Propagation Modes in Multimode Graded-Index Fibers

Transcription

1 International Journal o Science and Research (IJSR) ISSN (Online): Index Copernicus Value (013): 6.14 Impact Factor (015): Propaation Modes in Multie Graded-Index Fibers Zaman Hameed Kareem 1, Ali Hadi Hassan, Hassan Abid Yasser 3 1, Department o Physics, Collee o Education, Al-Mustansiriyah University, P.O.Box:4619,Bahdad, Iraq. 3 Thi-Qar University, Collee o Science, Department o Physics Abstract: In this research some important parameters o raded index iber have been studied such as numerical aperture, the normalied requency and their eects on the al dispersion. The al dispersion and the propaation delay have been calculated and plottedas unctions o radial number and aimuthal number. Keywords: raded index iber, Step index ibers,al dispersion, radial number, Aimuthal number 1. Introduction As its name implies, multie ibers propaate more than one e. Multie ibers can propaate over 100 es. The number o propaated es depends on the core sie and numerical aperture NA. However,when the core sie and NA increase, the number o es increases. Typical values o iber core sie, and NA are 50 to 100 m and 0.0 to 0.9, respectively.launched into a multie iber with more ease. The hiher NA and the larer core sie make it easier to make iber connections. Durin iber splicin, core-to-core alinment becomes less critical. Another advantae is that multie ibers permit the utiliation o liht-emittin diodes LEDs. Sinle e ibers typically must use laser diodes. LEDs are cheaper, less complex, and last loner. LEDs are preerred or most applications. Multie ibers also have some disadvantaes. As the number o es increases, the eect o al dispersion increases [1]. Modal dispersion (interal dispersion) means that es arrive at the iber end at slihtly dierent times. This time dierence causes the liht pulse to spread. Modal dispersion aects system bandwidth. Fiber manuacturers adjust the core diameter, NA, and index proile properties o multie ibers to maximie system bandwidth. Then multie optical ibers are cateoried into two types accordin to their structure, Step index ibers (SIF S ) and Graded index iber (GIF S )[].. Parameters o Graded Index Fiber the liht incident on the iber core at position R will propaate as a uided e only i it is within the local numerical aperture NA(R) at that point. The local numerical aperture is deined as [3] Whereq the raded order, the axial local numerical aperture NA(0) is deined as NA(0) n1.where is the core-claddin index dierence? It is clear that NA o a 'raded' index iber; decreases rom NA(0) to ero as moves rom the iber axis to the core claddin boundary. As the use o al distribution (R), the undamental e in the [step index; iber] is mostly 'approximated' with Gaussian distribution o the orm [4] where the width parameter is determined by curve ittin or by ollowin a variational procedure. The quality o it is enerally quite ood or values o V (normalied requency) the neihborhood o. The spot sie can be determined rom an analytical approximation accurate to within 1% or 1. V. 405 and iven by [5] The spot sie w is dierent or each q, where the smaller q is the larer spot sie w. V, V is proportional to From the deinition The determination o NA or "raded; index iber" is more complex than that or "step index iber.in raded index iber, 1 i is relatively independent o V and 1. In this NAis a unction o position across the core end ace. This is case, in contrast to the step index iber, where NA is constant across the core. Geometrical optics consideration shows that 1 1 (4) Now, usin the deinition 1 a V and chain rule, one may be ound 1 a Volume 5 Issue 11, November 016 Paper ID: ART DOI: /ART

2 Then we can write International Journal o Science and Research (IJSR) ISSN (Online): Index Copernicus Value (013): 6.14 Impact Factor (015): V V a, a a V V V V V V V V 1 V 1 terms within the brackets can be estimated rom the curves o 1 the / versusv. For small V and the undamental e, the bracketed term can be positive (or neative waveuide dispersion), so it is possible to use small V or a to cancel the material dispersion or to have ero chromatic dispersion. This is the basic principal o / dispersion-shited ibers. Note that, the curve o 1 and V or the undamental e is dierent or dierent raded order. That is; the waveuide dispersions o" the sinle e iber is aected by the raded order. In turn, the waveuide dispersion can be controlled by the iber characteristics such as iber radius and reractive indices as well as the index proile. 3. Graded-index Fibers To perorm the dierentiation, note that F is a unction o w. Let where A ( c )/( a n1 ) that may be assumed does n w because not depend onw, inorin the eect o 1 / it much smaller than the al dispersion eect. As a consequence Usin Eq.(8) into (6), the unit propaation delay or e will be The al dispersion in raded index ibers is computed by a complicated procedure that depends on many approximations. In the present work, we are attempted to deduce the al dispersion o raded-; index ibers. Multie optical ibers described by i (, r, ) A (, r ) e i i i( t ) support a lare but inite number o es which are particular solutions o Maxwell's equations. Each e propaates at its own velocity resultin rom its particular propaation constant. From the WKB approximation, the al propaation constant was approximated by [6] Eq.(9) may be rearraned to explain The binomial expansion or 1 S will ive Substitutin this expansion into Eq.(10) will obtain where q / ( q ), m 1 stands or the principal e number and F represents the total number o propaation es iven by Eq.(5), which may be rewritten as F ank 1 o. Note that, the pair (, m ) represent the radial and aimuthal numbers. Physically, (, m ) have the meanin that they count the number o maximum intensities that may appear in the radial and aimuthal directions in the ield intensities o a iven e. In a strict sense, the e number is a discrete inteer parameter which takes values ranin rom unity or LP01 e to the total number o e roups. However, very oten can be treated as a continuous variable. This approximation is o reat interest because it allows one to replace the discrete e spectrum by a el continuum. As a result, the WKB method can readily be used and es sums can be converted to interals that are easier to handle [3,6]. From Eq. (5), the unit propaation delay is Inorin the terms with 3 and hiher, the last equation will be To calculate the al dispersion, we ind the propaation es that ive the maximum and minimum roup delay. The minimum dispersion occurs when the delays at the two endpoints are S (1 / F) ands 1. That is; are the same, where For multie ibers F 1, such that 0. In this case, the optimum may be ound by equaliin the results in Eq.(14) to et Volume 5 Issue 11, November 016 Paper ID: ART DOI: /ART

3 Eq.(16) may be reormed to obtain International Journal o Science and Research (IJSR) ISSN (Online): Index Copernicus Value (013): 6.14 Impact Factor (015): Usin the deinition q / ( q ) and Eq.(17), we will et Under the above conditions, al dispersion can be explained as the dierence between and. That is; the al dispersion 1 F D represents the dierence Substitutin Eq.(14) into (19) and usin the result in Eq.(18), the al dispersion will be D 0, while the approximation qopt (1 ) will make Like that D or raded-index iber is proportional to, which is much smaller than D n1 / c, or step;- index iber. It is important to note that the above approximations are accurate only i the al dispersion is considered and the chromatic dispersion is inored. This may be attributed to the requency dependence o and n that assumed constants in the above derivation. 1 In eneral, the al dispersion is computed usin Eq.(13), m where or any e and any raded order may be determined. The dispersion between two es will be the dierence. For the step;-index iber that is a special case o raded;-index iber with, we have q and 0 S 0 or the minimum e order and hence 1 n1 / c, while the maximum e order has S 1 that ives be non-limited. The iure shows that the mentioned actor be positive with a rane o V. Thereore, Dwav 0 bein within this rane as a result it may be usin the neative value to control in D manitude to havin a shited ch custom made. It is noted that rom the iure the "step index" iber case; the intentional rane is (0.-.) while the other cases o "raded index" iber causes this rane to be shited to the riht. Thereore, D depends on raded order basically as well as to its dependency on V. In other words, the controllin in chromatic dispersion will depends on q. From the other hand, as lon as D is important only in the sinle eiber, so that there is not needin to plot the other es. Fi. () shows the behavior o D with the radial number or many cases o q where it is usin the disparity between both the 'maximum and minimum' values which revert to the same aimuthal number.however, rom the iure, one can see that whenq 0 the parameter D equates to ero, wav wav D while or the cases in which q less than, the has neative values. So, the D increases with increasinq until arrivin to its maximum value at q 4 ater which, it is turnabout to decreasin because o the nature o the minimum and maximum values that is revert to a speciic aimuthal number. The physical causation behind this behavior reverts to the es nature, where there is a commutation takes place amon them whenq and this siniicant is because there is an antecedence or slowness or the e relative to the other e. The continuation in q increasin return in D value to a stationary case deerence on ero which is so lare in comparison with the case q or in comparison with the chromatic dispersion. Fi. (3) Shows the behavior o D with the aimuthal number, where the dierence ave between the minimum and maximum values relatin to a speciic radial number. From the two iures comparison we notice a variation behavior symmetrical with D values varyin. However, the al dispersion or step index iber will be 1 F D n c. 1 / 4. Results and Discussion Fi. (1) Illustrates the variation o the actor V V V 1 V 1 as a unction o V or all collection cases o q. All the descriptions pertains the LP 01 e because it is the alone e that is be limited within a limited rane o V, while the remainder es will Fi. (4) Illustrates as a unction o radial numberor a number o q cases and or aimuthal number divers values. From iure it isclariied that has a sliht start point and a hih end point. The perceptive dierence in antecedent iures in the disparity between 'these two points' except the case q 1in which one can see the reverse occurrence. We note that the increasin o L mean the curve elevation except the case q 1occurrences the reverse too. The existence o q conversed caused existence D 0 this is apparent in the two antecedent iures. Volume 5 Issue 11, November 016 Paper ID: ART DOI: /ART

4 International Journal o Science and Research (IJSR) ISSN (Online): Index Copernicus Value (013): 6.14 Impact Factor (015): V 1 Fiure 1: The actorv V V 1 as a unction ov or dierent raded orders. Fiure : D as a unction o radial number or dierent raded orders Volume 5 Issue 11, November 016 Paper ID: ART DOI: /ART

5 International Journal o Science and Research (IJSR) ISSN (Online): Index Copernicus Value (013): 6.14 Impact Factor (015): Fiure 3: D as a unction o aimuthal number or dierent raded orders Fiure 4: as a unction o radial number or dierent aimuthal numbers and dierent raded orders. Volume 5 Issue 11, November 016 Paper ID: ART DOI: /ART

6 5. Conclusion International Journal o Science and Research (IJSR) ISSN (Online): Index Copernicus Value (013): 6.14 Impact Factor (015): In raded index iber some parameters have the ollowin behaviors 1) LP 01 e is the alone e that is be limited within a limited rane o V. So, the mentioned actor be positive with a rane o V. Thereore, Dwav 0 bein within this rane ) Dwav dependon raded order and V. 3) Whenq 0 the parameter D equates to ero, while D or the cases in which q less than, the has neative values. D increases with increasin q until arrivin to its 4) maximum value at q 4 ater which, it is turnabout to decreasin. 5) The continuation in q increasin return in D value to a stationary case deerence on ero which is so lare in comparison with the case q or in comparison with the chromatic dispersion. 6) The varyin o D with the radial number is symmetrical with D values varyin with the aimuthal number. 7) have a sliht start point and hih end point. Reerences [1] Hattori H., "Low Nonlinearity Optical Fibers or Broadband and Lon Distance Communications", Ph. D. Thesis, Virinia Polytechnic Institute and state University, [] Cheo P., Fiber Optics and Optoelectronics (Prentice Hall Series In Solid Physical Electronics, Second Edition, 1990) Ch.4. [3] Keiser G., "Optical Fiber Communications", Second Edition, McGraw-Hill, Inc., [4] Dutta I. et al., Chowdhury A. and Kumbhakar D., "Accurate Estimate o Some Propaation Characteristics or the First Hiher Order Mode in Graded Index Fiber with Simple Analytic Chebysher Method", Journal o Optical Communications, Vol.34, No.1, 013, pp [5] Arawal G., "Fiber-Optic Communication Systems", Third Edition, John Wiley and Sons, Inc., 00. [6] Bell W., "Special Functions or Scientists and Enineers", D. Van Nostrand Company, Ltd, Volume 5 Issue 11, November 016 Paper ID: ART DOI: /ART