Overlap Relation Between Free Space Laguerre Gaussian Modes and Step Index Fiber Modes

Transcription

1 Reserch Article Journl of the Opticl Society of Americ A Overlp Reltion Between Free Spce Lguerre Gussin Modes nd Step Index Fiber Modes ROBERT BRÜNING, YINGWEN ZHANG, MELANIE MCLAREN 3, MICHAEL DUPARRÉ, AND ANDREW FORBES 3,* Institute of Applied Optics, Abbe Center of Photonics, Friedrich Schiller University Jen, Fröbelstieg, D-7743 Jen, Germny Council for Scientific nd Industril Reserch, Ntionl Lser Center, P.O. Box 395, Pretori, South Afric 3 School of Physics, University of the Witwtersrnd, Privte Bg 3, Johnnesburg 5, South Afric * Corresponding uthor: ndrew.forbes@wits.c.z Compiled June 4, 5 We investigted the overlp reltion of the free spce Lguerre Gussin modes to the corresponding linerly polrized modes of step index fiber. To mximize the overlp for n efficient coupling of the free spce modes into fiber, the scle dependent overlp ws theoreticlly nd experimentlly determined. The presented studies pves the wy for further improvement of free spce to fiber opticl connections. 5 Opticl Society of Americ OCIS codes: (4.349) Lsers, distributed feedbck; (6.4) Fibers, polriztion-mintining;(6.3735) Fiber Brgg grtings. INTRODUCTION Multi-mode fibers (MMF) re widely used in plethor of pplictions such s opticl sensors ], fiber lsers ] nd opticl communiction 3]. Beside the possibility of ll-fiber devices the connection between the fiber nd its supported modes with the free spce modes is importnt to control the opticl properties of the emerging bem or the excited field distributions t the fiber input. The simplest connection is given by fibers with n prbolic refrctive index profile, typicl of grded-index fibers, whose modes cn be described by Lguerre Gussin functions nd re lso solutions of the free spce wve eqution 4]. However, in ll other cses the fiber modes do not mtch the free spce solutions, resulting in no stble propgtion of the emerging bems in free spce. Also, in generl free spce modes re not suitble to excite selective pure fiber modes nd dditionl bem shping techniques re required, such s the use of computer generted hologrms 5]. The min disdvntges of such bem shping techniques re the low efficiency nd resulting high trnsformtion losses. For the common type of step-index fibers, which show the sme cylindricl symmetry s in free spce, it is possible to pproximte the fiber modes by suitble free spce modes 6]. In this pper, we investigte the pproximtion of step-index fiber modes by free spce modes theoreticlly s well s experimentlly to evlute the qulity nd the limittion of this pproch. We determine the overlp reltion between both mode sets s function of the scle prmeter for the free spce s well s the fiber prmeter, V. For the theoreticl investigtions we develop n nlyticl solution for this overlp problem, which llows us to study wide prmeter rnge. Additionlly we investigte experimentlly the overlp reltion by pplying the correltion filter method 7] nd verify the nlyticl solution. Our results will be of interest to studies where fiber to free spce links re necessry.. FUNDAMENTALS At the trnsition from free spce to fiber, nd vice vers, one lwys hs mode coupling between the free spce modes on the one side nd the fiber modes on the other. To chieve mximized coupling efficiency nd low crosstlk between modes of different order, the scling of the free-spce bem hs to be dpted to the fiber. For the cse of wekly guiding step index fiber the fiber modes re given by the linerly polrized (LP) mode set. For tht the field distribution F lp (r, ϕ) is given by the solution of the sclr Helmholtz-eqution. Considering cylindriclly symmetric fiber with core rdius, the solution is given by 4] LP lp (r, ϕ) = N lp J l ( ν lpr )/J l(ν lp ) for r< K l ( µ lpr )/K l(µ lp ) for r eilϕ, () where N lp is normliztion constnt, J l denotes the lth order Bessel-function of the first kind nd K l denotes the lth order modified Bessel-function of the second kind, with ν lp nd µ lp the normlized propgtion constnts of the core nd cldding, respectively. The exct expression for the normliztion constnt

2 Reserch Article Journl of the Opticl Society of Americ A is derived in the ppendix. For the description of the LP modes dditionl the fiber prmeter V, which is defined s V = ν + µ = ( ) πλ (n core n cldding ) () is necessry to define the mount of modes nd their propgtion constnts. Since the step index fiber hs cylindricl symmetry the dpted free spce mode set is given by the Lguerre-Gussin (LG) modes, which re the solutions of the prxil Helmholtzeqution in cylindriclly symmetric coordinte system. For tht the solution t the wist position is given by ( ) l LG pl (r, ϕ) = M r pl w L l p ( r w ) e r w e ilϕ, (3) where L l p re the ssocited Lguerre polynomils nd M pl = ( ) p! w is normliztion fctor. π(l+p)! Compring the modes described by Eq. () nd Eq. (3), we notice tht they show the sme zimuthl dependence nd chrcteriztion of the rdil order by the mount of root points in the intensity distribution. Hence corresponding modes cn be found by choosing the sme zimuthl order nd the field functions with the sme mount of roots in rdil direction. Since both mode sets differ in the ctul shpe of the rdil function nd dption of the scle prmeters is needed for the best possible mtching of both mode sets. The mtching of the scle cn be evluted by the overlp reltion η n = LG n (r, ϕ)lpn (r, ϕ)da, (4) where η n defines the mount of power which is coupled from on mode into the over t the trnsition between both mode sets. It cn rech vlues between one when the fields re perfectly mtched nd zero for orthogonl fields. The mount of power η n which is not coupled into the desired mode goes in other non-orthogonl, reflecting or rditing modes to stisfy energy conservtion. 3. ANALYTICAL ANALYSIS In order to optimize the mode overlp we derive n nlyticl express for the overlp reltion s function of the mode prmeters. First we seprte the problem into core nd cldding prt, corresponding to the solution of the fiber modes Eq.. The e ilϕ term insures tht the LP nd LG modes must hve the sme zimuthl index l. Hence to solve the overlp reltion the rdil prt of Eq. 4 becomes the importnt one. For deriving the solution for the core region we used tht the Bessel function of the first kind cn be written s infinite sum J l (x) = ( ) m ( x ) m+l (5) m!γ(m + l + ) m= with Γ the gmm function, wheres the Lguerre polynomil cn be written s it s generting function L l p (x) = ( ) d p ( ) xξ exp. (6) p! dξ ( ξ) l + ξ ξ= Using Eq. 5 nd 6 for the description of the mode function Eq. nd Eq. 3 respectively the overlp of both modes set within the core is given by ηlp core = N M J l (ν) p! ( ) l ( ) p d dξ ( ξ) l + (7) ( ( ) m νw ) m+l ( ) ξ m+l+ m!γ(m+l+) +ξ m= ]} Γ(m + l + ) Γ(m + l +, +ξ w ξ ) ξ=, with Γ(, z) the incomplete gmm function, N nd M the normliztion constnt of the LP nd LG mode respectively, the fiber core rdius nd w the Gussin rdius. For the cldding region lso nlyticl representtion for the overlp integrl cn be found by similr pproch. For tht we mke use of following reltion between modified Bessel functions of the second kind K l nd the modified Bessel functions of first kind I l π K l (x) = lim n l sin(nπ) I n(x) I n (x)], (8) wheres the I n cn be written s n infinite sum I n (x) = ( x ) m+n. (9) m!γ(m + n + ) m= Using Eq. (8) nd (9) to describe the field distribution of the fiber modes in the cldding region together with the previously used representtion of the LG modes given by Eq. (3) nd (6) the overlp of the cldding prt becomes ηlp cl = lim π N M n l sin(nπ) K n (µ) p! ( ξ) l + m! m= ( ) Γ m + n +, +ξ w ξ Γ(m + n + ) ( ) n ( d dξ ) p () ( µw ( µw ( ) Γ m +, +ξ w ξ Γ(m n + ). ξ= ) ( ) m l ξ m+ + ξ ) ( ) n ξ n + ξ The complete overlp reltion of core nd cldding is thn given by the sum of Eq. (7) nd (). 4. CORRELATION FILTER ANALYSIS For direct urement of overlp reltion between the LP nd the corresponding LG modes the correltion filter method (CFM) ws used. This performs ll opticlly the integrl reltion given by Eq. 4 nd llows the experimentl investigtion of the overlp between LP nd LG modes 7]. The experimentl setup used for the uring of the scle depending overlp reltion is shown in Fig.??. A plne wve hs illuminted phse only sptil light modultor (SLM). Applying the phse only coding technique proposed by Arrizòn et l.? ] enbles the shping of rbitrrily scled LG modes. The generted LG modes were then imged by telescopic 4f setup onto the CF, which hd the field distribution of the LP modes implemented s the trnsmission function. After n opticl Fourier trnsformtion with lens in f configurtion the correltion signl ws ccessible t the CCD sensor, which ws proportionl to the overlp reltion between both modes.

3 Reserch Article Journl of the Opticl Society of Americ A 3 PW SLM L L CF L 3.8 () LP /LG (b) LP /LG.8 CCD Fig.. The experimentl setup for the urement of the overlp reltion between LP nd LG modes, where the LG modes re generted with the SLM nd decomposed by the CFM. PW - plne wve; SLM - sptil light modultor; L 3 - lenses; CCD - cmer overlp η.4. overlp η.4. Since the encoding condition of the generted fields requires the normliztion to unit mplitude the energy conservtion is violted by scling the LG mode size. To ensure the comprbility of the results, correction prmeter for ech encoded field ws introduced, given by the mximl mplitude α mx of the orthonorml field distribution. An pproprited power scling nd correction of the ured intensities I mes ws chievble by subsequently pplying of the correction coefficient for to normlized intensity I norm = I mess α mx 5]. As the lst step the ured nd corrected overlp vlues hs to be normlized to one, which is given by the overlp between the LP mode nd itself. For tht the LP mode encoded in the correltion filter ws generted with the SLM nd the ured overlp reltion ws used to normlized the reltions obtined for the different scled LG modes. This procedure for the genertion of dynmic scled LG modes nd the evlution of the overlp regrding fixed scled set of LP modes enbled the urement of the scle depending overlp reltion between both mode sets. 5. RESULTS We pplied the correltion filter method onto LP mode set with underlying V prmeter of 4.7 resulting in the ppernce of six guided modes. In Fig. resulting scle depending overlp re depicted together with the theoreticlly curve clculted by our nlyticl solution. It cn be seen tht for ll LP modes of the investigted exmple the corresponding LG modes re good pproximtion for rtio between bem with nd core rdius of.75. In this exmple the overlp reltions re bout.99 for the LP, LP nd, LP modes nd bout.98 for the LP mode. Compring the scle depending overlp reltions of the four fiber modes show explicit differences between them. Noticeble re the difference in the decrese of the overlp for not idel scled LG modes, wheres for the fundmentl mode LP the overlp flls reltively slow, while the higher order modes show obvious stronger decline. Additionl the optiml rtio between bem width nd core rdius, for mximized overlp, chnges slightly for the investigted modes from.7 for the LP, see Fig. (d), up to.8 for the fundmentl mode LP, see Fig. (). Also is the mximl rechble overlp for the higher order rdil LP mode, see Fig. (b), with bout.98 slightly lower compred to the other modes. Since the comprison between the urement results nd the theoreticl curved chieved by the nlyticl solution showed very high complince nd proof the relibility of our method, we used the theoreticl vlues to study further the overlp η.8.4. (c) LP /LG (d) LP /LG.8 overlp η.4. Fig.. Comprission between theoreticl nd experimentl determined overlp reltion influences of the mode order or the V prmeter on the overlp reltion. As seen in Fig. the overlp reltion chnge in shpe, mximum position nd mximum vlue for different modes. This mens tht there is lwys n optiml mtch, bsed on the fundmentl Gussin bem size, for the corresponding LG mode set. Applying our nlyticl model we hve determined the scle depending overlp for different higher order mode for fiber with V = 5 to demonstrte some effects. As shown in Fig. 3 for higher order rdil modes with zimuthl index l = nd higher order zimuthl mode with rdil index p =, the position of the best fitting bem size shifts to smller Gussin bem widths with incresing mode order. This behvior becomes intuitive since the bem width of LG modes of higher order follows w = w p + l +, with w the Gussin width. On the other side the LP modes re well confined within the core resulting in decrese of the needed Gussin width to mtch the scle of the corresponding modes. A second effect which cn be seen is tht the scle region with high overlp shrinks with incresing mode order nd good dption of the scle prmeters become more crucil. Especilly for modes of higher rdil order the mximl chievble overlp between both mode sets decrese with incresing mode order indicting the limits of the pplied pproximtion. Hence the pproximtion of LP modes of the step index fiber by LG modes cn not extend for rbitrry higher order modes nd hs to be considered in possible ppliction with respect to the cceptble coupling losses introduced by this mismtch. Nevertheless this results demonstrte the bility to pproximte lower order LP modes efficiently by LG modes. Additionlly we investigted the influence of the underlying V-prmeter on the qulity of our suggested pproximtion.

4 Reserch Article Journl of the Opticl Society of Americ A rdil order p 3 4 zimuthl order l Fig. 3. Size depending overlp reltion for modes with incresing rdil nd zimuthl order. The trend of the best fitting bems size is highlighted by the green line () best fit bem size reltion LP LP LP LP LP 3 LP V prmter (b) mximl overlp V prmter Fig. 4. The V prmeter dependence of the overlp reltion between LP corresponding LG modes. () The best reltion between Gussin bem width nd core rdius for mximized overlp nd (b) the mximl chievble overlp in dependence of V For tht we hve clculted the best fitting bem size nd the mximl chievble overlp for different modes s function of the V prmeter s shown in Fig. 4. In Fig. 4 () the chnge of the best fitting bem size is depicted. It cn be seen tht for low V vlues, in cse of LP, or V vlues close to the cutoff of mode the size reltion becomes lrger one indicting the bd confinement of the mode inside the core. For incresing V vlues the reltion strts rpidly to decrese. A second effect occurs for week confined LP modes in fct tht the overlp reltion becomes reltively low nd increses with incresing distnce from the cut off up to some mximl vlue. Afterwrds the highest chievble overlp strts to decrese, whereby the it drops fster for with incresing mode order. Finlly we hve lso compred the intensity profiles for the best fitting LG modes with the corresponding LP modes. Four exmples for fiber with V prmeter of 4.7 cn be seen in Fig. 5, where the different mode profiles re normlized to unit intensity for better comprbility. It cn be clerly seen tht the conformity decrese with incresing mode order. Figure 5 () illustrting the fundmentl modes LP nd LG respectively re nerly mtched regrding their rdil behvior wheres the distinctions become more significnt for the higher order modes. For modes of higher zimuthl order like the LP nd LG mode respectively, depicted in Fig. 5 (b), the mximums of ring like intensity pttern re slightly displced. Also for () (c) - - (b) LP LP LG LP LG (d) - - LG LP LG Fig. 5. Comprison of the intensity profiles between the LP modes nd the corresponding LG modes for the mximized overlp. higher order rdil modes structurl differences occur like the displcement of the roots nd the side lobe mxim, see Fig. 5 (c). This distinctions become more relevnt with incresing mode order s shown in Fig. 5 (d) for the higher order rdil s well s zimuthl modes LP nd LG respectively, where especilly the side lobes show pregnnt devitions resulting in lower chievble overlp reltion. 6. CONCLUSION We hve shown tht the LP modes of step-index fiber cn be pproximted by n scle dpted set of LG free spce modes. The hve proof the vlidity of this pproximtion experimentlly for exmple fiber with V prmeter of 4.7. Further we used nlyticl solution of the overlp problem to investigte the limits of the pproch for wide spectrum of LP modes nd V prmeters. We hve found tht the pproximtion cn used within certin limits. In order to provide high overlp the LP modes hve to be fr from there cutoff condition nd of low rdil order. Additionlly the best fitting bem size chnge for ech corresponding mode pir. For the few mode cse like the exmple fiber, this dependence is week nd good pproximtion of ll modes cn be found for pproprited dpted LG mode set. In the cse of highly multi mode fibers the used LG mode set hs to be optimized individully for ech corresponding mode pir or t lest for different groups of them. 7. ACKNOWLEDGMENT The uthors thnk Drryl Nidoo for technicl ssistnce nd Sigmund Schröter for fbriction of the correltion filter. 8. APPENDIX In order to evlute the overlp reltion between the LG nd LP modes we need lso nlyticl expression for the normliztion constnt N lp in Eq. (). For tht we use the normliztion

5 Reserch Article Journl of the Opticl Society of Americ A 5 condition = LP lp (r, ϕ)lplp (r, ϕ)da () nd gives N lp = π J l ( ν ] lpr ) rdr J l (ν lp ) + K l ( µ lpr K l (µ lp ) ] ) rdr () The integrls re evluted to be π nd π Jl ( νr )] π rdr = νj ν l (ν) lj l (ν)j l+ (ν) (3) lp +νj l+ (ν) ] Kl ( µr )] π rdr = lim n l 4µ sin(nπ) πµ I n (µ ) (4) + πµi n (µ) µi n (µ) + ni n (µ)] πµi n (µ) + π nµi n (µ)i n (µ) (4n + µ )I n (µ) + µ I +n (µ) ] + µi n (µ) nk n (µ) sin(nπ) +πµi n (µ)]}. with I n (x) being the modified Bessel function of the first kind. Inserting Eq. (3) nd Eq. (4) into Eq. () yields the nlyticl expression for the normliztion constnt for the solution of the overlp reltion for the core nd cldding region Eq. (7) nd Eq. (), respectively. REFERENCES. B. Lee, Review of the present sttus of opticl fiber sensors, Opticl Fiber Technology 9, (3).. S. Rmchndrn, J. Fini, M. Mermelstein, J. Nicholson, S. Ghlmi, nd M. Yn, Ultr-lrge effective-re, higher-order mode fibers: new strtegy for high-power lsers, Lser & Photonics Reviews, (8). 3. G. Li, N. Bi, N. Zho, nd C. Xi, Spce-division multiplexing: the next frontier in opticl communiction, Adv. Opt. Photon. 6, (4). 4. A. W. Snyder nd J. D. Love, Opticl Wveguide Theory (Chpmn & Hll, 996). 5. D. Flmm, C. Schulze, D. Nidoo, S. Schröter, A. Forbes, nd M. Duprré, All-digitl hologrphic tool for mode excittion nd nlysis in opticl fibers, J. Lightwve Technol. 3, 3 3 (3). 6. R. Brüning, S. Ngcobo, M. Duprré, nd A. Forbes, Direct fiber excittion with digitlly controlled solid stte lser source, Opt. Lett. 4, (5). 7. T. Kiser, D. Flmm, S. Schröter, nd M. Duprré, Complete modl decomposition for opticl fibers using CGH-bsed correltion filters, Opt. Express 7, (9).