ILSC 003 Conerence Proceeings Beam Propagation Hazar Calculations or Telescopic Vieing o Laser Beams Ulrie Grabner, Georg Vees an Karl Schulmeister Please register to receive our Laser, LED & Lamp Saety NEWSLETTE (about times a year) ith inormation on ne onloas: http://laser-le-lamp-saety.seibersor-laboratories.at/nesletter This ILSC proceeings paper as mae available as p-reprint by Seibersor Laboratories ith permission rom the Laser Institute o America. Thir party istribution o the p-reprint is not permitte. This ILSC proceeings reprint can be onloae rom http://laser-le-lamp-saety.seibersor-laboratories.at eerence inormation or this proceeings paper Title: Beam Propagation Hazar Calculations or Telescopic Vieing o Laser Beams Authors: Grabner U, Vees G, Schulmeister K Proceeing o the International Laser Saety Conerence, March 0-3 th 003 Jacksonville, Floria, USA Page - Publishe by the Laser Institute o America, 003 Orlano, Floria, USA.lia.org
Beam Propagation Hazar Calculations or Telescopic Vieing o Laser Beams Ulrie Grabner, Georg Vees an Karl Schulmeister, AC Seibersor research, A- Seibersor, Austria ABSTACT We have perorme beam propagation calculations hich can be use to characterize the retinal spot size o laser beams or telescopic vieing. The results sho that the minimal retinal spot size prouce on the retina by using a telescope compare to that o the nake eye is increase at least by a actor that is equal to the magniying poer o the telescope an an increase o C (or C E ) is applicable. The hazar evaluation metho use is base on the calculation o the maximum level o thermal hazar that is eine as the ratio o the poer that enters the eye an the retinal spot iameter. Beam propagation principles ere use to calculate the istance beteen the beam aist an the eye lens that provies the maximum hazar. The only to input parameters necessary or these calculations are the aist iameter an the ar-iel ivergence o the beam. The beam ith on the retina at the most hazarous vieing istance is consequently use to etermine the angular subtense o the apparent source that is neee to calculate the MPEs an AEL values. Both the angular subtense o the apparent source an the most hazarous vieing istance have been calculate or a ie range o beam aist iths an ar-iel ivergences.. INTODUCTION The correction actor C that is containe in the MPEs or retinal thermal injury characterises that the hazar rom extene sources is loer than rom collimate laser beams or other point sources., The actor C epens on the plane angle α subtene by the iameter o the retinal image at the lens o the eye an measure in mra. This angle is equivalent to the size o the image on the retina an this in turn is irectly relate to the source size as shon in igure. In the saety stanars, a minimal angular subtense, α min, is eine as. mra, an angle that correspons to a retinal iameter o µm or a ocal length o the eye o 7 mm. Figure: The angular subtense α o an object For laser raiation the angular subtense α o a source is oten reerre to as the apparent source size, to enote that it is not relate to the physical imension o the emitter such as the beam iameter or the exit mirror. The size an location o the apparent source, hich is eine as the real or virtual object that orms the smallest possible retinal image by eye ocusing, can be thought o as the size an the location o a conventional source, hich leas to the same image size on the retina. As the angular subtense o the apparent source is irectly relate to the iameter o the irraiate area on the retina, together ith the energy or poer hich is passing through the pupil an is incient on the retina these to parameters etermine the exposure or irraiance at the retina. Generally, or a given energy or poer, an increase in image size reuces the retinal exposure (J/m ) an consequently ecreases the injury potential. The correction actor C is eine in the IEC 08- an ANSI Z3. as being the ratio o the angular extent α o the apparent source to α min : C α = () α min C is limite to values beteen an.7, as or α greater 00 mra the epenence o the heat lo on the image size oes no longer apply an the limit coul be expresse as constant raiance limit, hich is relecte by setting the image size constant as ell as by speciying a iel o vie o 00 mra or the irraiance measurement, hich limits the measure part o the source to 00 mra. Base on the epenence o the thermal retinal MPE on α, e have evelope a beam propagation moel to characterize the hazar rom exposure to laser beams hile using 7 x 0 binoculars. ILSC 003 CONFEENCE POCEEDINGS
. THEMAL HAZAD Folloing the linear epenence o thermal injury on the angular subtense o the source, hich is again a linear unction o the image iameter on the retina, the quantity that escribes the level or thermal hazar, is the beam poer or energy that alls on the retina ivie by its ith at the retina. Since, or a given avelength, the beam poer that is incient on the retina is proportional to the poer that enters the eye P r (in the sense o the part o the poer in beam that is incient on the eye an that passes through the pupil, usually reerre to as intraocular poer), the calculations are base on this value. The thermal hazar H can then be eine as: H P r = () r here r is the ith or the iameter o the beam at the retina. Folloing the concept evelope by Brooke War, the maximum hazar or retinal amage is oun by maximizing the parameter H. 3 3. HUMAN EYE MODEL eraction in the eye takes place mostly at the cornea hereas the eye lens serves the purpose o ocusing (accommoation). The range o accommoation o the eye varies greatly rom one person to another an in each person, ith age. It is maximum or young people an reuces by age. The eye moel use in this analysis consists o a single thin lens ith a variable ocal length, combining the reraction on the cornea an the eye lens, an a ixe lens to retina spacing o 7 mm. Although this is a strong simpliication, it escribes the complexity o the eye in a suicient manner. To consier the accommoation range o most people, a stanar eye is use having a ocal length in a range o e,min =.3 mm to e,max = 7 mm, giving a minimum an maximum ocusable istance o 00 mm an ininity, respectively. The pupil iameter in this eye moel is ixe to 7 mm.. BEAM POPAGATION The propagation o an ieal TEM 00 beam in ree space is escribe by the olloing equations: z 0( z) = + z (3) z z = z+ () z z π = () θ = () z r 0 ( z) I(, r z) = I( z) e (7) 0 here 0 (z) is the beam raius ithin hich 8.% o the total energy/poer is enclose an the intensity has reuce to /e o the peak intensity, is the beam aist raius, (z) is the curvature o the ave ront at istance z, z is the ayleigh length, is the avelength o the raiation an θ the ar-iel ivergence. Figure. Characteristics o a gaussian beam ILSC 003 CONFEENCE POCEEDINGS 7
Figure 3: aial intensity istribution o gaussian beam The TEM 00 beam is only the loest-orer solution in ree space. In general, a laser beam consists o urther higher moes. In this case the equations 3,, an 7 are still applicable, but the ormula or the ayleigh length has to be moiie to z π = (8) M M is the beam propagation ratio, hich is eine using an invariant o propagation that is not change by any linear, iraction limite optical system: the prouct o the beam aist raius an the ar-iel ivergence. M is eine as the ratio o the value o this invariant or the real beam to its value or a TEM 00 beam o the same avelength. M is equal to or TEM 00 beams an greater than or higher orer moes. M = θ θ, TEM00 TEM00 (9). Transormation o a gaussian beam by any iraction limite optical system To escribe the propagation o a laser beam through any optical system the complex beam parameter q(z) as use, hich or a TEM 00 beam is eine as. = i qz z π ( z) 0 q(z) is a complex unction an part o the loest-orer solution o the paraxial Helmholtz equation (0) U U U + ik = 0 x y z () or the complex iel amplitue i { p ( z ) k r q ( z ) } Urz (, ) = Ee + () o a rotationally symetric electrical iel Er = U( rz, ) e ikz (3) The relation beteen the complex unctions q(z) an p(z) can be erive by inserting Equ. into Equ. pz i qz = = () z qz z Inserting Equ. 0 into Equ. gives the solution or the electrical iel o a TEM 00 beam here an r 0 ( z) ik r ( z) i[ kz p( z) ] Exyz (,, ) = Ee e e + () k r z is the transversal phase actor z pz = arctan z is the longituinal phase actor o the gaussian ave. 8 ILSC 003 CONFEENCE POCEEDINGS
As the ave ront at the beam aist is planar (( z = 0) ), a beam aist is assume to be locate at z = 0. Consiering qz ( = 0) = q an = i = i qz z ( = 0) π ( = 0) π 0 () the complex beam parameter o the beam aist can be evaluate: π q = i = i z (7) Integrating Equ. gives the complex beam parameter qz = z+ q = z+ i z (8) q(z) is a linear unction o z an transorms in the same ay as the raius o curvature o spherical aves coming rom a point source A q + B ABCD-la or q = (9) gaussian beams C q + D here q is the complex beam parameter at z = z, q is the complex beam parameter at z = z an A B M = (0) C D is the matrix o the optical system... Transormation o a gaussian beam by a thin lens Assuming a gaussian beam ith a aist raius at z = 0 in ront o a thin lens, that has a ocal length an is locate at z =, leas to igure : Figure : Transormation o a gaussian beam by a thin lens To obtain the complex beam parameter q = q ( z = + ) at the istance behin the lens the matrix M has to be evaluate Inserting C D 0 0 A B 0 M = = = () π q( z = 0) = q = i z = i () ILSC 003 CONFEENCE POCEEDINGS 9
into Equ. 9 the olloing equation or q can be oun: ACz + BD + i z AD BC q + = = = = i Az + B BD + i z ( ) i Cz + D C z + D D = + + z i z z + Assuming that a ne beam aste ith raius is orme at z = + results in q + = q () ith e[ q] = 0 = z + () Solving Equ. leas to as a unction o, z an : here ( (, )) = + V () V (, ) = (7) + z Inserting into Equ. 3 gives z an : Im = z = (8) [ q ] π (3) = = V(, ) + z (9).. Transormation o a gaussian beam by a telescope The beam propagation moel as evelope or an astronomical reracting telescope that consists o to convergent lenses. The results o the simulation are hoever also applicable to a binocular ith corresponing ocal lengths. In the normal state o ajustment the secon ocal plane o the irst lens (the objective), coincies ith the irst ocal plane o secon lens (the eyepiece), so that an incient pencil o parallel rays emerges as a parallel pencil. For the calculations, the aist o a gaussian beam ith a aist raius is assume to be at z = 0. The objective o the telescope ith a ocal length is locate at z = at a istance + in ront o the eyepiece ith a ocal length. To receive the complex beam parameter q = q ( z = + + + ) at istance rom the lens the matrix M has to be calculate + + M = (30) 0 Doing calculations similar to that one in the case o the thin lens can be evaluate as a unction o, an : = + ( + ) (3) 0 ILSC 003 CONFEENCE POCEEDINGS
It shoul be note that oes not epen on z, an has a maximum or = 0 hich is equal to the istance beteen the exit pupil an the eyepiece. can also be negative, creating a virtual beam aist insie the telescope: = + or = 0 max 0 or + z an can be erive to: z π = = z (3) (33) = Note that the raius o the beam aist prouce by the telescope is not a unction o an epens only on the ratio o an, hich is eine as the magniying poer o the telescope. Equ., 7, 8, 9 an Equ. 3, 33, 3 have been erive consiering a TEM 00 beam. Again in case o a real laser beam these equations ill still be applicable, i the ormula or the ayleigh length is moiie to z π = M. DETEMINATION OF THE MINIMAL ETINAL SPOT SIZE The eye moel use in the olloing analysis consists o a single thin lens ith a variable ocal length an an image plane (retina) ixe behin the lens at a constant istance o e = 7mm rom the lens. This case seems quite similar to that o a single thin lens ith a ixe ocal length proucing a beam aist in a istance that can be etermine by Equ.. Hence one coul presume that the minimal spot size is a beam aist that is locate at the retina, but this is not the case. In general, to prouce a minimal spot size on the retina, a beam aist is orme in a very short istance in ront o the retina, resulting in a lens ocal length that is ierent to that orming a beam aist on the retina. To minimize the spots size on the retina an equation or the beam propagation behin the eye lens has to be erive. Taking into account that the ne beam aist is locate at the position z = +, Equ. an Equ. 9 can be inserte into z z z + 0 = 0 (,,, ) = + z By setting z = z, the beam raius behin the eye lens can be evaluate as a unction o the istance rom the eye lens. (3) (3) z (,,, z ) = + z - 0 e + + = z z z e + z + z (3) Minimizing Equ. 3 or given values o, an z = e = 7 mm gives us an equation or the eye ocal length me, that prouces the minimal beam iameter on the retina - + = 0 z e me + e( + z ) me = = + z + + ( ) e here ( ) is the raius o curvature o the input beam as a unction o the istance rom its beam aist. e (37) ILSC 003 CONFEENCE POCEEDINGS
I me is ithin the eye accommoation range the minimal spot size obtainable by eye ocusing is (, ) = (,, = ) = = e e me e e me π 0 ( + z ) here 0 ( ) is the raius o the input beam at the lens: 0 = + z Hence the beam raius behin the lens as a unction o the istance z rom the lens can be calculate ith z 0 π 0 z e e ( z ) = + - π (38) (39) (0) It shoul be note that as long as the eye is able to accommoate, to ierent gaussian beams ith equal avelengths an the same beam iameters on the lens are transorme to ientically propagating beams behin the eye lens, hoever, the eye ocal length is ierent in each case. So ar, the calculations i not take into account that the ocal length me has to be ithin the accommoation range o the eye. As it can be seen in Equ. 3 the spot raius on the retina e ( e ) as unction o the eye ocal length has only one minimum at e = me. Consequently it is an increasing unction or e > me an a ecreasing unction or e < me. Thereore in the case o me < e,min =.3 mm the minimum spot raius achievable by eye ocusing is obtaine ith e = e,min hereas in the case o me > e,max = 7 mm the minimum spot raius is obtaine ith e = e,max. me z 000 e = + + + ( + z ) - z.3 z + e i me < e,min =.3 mm me = ( + z ) e i.3 mm me 7 mm () me z 000 e = + + + ( + z ) - z 7 z + e i me > e,max = 7 mm.. Minimal retinal spot size in case o telescopic vieing The telescope use in the olloing calculations is an astronomical reracting telescope ith a ocal length o the objective o = cm, a ocal length o the eyepiece o = cm an a istance beteen the to lenses o T = + = cm, proviing a magniying poer o 7. For these calculations the iameter o the lenses are not relevant. To obtain the minimal spot size on the retina in case o using a telescope, Equ. 3, 33 an 3 have to be combine ith Equ.. In aition it has to be taking into account that the eye lens is locate at the exit pupil o the telescope at a istance EP = ( + ) / behin the eyepiece, to ensure that all the light entering the objective at ierent o-axis angles ill reach the eye. Figure shos the minimal spot size o the nake eye me or a number o beams ith a common aist raius o mm in comparison to that prouce on the retina by using the telescope T me or istances beteen the input beam aist an the position o the objective up to 30 m. At large istances the ratio T me / me converges to 7, the value o the magniying poer o the telescope, hereas i becomes smaller, the ratio at irst reaches ILSC 003 CONFEENCE POCEEDINGS
a maximum to ecrease again or even closer istances to a value o almost 7 or = 0. It has to be reemphasise, that the ratio is alays bigger than 7. Figure : atio T me / me o the minimal retinal beam raii ith an ithout telescopic vieing. MAXIMUM THEMAL HAZAD, MOST HAZADOUS DISTANCE AND ANGULA SUBTENSE Assuming a gaussian poer istribution, e obtain the olloing expression or the poer P(u) transmitte through a pupil ith iameter u Pu = Ptot e u 0 here P tot is the total beam poer an 0 is the /e beam raius at the pupil. It shoul be mentione that the use o Equ. in the olloing calculations ill overestimate the level o hazar as the gaussian istribution provies the greatest threat compare to other istributions an as eects o iraction on the beam iameter at the pupil are neglecte. 3 (). Nake eye In the case o the human eye the pupil iameter is 7 mm. I a laser ith a certain beam iameter an ar-iel ivergence moves aay rom the eye, the iameter on the lens increases reucing the transmitte poer incient on the retina. At the same time the minimal spot size on the retina ecreases an there is a certain istance here the thermal hazar H becomes a maximum. This istance can be reerre to as the most hazarous vieing istance. To obtain the corresponing angular subtense, the /e spot iameter at the retina has to be ivie by the lens-to-retina spacing. For the nake eye, the most hazarous istance an the corresponing angular subtense have been calculate by Brooke War or a large range o beam characteristics. Beams ith a ar-iel ivergence up to 00 mra an aist iths up to mm have been analyse to reveal the locations o the peak hazar as ell as the value o the beam ith at the retina at that vieing conition. One o the results as that there exists a range o lo ivergence beams here the most hazarous vieing istance is signiicantly greater than the stanar o 00 mm. In contrast to this there are also some high ivergence source conitions that represent the highest hazar hen place closer than 00 mm rom the eye, espite the eye is outsie its accommoation range. 3. Telescope In the case o using a telescope it has to be taken into account that there is another aperture at the objective an at the eyepiece, hich aitionally truncates the input beam. Consiering this, the poer entering the eye can be evaluate ith DOB DEP DEL OB,0 EP,0 EL,0 Pr = Ptot e e e (3) here D OB, OB,0, D EP, EP,0, D EL an EL,0 are the raii o an the /e beam raii at the objective, the eyepiece an the eye pupil, respectively. ILSC 003 CONFEENCE POCEEDINGS 3
The telescope use in the olloing analysis has an objective ocal length o = cm, a ocal length o the eyepiece o = cm an a istance beteen the to lenses o T = + = cm, as in the calculations or igure. While the objective iameter has to be 0 mm to ensure an exit pupil o the same size as the eye pupil, the iameter o the eyepiece as chosen to be mm. The most hazarous vieing istance an the corresponing angular subtense ere calculate or telescopic vieing o a laser beam. To account or a minimal /e beam iameter o µm on the retina, the beam iameter as restricte to that value uring the entire evaluation. The results are shon in the igures, 7, 8 an 9. 0,9 0,8 0 MHV [m] 8 beam aist iameter [mm] 3 0,7 0, 0, 0000 ar iel ivergence [mra] 00 0300 30 00 3 beam aist iameter [mm] 0, 0 00 0 00 0 300 30 00 0,3 ar iel ivergence [mra] Figure : Most hazarous vieing istance MHV as a unction o the aist iameter an ivergence o the beam Figure 7: Contours o the most hazarous vieing istance MHV [m] as a unction o the aist iameter an ivergence o the beam As it can be seen in igures an 7, the most hazarous vieing istance as ell as the relaxation actor C mainly epen on the ar-iel ivergence. For beams ith a ar-iel ivergence o more than 0 mra the most hazarous vieing istance lies beteen cm an m. That means that the beam aist is locate close to the objective o the telescope. I the ar-iel ivergence becomes smaller than 0 mra the most hazarous vieing istance rapily increases up to a value o about m or a beam ith a very lo ivergence an a small aist iameter. This is because the poer transmitte through the telescope an the eye pupil ecreases much less than the spot size on the retina. 70 00 0 0 0 30 0 C beam aist iameter [mm] 3 00 3 beam aist iameter [mm] 3 0 0 00 30030 0 00 0 00 ar iel ivergence [mra], 00 0 00 0 00 0 300 30 00 ar iel ivergence [mra] Figure 8: Thermal elaxation actor C as a unction o the aist iameter an ivergence o a beam Figure 9: Contours o the angular subtense o the apparent source as a unction o the aist iameter an ivergence o a beam 0 mra also marks a signiicant borer line in igures 8 an 9. All beams ith a ar-iel ivergence o more than 0 mra prouce beam iameters at the retina that correspon to an angular subtense equal or even bigger ILSC 003 CONFEENCE POCEEDINGS
than the maximum α max = 00 mra hereas the angular subtense or beams ith a ar-iel ivergence beteen mra an 0 mra oes not excee 7 mra. 7. CONCLUSIONS The calculations presente above sho that the minimal retinal spot size prouce on the retina by using a telescope compare to that o the nake eye is increase by a actor that is at least equal to the magniying poer o the telescope. In aition it coul be emonstrate that even in the case o telescopic vieing, the angular subtense o the apparent source can be obtaine by the knolege o the aist iameter an the ar-iel ivergence o the beam. The analysis o the most hazarous vieing istance an the corresponing angular subtense inclue a ie range o beam aist iameters ( mm to mm) an ar-iel ivergences ( mra to 00 mra) covering beam propagation ratios M up to 3000. It as shon, that both the most hazarous vieing istance an the angular subtense are primarily unctions o the ar-iel ivergence. The computations reveal that all beams ith a iel ivergence o more than 0 mra have an angular subtense o α max = 00 mra hereas the angular subtense or beams ith ar-iel ivergences beteen mra an 0 mra oes not excee 7 mra. The analysis also inicates that the most hazarous vieing istance or beams ith a iel ivergence above 0 mra is less than m. 8. EFEENCES ANSI Z3. IEC 08-3 B. War, WP Dat.p M. Born, E. Wol, Principles o Optics, Cambrige University Press, Cambrige, 999 E. Galbiati, Evaluation o apparent source in laser saety, Journal o Laser Applications, Vol. 3, Nr., 00 F. Perotti, an L. Perotti, Introuction to Optics, n Eition, Engleoo Clis, Ne Jersey: Prentice Hall, Inc., 993 ILSC 003 CONFEENCE POCEEDINGS