" ike rather than spherica wave emanating from a spot: now we know when we can use simpe geometrica optics and when we shoud not when deaing with Gaussian beam: the Rayeigh ranges << f or the waist is very sma Gaussian beam focusing or mode matching Now we have the other extreme: our beam is penci ike, which means it has a arge Rayeigh range: >> f. Aso, the beam waist shoud aso be cose to the ens, in other words, z is NOT >> f. What wi happen? Let's ook at beam waist: W ' f = W = W z - f + f z - f + = f p W z - f +

17 LectSet - Gaussian beam basic 5358_p.nb 7 Since we assume that >> z - f, we can write: W ' º p W f or the famous Gaussian beam transform formua: p W W ' =f (4.3.) Notice how the beam waists are inverse of each other: input a arge beam, we have a very sma focused spot. Input a sma spot, we have a arge focus spot Foca-foca conjugate Now suppose we want the "symmetric case". Can we make a beam such that the input is simiar to the output except for a dispacement of the waist? What if we just put a Gaussian waist at the foca pane of a ens? That is: z = f or z - f =? From the above: d = f + z - f M = f This means that the output beam waist is at the other foca pane. Sure, this is simpe enough to understand: if we put a beam waist at one foca pane, the ens just bend the phase such that the output is at the other foca pane. But what happens to beam waist size? For that, we need: M = f z - f + = f If we want the two beams aso to be neary equa in waist, we need f ~, in other words, the Rayeigh range of the beam has to be neary equa to the ens f. So we can't arbitrary make beams ike that for a given ens. Either we choose the ens f or we must have beam with proper waist!

18 8 LectSet - Gaussian beam basic 5358_p.nb =. ; f = 5. ; ens = Graphics Hue.45,,,, Disk,, 5, 5, Hue,,,, Thickness.5, Line -,,,, Purpe, PointSize., Point - f,, Point f, ; Manipuate z = p w ; M = f -z - f +z ; d = f - z + f M ;w= M w ; z = p w ; profie = Pot -w z - z +,w + z z - z z, z, -,, PotRange Æ A, PotStye Æ RGBCoor.,.,, Fiing Æ Æ, Hue,.7,,. ; waist = Graphics Red, Thickness.5, Line z, -w, z, w ; profie = If d., Pot -w z - d +,w + z - d z z, z,,, PotRange Æ A, PotStye Æ RGBCoor.,.,, Fiing Æ Æ, Hue.6,,,. z - d, Pot -w +,w + z z - d z, z, d,, PotRange Æ A, PotStye Æ RGBCoor.,.,, Fiing Æ Æ, Hue.6,,,.3 ; waist = Graphics Bue, Thickness.5, Line d, -w, d, w ; Show ens, profie, waist, profie, waist, PotRange Æ -,, -5, 5, z, -,, w, 3,.5, 5 Consider focusing a beam: From the formua: p W W ' =f (4.3.) W ' = f p W If we want a very sma focused spot, we want W as arge as possibe (and f as sma as possibe). But the input beam can t be arger that the ens! otherwise, the power wi not be coected. Hence, we can say that W is imited by the radius of the ens r. Hence: W ' ~ f p r

19 LectSet - Gaussian beam basic 5358_p.nb 9 But the ratio f ~, which is aso caed nunerica aperture (NA) of the ens. We see that it doesn t matter arge r Sin q ens or sma ens, ong f or short f, the ony thing that matter is the NA=Sin[q] The above enses give exacty the same spot size when focusing at optima beam width. W ' ~ (4.3.) p NA That s why NA is the ony rating that matter in enses when it comes to resoution: how fine the detais we can see or how sma a spot we can focus Off-axis beam (advanced ony) Linear decomposition approach (advanced ony) BeamOut beamin_list, ensparm_list, D_, x_, y_, z_ := Modue, W, z, Wz, Rz, R, f,z,k, P, Ta, f,z = ensparm;, W = beamin ; k = *Pi ;Ta=D f ; z = k W ; Wz = W + z z z Rz = ; z + z R = Rz - f ; P = R + ; k Wz ; p P z + P kz kx k y-d-p Ta z+p z+p - k P Ta

20 LectSet - Gaussian beam basic 5358_p.nb BeamIntOut beamin_list, ensparm_list, D_, x_, y_, z_ := Modue, W, z, Wz, Rz, R, f,z,k, P, Ta, f,z = ensparm;, W = beamin ; k = *Pi ;Ta=D f ; z = k W ; Wz = W + z z z Rz = ; z + z R = Rz - f ; R - P = R + k Wz ; 4 k Wz 4 ; R - z + R + k Wz = 4 k Wz 4 z R R - k Wz 4 k Wz R + 4 k Wz 4 Abs p P z + P kx k y-p Ta z+p z+p ens =, *^4 ; * unit micron * beamin =.5,.3*^4 * unit micron * ; Pot3D Re BeamOut beamin, ens,,, y, z, z, 998,, y, -,, PotPoints ->,, BoxRatios ->,,.5, Mesh -> Fase, PotRange -> A, ViewPoint ->,, Ü SurfaceGraphics Ü

21 LectSet - Gaussian beam basic 5358_p.nb ens =, *^4 ; * unit micron * beamin =.5,.3*^4 * unit micron * ; Pot3D Re BeamOut beamin, ens, 3,, y, z, z, 998,, y, -,, PotPoints ->,, BoxRatios ->,,.5, Mesh -> Fase, PotRange -> A, ViewPoint ->,, Ü SurfaceGraphics Ü ens =, *^4 ; * unit micron * beamin =.5,.3*^4 * unit micron * ; Pot3D Re BeamOut beamin, ens,,, y, z, z, 998,, y, -,, PotPoints ->,, BoxRatios ->,,.5, Mesh -> Fase, PotRange -> A, ViewPoint ->,, Ü SurfaceGraphics Ü ens =, *^4 ; * unit micron * beamin =.5,.5 * unit micron * ; Pot3D Re BeamOut beamin, ens,,, y, z, z, 998,, y, -,, PotPoints ->,, BoxRatios ->,,.5, Mesh -> Fase, PotRange -> A, ViewPoint ->,, Ü SurfaceGraphics Ü

22 LectSet - Gaussian beam basic 5358_p.nb ens =, *^4 ; * unit micron * beamin =.5,.5 * unit micron * ; Pot3D Re BeamOut beamin, ens,,, y, z, z, 9 98,, y, -,, PotPoints ->,, BoxRatios ->,,.5, Mesh -> Fase, PotRange -> A, ViewPoint ->,, Ü SurfaceGraphics Ü ens =, *^4 ; * unit micron * beamin =.5,.5 * unit micron * ; Pot3D Re BeamOut beamin, ens, 3,, y, z, z, 9 98,, y, -3, -99, PotPoints ->,, BoxRatios ->,,.5, Mesh -> Fase, PotRange -> A, ViewPoint ->,, Ü SurfaceGraphics Ü ens =.5, 4.9 *^4 ; * unit micron * beamin = 4.8, * unit micron * ; Pot3D Abs BeamOut beamin, ens,,, y, z ^, z, 5, 5 5, y, -,, PotPoints ->, 5, BoxRatios -> 5, 4, 4, Mesh -> Fase, PotRange -> A, ViewPoint ->,, Ü SurfaceGraphics Ü

23 LectSet - Gaussian beam basic 5358_p.nb 3 When you go into the ab and try to focus a beam to as a sma spot as possibe, what do you do? Project: use whatever enses you can find in the ab, see who can obtain the smaest focused spot. Dr. Bujin Guo is using a pinhoe and try to focus a beam through. Obtain from him necessary parameters to cacuate how much the throughput is (% of power that gets through the pin hoe). Look at the diode beam output. Why do you think the beam diverges in one dimension so much more than the other? Write a theory for x-y asymmetric beam. Diode beam focus: how do you coupe a beam from a diode into a rectanguar waveguide? (which orientation to use?) Where is the beam focused? d - f = z - f d = f + z - f f z very near the foca point. f = z z - f +z - f f º f z z - f z + º z - f f If we want to coimate and expand the beam to shoot far away, where do we put the beam? z f use the reation: d - f = z - f. If the input beam waist is right at the foca point: z z - f +z = f, then: d - f = : the output beam waist is at the other foca point Numerica aperture of a ens If we want a ens of focus or coimate a beam, how do we order it? Let's ook at the externa cavity: Grating: N = g/mm, 5 5 6mm 3 /N [(sin + sin )] = = 75 º, sin =.966, = 5.um sin =.6, = 3.º Lens:.7mm ZnSe f =.7 mm, NA= º 3. º 3 º.5 º w x w y L L ink

24 4 LectSet - Gaussian beam basic 5358_p.nb Does the ens diameter matter? Why do we want "aspheric" ens? what is the reation between distance L and L and foca ength f? Where shoud the ocation of the ens shoud be if we are given a fixed distance L +L. What can the atera sit do for us? Summary Now we know why Gaussian beam is interesting and usefu: it remains Gaussian (athough transformed) under enses or spherica/parabooida mirrors. (A the above resuts for ens are aso appicabe with mirrors). With appropriate condition, we can find a set of Gaussian soutions that remain invariant in a system of optics: aser cavity. We' see that Gaussian turns out to be just one soution among many cavity modes; not surprisingy, it is the simpest soution, the owest order soution of the Hermite-Gaussian function famiy. More importanty: many concepts above are NOT excusive to the Gaussian beam. Even non- Gaussian beams have simiar behaviors: this is because they a are governed by diffraction behavior. What we earn is the diffraction of paraxia ight beam. Gaussian is just a specia case with neat mathematica expression. In the vertica dimension, a diode beam coming out of its facet has a fat wavefront and an ampitude profie ike this: a -a x cos[ k a] e z cos[ k a] e x a cos[ k x] x a Pot the wave profie of the beam as it eaves the diode facet aong the x and z dimension. Use resonabe assumption of everything. (you must figure it out yoursef). Assignment: do a the excersizes in section 4.3 above and this: John measured a beam with a Gaussian intensity profie - r w and obtained w=.5 mm. The waveength was = mm. He then measured the beam divergence which has a fu width haf max (FWHM) of 35 degree. Is this posibe? TanT = p w. w ->.5. ->.73

25 LectSet - Gaussian beam basic 5358_p.nb 5 Pot - Tan q*p 8.7 ^, q, -4, 4, PotRange -> A Advanced demonstration: Large NA beam More accurate cacuation needed for very arge NA beam. Poarization needs be incuded. Iustration: