Chapter 3: Dielectric Waveguides and Optical Fibers

Transcription

1 Chapter 3: Dielectric Waveguides and Optical Fibers Syetric planar dielectric slab waveguides Modal and waveguide dispersion in planar waveguides Step index fibers Nuerical aperture Dispersion in single ode fibers Bit-rate, dispersion, and optical bandwidth Graded index optical fibers Light absorption and scattering Attenuation in optical fibers PHY430 Chapter Three

2 Professor Charles Kao, who has been recognized as the inventor of fiber optics, left, receiving an IEE prize fro Professor John Midwinter (998 at IEE Savoy Place, London, UK; courtesy of IEE). Professor Charles Kao served as Vice Chancellor (President) of the Chinese University of Hong Kong fro 987 to 996. The introduction of optical fiber systes will revolutionize the counications network. The low-transission loss and the large bandwidth capability of the fiber syste allow signals to be transitted for establishing counications contacts over large distances with few or no provisions of interediate aplification. Charles K. Kao PHY430 Chapter Three

3 Prof. Charles Kuen Kao ( 高錕 ) Dr. Charles Kuen Kao is a pioneer in the use of fiber optics in telecounications. He is recognized internationally as the Father of Fiber Optic Counications. He was born in Shanghai on Noveber 4, 933, and was awarded BSc in 957 and PhD in 965, both in electrical engineering, fro the University of London. He joined ITT in 957 as an engineer at Standard Telephones and Cables Ltd., an ITT subsidiary in the United Kingdo. In 960, he joined Standard Telecounications Laboratories Ltd., UK, ITT s central research facility in Europe. It was during this period that Dr. Kao ade his pioneering contributions to the field of optical fibers for counications. After a four years leave of absence spent at The Chinese University of Hong Kong, Kao returned to ITT in 974 when the field of optical fibers was ready for the pre-product phase. In 98, in recognition of his outstanding research and anageent abilities, ITT naed hi the first ITT executive scientist. Fro 987 until 996, Dr. Kao served as vice chancellor (president) of The Chinese University of Hong Kong. PHY430 Chapter Three 3

4 Professor Charles Kao Engineer and Inventor of Fibre Optics Father of Fiber Optic Counications He received L.M. Ericsson International Prize, Marconi International Fellowship, 996 Prince Philip Medal of the Royal Acadey of Engineers*, and 999 Charles Stark Draper Prize**. He was elected a eber of the National Acadey of Engineering of USA in 990. *The Royal Acadey of Engineering Medals The Fellowship of Engineering Prince Philip Medal (solid gold) For his pioneering work which led to the invention of optical fibers and for his leadership in its engineering and coercial realization; and for his distinguished contribution to higher education in Hong Kong. In 989 HRH The Prince Philip, Duke of Edinburgh, Senior Fellow of The Fellowship of Engineering, agreed to the coissioning of a gold edal to be awarded periodically to an engineer of any nationality who has ade an exceptional contribution to engineering as a whole through practice, anageent or education, to be known as The Fellowship of Engineering Prince Philip Medal. PHY430 Chapter Three 4

5 **NAE Awards One of the NAE s goals is to recognize the superior achieveents of engineers. Accordingly, election to Acadey ebership is one of the highest honors an engineer can receive. Beyond this, the NAE presents four awards to honor extraordinary contributions to engineering and society. For the conception and invention of optical fibers for counications and for the developent of anufacturing processes that ade the telecounications revolution possible. The developent of optical fiber technology was a watershed event in the global telecounications and inforation technology revolution. Many of us today take for granted our ability to counicate on deand, uch as earlier generations quickly took for granted the availability of electricity. But this draatic and rapid revolution would siply not be possible but for the developent of silica fibers as a high bandwidth, light-carrying ediu for the transport of voice, video, and data. The silica fiber is now as fundaental to counication as the silicon integrated circuit is to coputing. Optical fiber is the concrete of the inforation superhighway. By the end of 998, there were ore than 5 illion kiloeters of optical fibers installed for counications worldwide. Through their efforts, Kao, Maurer, and MacChesney created the basis of odern fiber optic counications. Their creative application of aterials science and engineering and cheical engineering to every aspect of fiber aterials coposition, characterization, and anufacturing, their understanding of the stringent aterials requireents placed on the fiber by the perforance needs of the telecounications syste, and, above all, their dedication to achieving their vision, were all critical to their success. PHY430 Chapter Three 5

6 Chapter 3: Dielectric Waveguides and Optical Fibers Syetric planar dielectric slab waveguides Modal and waveguide dispersion in planar waveguides Step index fibers Nuerical aperture Dispersion in single ode fibers Bit-rate, dispersion, and optical bandwidth Graded index optical fibers Light absorption and scattering Attenuation in optical fibers PHY430 Chapter Three 6

7 Syetric Planar Dielectric Slab Waveguides The region of higher refractive index (n ) is called the core, and the region of lower refractive index (n ) sandwiching the core is called the cladding. Our goal is to find the conditions for light rays to propagate along such waveguides. PHY430 Chapter Three 7

8 Δφ k k k ( AC) k( AB + BC) [ BC cos( θ ) + BC] k d cosθ φ [ cos ] θ + φ φ [ d cosθ ] φ [ d cosθ ] φ ( π ), 0,,,... A and C are on the sae wavefront. They ust be in phase. Constructive interference occurs between A and C to achieve axial transitted light intensity. Waveguide condition: ( ) OR Δφ(AC) (π), 0,,, πn a θ φ λ cos φ is a function of θ. PHY430 Chapter Three 8 π

9 Δφ k k( AB) φ k( A' B' ) ( AB) k [ AB cos( π θ )] ( )( AB cos θ ) k d k π sin θ k φ [ cos ] θ φ φ [ d cosθ ] φ ( π ), 0,,,... B and B' are on the sae wavefront. They ust be in phase. OR Constructive interference occurs between B and B' to achieve axial transitted light intensity. Δφ(BB') (π), 0,,, Waveguide condition: πn λ ( a) cosθ φ π PHY430 Chapter Three 9

10 To obtain the waveguide condition and solve the propagation odes for the syetric planar dielectric waveguides: () The wave optics approach Solve Maxwell s equations. There is no approxiations and the results are rigorous. () The coefficient atrix approach Straightforward. Not suitable for ultilayer probles. (3) The transission atrix ethod Suitable for ultilayer waveguides. (4) The odified ray odel ethod It is siple, but provides less inforation. PHY430 Chapter Three 0

11 PHY430 Chapter Three Propagation Constant along the Waveguide n k n k θ λ π θ κ θ λ π θ β cos cos sin sin ( ) π φ θ λ π a n cos Larger leads to saller θ. κ : transverse propagation constant

12 PHY430 Chapter Three Electric Field Patterns ( ) ( ) ( ) ( ) [ ] ( ) y a k y a k AC k AC k AC k A C k AC k φ θ φ θ θ π φ θ φ θ π φ Φ cos cos sin cos cos ' ( ) ( ) a y y φ π π + Φ ( ) π φ θ λ π a n cos

13 Electric Field Patterns E E E ( y, z, t) E0 cos( ωt βz + κ y + Φ ) ( y z, t) E cos( ωt β z κ y), 0 ( y z, t) E + E E cos κ y + Φ cos ωt z + Φ, 0 β It is a stationary wave along the y-direction, which travels down the waveguide along z. PHY430 Chapter Three 3

14 Electric Field Patterns The lowest ode ( 0) has a axiu intensity at the center and oves along z with a propagation constant of β 0. There is a propagating evanescent wave in the cladding near the boundary. PHY430 Chapter Three 4

15 Electric Field Patterns PHY430 Chapter Three 5

16 Broadening of the Output Pulse Each light wave that satisfies the waveguide condition constitutes a ode of propagation. The integer identifies these odes and is called the ode nuber. θ is saller for larger. Higher odes exhibit ore reflections and penetrate ore into the cladding. Short-duration pulses of light transitted through the waveguide will be broadened in ters of tie duration. PHY430 Chapter Three 6

17 PHY430 Chapter Three 7 Single and Multiode Waveguides ( ) ( ) π φ λ π / / V n n a V V-nuber (V-paraeter, noralized thickness, noralized frequency) is a characteristic paraeter of the waveguide. When V < π/, 0 is the only possibility and only the fundaental ode ( 0) propagates along the waveguide. It is tered as the single ode waveguide. λ c that satisfies V π/ is the cut-off wavelength. Above this wavelength, only the fundaental ode will propagate. > > sin n n c θ θ θ ( ) π φ θ λ π a n cos ( ) / / 4 / 4 cos n n a n n an n n < + < + > λ π φ π λ π φ π θ

18 TE and TM Modes transverse electric (TE ) field odes transverse agnetic (TM ) field odes The phase change at TIR depends on the polarization of the electric field, and it is different for E and E //. These two fields require different angles θ to propagate along the waveguide. PHY430 Chapter Three 8

19 PHY430 Chapter Three 9 Exaple: waveguide odes Consider a planar waveguide with a core thickness 0 μ, n.455, n.440, light wavelength of 900 n. Given the waveguide condition and the phase change in TIR for the TE ode, find angles θ for all the odes. n n θ θ φ cos sin tan / n n n a θ θ π θ λ π cos sin cos tan / ( ) π φ θ λ π a n cos

20 LHS (left-hand side): RHS (right-hand side): f πn a π tan cosθ λ ( θ ) sin θ n n cosθ / versus θ versus θ PHY430 Chapter Three 0

21 Penetration depth of the evanescent wave: δ α n πn sin θ n λ θ δ δ in μ. An accurate solution of the angle for the fundaental TE ode is The angle for the fundaental TM ode is 89.70, which is alost identical to the angle for the TE ode. / PHY430 Chapter Three

22 Exaple: the nuber of odes Estiate the nuber of odes that can be supported in a planar dielectric waveguide that is 00 μ wide and has n.490, n.470. The freespace light wavelength is μ. V φ π V π πa ( ) / V n n λ π π There are about 49 odes. ( ) / M Int V π + Int (x) is the integer function. It reoves the decial fraction of x. PHY430 Chapter Three

23 Exaple: ode field width (MFW), w 0 The field distribution along y penetrates into the cladding. The extent of the electric field across the waveguide is therefore ore than a. The penetrating field is due to the evanescent wave and it decays exponentially according to δ E cladding cladding The penetration depth ( y' ) E ( 0) exp( α y' ) cladding cladding λ n sin i αcladding πn n For 0 ode (axial ode), θ i 90 The ode field width δ λ π [ n n ] / cladding ( V ) a + w 0 a + a V V a V θ / For optical fibers, it is called the ode field diaeter (MFD). PHY430 Chapter Three 3

24 Chapter 3: Dielectric Waveguides and Optical Fibers Syetric planar dielectric slab waveguides Modal and waveguide dispersion in planar waveguides Step index fibers Nuerical aperture Dispersion in single ode fibers Bit-rate, dispersion, and optical bandwidth Graded index optical fibers Light absorption and scattering Attenuation in optical fibers PHY430 Chapter Three 4

25 Group Velocity Group velocity v g dω dk Phase velocity ω k v c n The group velocity defines the speed with which energy or inforation is propagated because it defines the speed of the envelope of the aplitude variation. To find out the group velocity, we need to know the change of ω with respect to k. The ω versus k characteristics is called the dispersion relation or dispersion diagra. PHY430 Chapter Three 5

26 Waveguide Dispersion Diagra Waveguide condition πn λ ( a) cosθ φ θ depends on the waveguide properties (n, n, and a) and the light frequency, ω. β k πn sin λ θ sinθ π The slope dω/dβ at any frequency is the group velocity v g. ω is a function of β waveguide dispersion diagra. PHY430 Chapter Three 6

27 Interodal Dispersion The lowest ode ( 0) has the slowest group velocity, close to c/n, because the lowest ode is contained ainly in the core, which has a larger refractive index n. The highest ode has the highest group velocity, close to c/n because a portion of the field is carried by the cladding, which has the saller refractive index n. Different odes take different tie to travel the length of the fiber (for an exact onochroatic light wave). This phenoenon is called interodal dispersion. PHY430 Chapter Three 7

28 A direct consequence is that a short duration light pulse signal that is coupled into the waveguide will travel along the guide via the various allowed odes with different group velocities. The reconstruction of the light pulse at the receiving end fro the various odes will result in a broadened signal. The interodal dispersion can be estiated by Δτ v g in v L g in c n, v v L g ax g ax c n Δ τ L n c n n.48 n.46 Δτ 67ns / k L PHY430 Chapter Three 8

29 Intraodal Dispersion The group velocity dω/dβ of a single ode changes with the frequency ω. If the light source contains various frequencies (there is no perfect onochroatic wave), different frequencies will travel at different velocities. This is called waveguide dispersion. The refractive index of a aterial is usually a function of the light frequency. The n ω dependence also results in the change in the group velocity of a given ode. This is called aterial dispersion. Waveguide dispersion and aterial dispersion cobined together are called intraodal dispersion. PHY430 Chapter Three 9

30 Chapter 3: Dielectric Waveguides and Optical Fibers Syetric planar dielectric slab waveguides Modal and waveguide dispersion in planar waveguides Step index fibers Nuerical aperture Dispersion in single ode fibers Bit-rate, dispersion, and optical bandwidth Graded index optical fibers Light absorption and scattering Attenuation in optical fibers PHY430 Chapter Three 30

31 Step Index Fibers Δ n n n Noralized index difference for practical fibers, Δ << The general ideas for guided wave propagation in planar waveguides can be extended to step indexed optical fibers with certain odifications. The planar waveguide is bounded only in one diension. Distinct odes are labeled with one integer,. The cylindrical fiber is bounded in two diensions. Two integers, l and, are required to label all the possible guided odes. PHY430 Chapter Three 3

32 There are two types of light rays for cylindrical fibers. A eridional ray enters the fiber through the axis and also crosses the fiber axis on each reflection as it zigzags down the fiber. A skew ray enters the fiber off the fiber axis and zigzags down the fiber without crossing the axis. It has a helical path around the fiber axis. Guided eridional rays are either TE or TM type. Guided skew rays can have both electric and agnetic field coponents along z, which are called hybrid odes (EH or HE odes). PHY430 Chapter Three 3

33 We usually consider linearly polarized light waves that are guided in step index fibers. A guided ode along the fiber is represented by the propagation of an electric field distribution E l (r, ϕ) along z. E LP ( r, ϕ) [ j( ωt β z) ] E exp l l PHY430 Chapter Three 33

34 V-Nuber of Step Index Fibers πa V n n / λ ( ) Δ When the V-nuber is saller than.405, only the fundaental ode (LP 0 ) can propagate through the fiber core (single ode fiber). The cut-off wavelength λ c above which the fiber becoes single ode is given by V ( n ) n / n ( )/ n n + n V cut off π n > n Δ << πa πa n λ λ [( )( )] n + n n n / ( n Δ) / a λc ( ) / n n. 405 PHY430 Chapter Three 34

35 The nuber of odes M in a step index fiber can be estiated by b λβl n π n n M V Since the propagation constant β l depends on the waveguide properties and the wavelength, a noralized propagation constant is usually defined. This noralized propagation constant, b, depends only on the V-nuber. b 0 corresponds to β l πn /λ b corresponds to β l πn /λ PHY430 Chapter Three 35

36 Exaple: A Multiode Fiber A step index fiber has a core of refractive index.468 and diaeter 00 μ, a cladding of refractive index of.447. If the source wavelength is 850 n, calculate the nuber of odes that are allowed in this fiber. V πa λ π00 n n M V 48 Exaple: A Single Mode Fiber What should be the core radius of a single ode fiber that has a core refractive index of.468 and a cladding refractive index of.447, and is to be used for a source wavelength of.3 μ? V π λ πa n n a.0μ a.405 PHY430 Chapter Three 36

37 Exaple: Single Mode Cut-Off Wavelength What is the cut-off wavelength for single ode operation for a fiber that has a core with a diaeter of 7 μ, a refractive index of.458, and a cladding of refractive index of.45? What is the V-nuber and the ode field diaeter (MFD) when operating at λ.3 μ? V π λ π 7 n n λ a.405 λ.08μ When λ.3 μ: V π λ π 7 n n a.35 V w 0 V.35 ( a) 7 0.3μ PHY430 Chapter Three 37

38 Exaple: Group Velocity and Delay Consider a single ode fiber with core and cladding indices of.448 and.440, core radius of 3 μ, operating at.5 μ. Given that we can approxiate the fundaental ode noralized propagation constant by V b (.5 < V <.5) Calculate the propagation constant β. Change the operating wavelength to λ by a sall aount, say 0.0%, and then recalculate the new propagation constant β. Then deterine the group velocity of the fundaental ode at.5 μ, and the group delay τ g over k of fiber. V b πa λ ( / k) β n n n n n k ω π λ πc λ PHY430 Chapter Three 38

39 λ (μ) V k ( - ) ω (rad s - ) b β ( - ) The group velocity is v g ω ω β ' β ( ) ( ) 5 ' /s The group delay is τ g L v g μs PHY430 Chapter Three 39

40 Chapter 3: Dielectric Waveguides and Optical Fibers Syetric planar dielectric slab waveguides Modal and waveguide dispersion in planar waveguides Step index fibers Nuerical aperture Dispersion in single ode fibers Bit-rate, dispersion, and optical bandwidth Graded index optical fibers Light absorption and scattering Attenuation in optical fibers PHY430 Chapter Three 40

41 Nuerical Aperture Only the rays that fall within a certain cone at the input of the fiber can propagate through the optical fiber. Maxiu acceptance angle α ax is that which just gives TIR at the core-cladding interface. n 0 sinα ax n sin 90 sinα ( θ ) sinθ n / n c ( ) / n n n ax cosθc n0 n0 NA sinα ( n ) / n ax NA n 0 c NA: nuerical aperture α ax : axiu acceptance angle α ax : total acceptance angle PHY430 Chapter Three 4

42 Exaple: A Multiode Fiber and Total Acceptance Angle A step index fiber has a core diaeter of 00 μ and a refractive index of.480. The cladding has a refractive index of.460. Calculate the nuerical aperture of the fiber, acceptance angle fro air, and the nuber of odes sustained when the source wavelength is 850 n. NA n n NA sinα ax n 0 o α ax V πa λ πa π00 n n NA 0.45 λ M V PHY430 Chapter Three 4

43 Chapter 3: Dielectric Waveguides and Optical Fibers Syetric planar dielectric slab waveguides Modal and waveguide dispersion in planar waveguides Step index fibers Nuerical aperture Dispersion in single ode fibers Bit-rate, dispersion, and optical bandwidth Graded index optical fibers Light absorption and scattering Attenuation in optical fibers PHY430 Chapter Three 43

44 Material Dispersion There is no interodal dispersion in single-ode fibers. Material dispersion is due to the dependence of the refractive index on the free space wavelength. Δτ L D Δλ D is called the aterial dispersion coefficient. Dispersion is expressed as spread per unit length because slower waves fall further behind the faster waves over a longer distance. PHY430 Chapter Three 44

45 Material Dispersion D λ d n c dλ The transit tie τ of a light pulse represents a delay between the output and the input. The signal delay tie per unit distance, τ/l, is called the group delay (τ g ). τ τ L v g g dβ dω PHY430 Chapter Three 45

46 Waveguide Dispersion b λβl π n n n V πa λ n n Waveguide dispersion is due to that the group velocity dω/dβ varies as a function of λ. Δτ L D w Δλ D w is called the waveguide dispersion coefficient. PHY430 Chapter Three 46

47 D w Waveguide Dispersion.984N g for.5 < V <.4 ( ) πa cn D and D w have opposite tendencies. PHY430 Chapter Three 47

48 Profile Dispersion There is an additional dispersion echanis called the profile dispersion that arises because the group velocity of the fundaental ode, v g,0, also depends on the noralized index difference. Δ is dependent on the wavelength due to aterial dispersion characteristics, i.e., n versus λ and n versus λ behavior. Therefore, in reality, profile dispersion originates fro aterial dispersion. Δ n n n Δ τ L D p Δ λ D p is called the profile dispersion coefficient. D p is less than ps k - n -, uch saller than D and D w. PHY430 Chapter Three 48

49 Chroatic Dispersion In single-ode fibers, the dispersion of a propagating pulse arises because of the finite width Δλ of the source spectru. This type of dispersion caused by a range of source wavelengths is generally tered chroatic dispersion, including aterial, waveguide, and profile dispersion, since they are all dependent on Δλ. Δτ L D + D w + D p Δλ D D + D + ch w D p PHY430 Chapter Three 49

50 Polarization Dispersion Polarization dispersion arises when the refractive index is not isotropic. When the refractive index depends on the direction of the electric field, the propagation constant of a given ode depends on the polarization. The anisotropic n and n ay result fro the fabrication process (changes in the glass coposition, geoetry, and induced local strains). Typically, polarization dispersion is less than a fraction of ps k -. Polarization dispersion scales roughly with L. PHY430 Chapter Three 50

51 Dispersion Flattened Fibers Dispersion Flattened Fibers D w can be adjusted by changing the waveguide geoetry, for exaple, using fibers with ultiple cladding layers (such fibers are ore difficult to anufacture). It is often desirable to have iniu dispersion over a range of wavelengths. For exaple, fibers with dispersion of 3 ps k - n - over the wavelength range of.3.6 μ allow for wavelength ultiplexing, e.g., using a nuber of wavelengths as counication channels. PHY430 Chapter Three 5

52 Exaple: Material, Waveguide, and Chroatic Dispersion A single-ode fiber has a core of SiO -3.5%GeO for which the aterial and waveguide dispersion coefficients are shown in the figure. This fiber is excited fro a.5 μ laser source with a linewidth Δλ / of n. What is the dispersion per k of the fiber if the core diaeter a is 8 μ? What should be the core diaeter for zero chroatic dispersion at λ.5 μ? At λ.5 μ, D +0 ps k - n - With a 4 μ, D w 6 ps k - n - The chroatic dispersion coefficient is D ch Δτ D + Dw 0 6 The chroatic dispersion is / / L Dch Δλ/ 4psk n ( 4psk n )( n) 8psk The chroatic dispersion will be zero at.5 μ when D w D or when D w 0 ps k - n -. The core radius should therefore be about 3 μ. The dispersion is zero only at one wavelength. PHY430 Chapter Three 5

53 Chapter 3: Dielectric Waveguides and Optical Fibers Syetric planar dielectric slab waveguides Modal and waveguide dispersion in planar waveguides Step index fibers Nuerical aperture Dispersion in single ode fibers Bit-rate, dispersion, and optical bandwidth Graded index optical fibers Light absorption and scattering Attenuation in optical fibers PHY430 Chapter Three 53

54 Bit Rate and Dispersion In digital counications, signals are generally sent as light pulses along an optical fiber. Inforation is first converted to an electrical signal in the for of pulses that represent bits of inforation. The electrical signal drives a laser diode whose light output is coupled into a fiber for transission. The light output at the destination end of the fiber is coupled to a photodetector that converts the light signal back to an electrical signal. The inforation bits are then decoded fro this electrical signal. Engineers are interested in the axiu rate at which the digital data can be transitted along the fiber. This rate is called the bit rate capacity B (bits per second) of the fiber. PHY430 Chapter Three 54

55 Bit Rate and Dispersion Suppose we feed light pulses of short duration into the fiber. The output pulses will be broadened due to various dispersion echaniss. The dispersion is typically easured between half-power (or intensity) points and is called full width at half power (FWHP), or full width at half axiu (FWHM). To clearly distinguish two consecutive pulses, that is no intersybol interference, requires that they be separated fro peak to peak by at least Δτ / (intuitively). PHY430 Chapter Three 55

56 Bit Rate and Dispersion There are two types of bit rates. One is called the return-to-zero (RZ) bit rate, for which a pulse representing the binary inforation ust return to zero before the next binary inforation. The other is called the non-return-to-zero (NRZ) bit rate, for which two consecutive binary pulses don t have to return to zero in between, that is, the two pulses can be brought closer. In ost cases we refer to the RZ bit rate. PHY430 Chapter Three 56

57 Bit Rate and Dispersion The axiu bit rate depends on the input pulse shape, fiber dispersion characteristics (hence the output pulse shape), and the odulation schee of inforation bits. For Gaussian output light pulses h(t) centered at 0: h () t t σ e Standard deviation σ 0.45Δτ / πσ B σ Δτ / PHY430 Chapter Three 57

58 Bit Rate and Dispersion Dispersion increases with fiber length L and also with the wavelength range of the source, σ λ 0.45Δλ /. It is therefore ore appropriate to specify the product of the bit rate B and the fiber length L at the operating wavelength. BL 0.5L σ output 0.5L L D 0.5 ch σ λ,input Dch σ λ,input BL is a characteristic of the fiber, through D ch, and also of the wavelength range of the source. In specifications, the fiber length is taken as k and its unit is therefore Gb s - k. When both chroatic (intraodal) and interodal dispersion are present and needed to be taken into account. The overall dispersion can be found according to σ overall σ interodal + σ intraodal PHY430 Chapter Three 58

59 Optical and Electrical Bandwidth The input light intensity into the fiber can be odulated to be sinusoidal. The light output intensity at the fiber destination should also be sinusoidal with a phase shift due to the tie it takes for the signal to travel along the fiber. We can deterine the transfer characteristics of the fiber by feeding in light intensity signals with various frequencies. PHY430 Chapter Three 59

60 Optical and Electrical Bandwidth The response, as defined by P o /P i, is flat and falls with frequency when the frequency becoes too large so that dispersion effects sear out the light at the output. The frequency f op at which the output intensity is 50% below the flat region defines the optical bandwidth of the fiber and hence the useful frequency range in which odulated optical signals can be transferred along the fiber. For Gaussian output light pulses, we have f op The electrical signal fro the photodetector (current or voltage) is proportional to the fiber output light intensity. The electrical bandwidth, f el, is usually defined as the frequency at which the electrical signal is 70.7% of its low frequency value. 0.75B 0.9 σ PHY430 Chapter Three 60

61 Relationship between dispersion paraeters, axiu bit rates, and bandwidths Dispersed output pulse shape FWHM, Δτ / B (RZ) B (NRZ) f op f el Gaussian with standard deviation σ σ 0.45Δτ / 0.5/σ 0.5/σ 0.75B 0.9/σ 0.7f op 0.3/σ Rectangular with full width ΔT σ 0.9ΔT 0.9Δτ / 0.5/σ 0.5/σ 0.69B 0.7/σ 0.73f op 0.3/σ PHY430 Chapter Three 6

62 Exaple: Bit Rate and Dispersion Consider an optical fiber with a chroatic dispersion coefficient 8 ps k - n - at an operating wavelength of.5 μ. Calculate the bit rate-distance product (BL), and the optical and electrical bandwidths for a 0 k fiber if a laser diode source with a FWHP linewidth Δλ / of n is used. The FWHP dispersion at the output side is Δτ L D λ ( 8psk n )( n) 6 / / ch Δ / psk Assue a Gaussian light pulse shape, the RZ bit rate-distance product is f op BL 0.5L 0.5L Gbs k σ 0.45Δτ The optical and electrical bandwidths for a 0 k distance are / 0.9 / σ 0.9 / τ / ( 0.45Δ ) 0.9 /( ).8GHz f el 0.7 f op.0ghz PHY430 Chapter Three 6

63 Chapter 3: Dielectric Waveguides and Optical Fibers Syetric planar dielectric slab waveguides Modal and waveguide dispersion in planar waveguides Step index fibers Nuerical aperture Dispersion in single ode fibers Bit-rate, dispersion, and optical bandwidth Graded index optical fibers Light absorption and scattering Attenuation in optical fibers PHY430 Chapter Three 63

64 Drawback of Single Mode Step Index Fibers Single ode step index fibers have sall NA and the aount of light coupled into a fiber is liited. Multiode step index fibers have large NA and core diaeters, which allow for ore light power launched into a fiber. However, they suffer fro interodal dispersion. Intuitively, those rays that experience less reflections will arrive at the end of the fiber earlier. PHY430 Chapter Three 64

65 Graded Index (GRIN) Fibers In the graded index (GRIN) fiber, the refractive index is not constant within the core but decreases fro n at the center, as a power law, to n at the cladding. Such a refractive index profile is capable of iniizing interodal dispersion. Intuitively, the velocity, c/n, is not constant and increases away fro the center. A ray such as that has a longer path than ray experiences a larger velocity during a part of its journey to enable it to catch up with ray. Siilarly, ray 3 experiences a larger velocity than ray during part of its propagation to catch up with ray. PHY430 Chapter Three 65

66 Graded Index (GRIN) Fibers The refractive index profile can be described by a power law with an index γ, which is called the profile index (or the coefficient of index grating). n n n n [ Δ( r a) ] γ / / The interodal dispersion is iniized when (Δ is sall) 4 + Δ + Δ 3 γ + 3Δ 3 + Δ With the optial profile index, the dispersion in the output light pulse per unit length is given by ( + Δ) Δ ( Δ) σ interodal L r < a r a 0 n 3c Δ PHY430 Chapter Three 66

67 PHY430 Chapter Three 67

68 Chapter 3: Dielectric Waveguides and Optical Fibers Syetric planar dielectric slab waveguides Modal and waveguide dispersion in planar waveguides Step index fibers Nuerical aperture Dispersion in single ode fibers Bit-rate, dispersion, and optical bandwidth Graded index optical fibers Light absorption and scattering Attenuation in optical fibers PHY430 Chapter Three 68

69 Absorption In general, light propagating through a aterial becoes attenuated in the direction of propagation. In absorption, soe of the energy fro the propagating wave is converted to other fors of energy, for exaple, to heat by the generation of lattice vibrations. PHY430 Chapter Three 69

70 Scattering When a propagating wave encounters a sall dielectric particle or a sall inhoogeneous region whose refractive index is different fro the average refractive index of the ediu, the field forces dipole oscillations in the dielectric particle or region by polarizing it, leading to the eission of electroagnetic waves in any directions so that a portion of the light energy is directed away fro the incident bea. PHY430 Chapter Three 70

71 Scattering Whenever the size of a scattering region, whether an inhoogeneity or a sall particle, is uch saller (<λ/0) than the wavelength of the incident wave, the scattering process is generally tered Rayleigh scattering. Lord Rayleigh, an English physicist (877 99) and a Nobel laureate (904), ade a nuber of contributions to wave physics of sound and optics. PHY430 Chapter Three 7

72 blue sky yellow sun sunrise sunset PHY430 Chapter Three 7

73 Why the sky is blue? Why does the sun look yellow if we look at the sun directly? Why does the sky around the sun appear red during sunrise and sunset? PHY430 Chapter Three 73

74 Rayleigh scattering becoes ore severe as the frequency of light increases (the wavelength decreases). Blue light that has a shorter wavelength than red light is scattered ore strongly by particles in air. When we look at the sun directly, it appears yellow because the blue light has been scattered. When we look at the sky in any direction but the sun, our eyes receive scattered light, which appears blue. At sunrise and sunset, the rays fro the sun have to travel the longest distance through the atosphere and have the ost blue light scattered, which gives the sky around the sun its red color at these ties. PHY430 Chapter Three 74

75 Chapter 3: Dielectric Waveguides and Optical Fibers Syetric planar dielectric slab waveguides Modal and waveguide dispersion in planar waveguides Step index fibers Nuerical aperture Dispersion in single ode fibers Bit-rate, dispersion, and optical bandwidth Graded index optical fibers Light absorption and scattering Attenuation in optical fibers PHY430 Chapter Three 75

76 Attenuation in Optical Fibers Assue a fiber of length L. The input optical power is P in. The optical power is attenuated to P out at the end of the fiber. We define an attenuation coefficient α for the fiber. dp dp P α P out αpdx αdx L Pin ln P P exp in out ( αl) Optical power attenuation in optical fibers is generally expressed in ters of decibels per unit length of fiber, typically as db per k. α α db db 0log L 0 ln 0 ( ) α P P in out 4.34α PHY430 Chapter Three 76

77 Attenuation in Optical Fibers The sharp increase in attenuation at wavelengths beyond.6 μ is due to energy absorption by lattice vibrations of silica. Two peaks at.4 and.4 μ are due to OH ions in silica glasses. The overall background is due to Rayleigh scattering because of the aorphous structure of silica glasses (ipossible to eliinate Rayleigh scattering in glasses). PHY430 Chapter Three 77

78 Attenuation in Optical Fibers The attenuation α R in a single coponent glass due to Rayleigh scattering is approxiately given by α R 8 3 π λ 3 4 ( ) n β k T T B f λ is the free space wavelength. T f is called the fictive teperature, at which the liquid structure during the cooling of the fiber is frozen to becoe the glass structure. β T is the isotheral copressibility of the glass at T f. PHY430 Chapter Three 78

79 Microbending and Macrobending Losses Microbending is due to a sharp local bending of the fiber that changes the guide geoetry and refractive index profile locally, which leads to soe of the light energy radiating away fro the guiding direction. Local bending leads to an increase in the incidence angle, which induces either an increase in the penetration depth into the cladding or a loss of total internal reflection. PHY430 Chapter Three 79

80 Microbending and Macrobending Losses Measured icrobending loss for a 0-c fiber bent with different aounts of radius of curvature. Single ode fiber with a core diaeter of 3.9 μ, cladding radius 48 μ, Δ , NA 0., V.67 and.08. Microbending loss increases sharply with decreasing radius of curvature. Macrobending loss is due to sall changes in the refractive index of the fiber due to induced strain when it is bent during its use, for exaple, when it is cabled and laid. Typically, when the radius of curvature is close to a few centieters, acrobending loss crosses over into the regie of icrobending loss. PHY430 Chapter Three 80

81 PHY430 Chapter Three 8

82 How Sall A Fiber Can Be? cladding step index fiber E LP ( r, ϕ) [ j( ωt β z) ] E exp l l An electric field pattern E l (r, ϕ) propagates along the fiber. core LP 0 ode PHY430 Chapter Three 8

83 How Sall A Fiber Can Be? Due to the presence of the evanescent wave in the cladding, not all of the optical power propagating along the fiber is confined inside the core. The extent to which a propagating ode is confined to the fiber core can be easured by the ratio of the power carried in the cladding to the total power that propagates in the ode. ν P core P cladding + P cladding V πa λ ( n n ) / V cut-off (single ode).405 PHY430 Chapter Three 83

84 Consider a step index fiber with a silica core (n.45). The cladding of this fiber is siply air (n.0). The laser light source is fro an argon ion laser with a wavelength of 54.5 n (green light). If the radius of this fiber is equal to the light wavelength (a 54.5 n), then V How Sall A Fiber Can Be? πa π λ ( ) / n n ν 0.0 If the diaeter of this fiber is equal to the light wavelength (a 54.5 n), then V 3.3 ν 0. If the diaeter of this fiber is equal to half the light wavelength (4a 54.5 n), then V.6 ν 0.5 PHY430 Chapter Three 84

85 Subwavelength Waveguides SnO nanoribbons 350 n wide 45 n thick λ 0 ~55 n M. Law, D. J. Sirbuly, J. C. Johnson, J. Goldberger, R. J. Saykally, P. D. Yang, Science 004, 305, 69. For such sall fibers, difficulties lie in fabrication, precise positioning, and light coupling. Propagation loss ight not be a proble if our goal is to fabricate high-density integrated photonic circuits. PHY430 Chapter Three 85

86 Plasonic Waveguides There has been great interest in the use of optical interconnects to exchange digital inforation between electronic icroprocessors. One severe liit on the integration of optical and electronic circuits is their respective sizes. Electronic circuits can be fabricated with sizes below 00 n, while the iniu sizes of optical structures are liited by optical diffraction to the order of 000 n. Surface plason-based photonics, plasonics, ay offer a solution to this dilea. The resonant interaction between the electron oscillations near the etallic surface and the electroagnetic field of light creates surface plasons, which are bound to the etallic surface with exponentially decaying fields in both neighboring edia. This feature of surface plasons provides the possibility of the localization and guiding of light in sub-wavelength etallic structures, and it can be used to construct iniaturized optoelectronic circuits with sub-wavelength coponents. PHY430 Chapter Three 86

87 Plasonic Waveguides Thin Metal Fils 40-n thick and.5-μ wide Au stripes lying on a glass substrate. One of the surface plason eigen odes is excited by total internal reflection illuination at a wavelength of 800 n. W. L. Barnes, A. Dereux, T. W. Ebbesen, Nature 003, 44, 84; E. Ozbay, Science 006, 3, 89. PHY430 Chapter Three 87

88 Plasonic Waveguides Thin Metal Fils The guided surface plason ode has three axia. Both the height and the square root of the waveguide cross section features a sub-wavelength size, iplying that the guided ode is essentially bound to the etal surface rather than being a standing wave confined inside the etal volue. Advantages: () Optical signals and electric currents can be carried through the sae thin etal guides. () Light can be transported along these etallic thin fils over a distance fro several icroeters to several illieters, depending the sizes of etallic structures and light wavelengths. Disadvantages: There exists ohic resistive heating within the etal, which liits the axiu propagation length within these structures. PHY430 Chapter Three 88

89 Plasons Plasonic behavior is a physical concept that describes the collective oscillation of conduction electrons in a etal. Many etals can be treated as free-electron systes whose electronic and optical properties are deterined by the conduction electrons alone. In the Drude Lorentz odel, such a etal is denoted as a plasa, because it contains equal nubers of positive ions (fixed in position) and conduction electrons (free and highly obile). Under the irradiation of an electroagnetic wave, the free electrons are driven by the electric field to coherently oscillate at a plasa frequency of ω p relative to the lattice of positive ions. For a bulk etal with infinite sizes in all three diensions, ω p is related to the nuber density of electrons. Quantized plasa oscillations are called plasons. ω p ε Ne 0 e / Metal Na Al Ag Au Theoretical λ p 00 n 77 n 40 n 40 n PHY430 Chapter Three 89

90 Propagating Surface Plasons (PSPs( PSPs) In reality, etallic structures are of finite diensions and are surrounded by aterials with different dielectric properties. Since an EM wave ipinging on a etal surface only has a certain penetration depth, just the electrons on the surface are the ost significant. Their collective oscillations are properly tered surface plason polaritons (SPPs), but are often referred to as surface plasons (SPs). For a etal vacuu interface, application of the boundary condition results in an SP ode of ω p / / in frequency. Such an SP ode represents a longitudinal surface charge density wave that can travel across the surface. For this reason, these SPs are also widely known as propagating SPs (or PSPs). PHY430 Chapter Three 90

91 Propagating Surface Plasons Propagating surface plasons at a etal-dielectric interface have a cobined electroagnetic wave and surface charge character. There is an enhanced field coponent perpendicular to the interface and decaying exponentially away fro the interface. This evanescent wave reflects the bound, non-radiative nature of surface plasons and prevents power fro propagating away fro the interface. PHY430 Chapter Three 9

92 Decay Lengths of PSPs The decay length, δ d, in the dielectric ediu above the etal, typically air or glass, is of the order of half the wavelength of involved light. The decay length in the etal, δ, is typically between one and two orders of agnitude saller than the wavelength involved, which highlights the need for good control of fabrication of surface plason-based devices at the nanoeter scale. PHY430 Chapter Three 9

93 Propagation Length of PSPs δ SP c ω ' ' ε + ε ε ε d d 3 ( ' ) ε ε " ε d is the dielectric constant of the dielectric aterial. ε ε +ε is the dielectric function of the etal. Silver has the lowest absorption losses (sallest ε ) in the visible spectru. The propagation length for silver is typically in the range of 0 00 μ, and increases to as the wavelength oves into the.5 μ near-infrared telecounication band. In the past, absorption by etals was seen as such a significant proble that surface plasons were not considered as viable for photonic eleents. This view is now changing due priarily to recent deonstrations of surface plasonbased coponents that are significantly saller than the propagation length. Such developents open the way to integrate surface plason-based devices into circuits before propagation losses becoe too significant. PHY430 Chapter Three 93

94 Characteristic Length Scales of PSPs These characteristic length scales are iportant for PSP-based photonics in addition to the associated light wavelength. The propagation length sets the upper size liit for any photonic circuit based on PSPs. The decay length in the dielectric aterial, δ d, dictates the axiu heights of any individual feature that ight be used to control surface plasons. The decay length in the etal, δ, deterines the iniu feature size that can be used. PHY430 Chapter Three 94

95 Moentu Misatch between PSPs and Free- Space Light Wave k SP k 0 ε ε d d ε + ε The interaction between the surface charge density and the electroagnetic field results in the oentu of the surface plason ode, ħk sp, being larger than that of a free-space photon of the sae frequency, ħk 0. The oentu isatch proble ust be overcoe in order to couple light and surface plasons together. PHY430 Chapter Three 95

96 Plasonic Waveguides Arrays of Metal Nanoparticles One can use an array of etal nanoparticle resonators in order to avoid ohic heating. The resonant structures of nanoparticles can be used to guide light, whereas the reduced etallic volue eans a substantial reduction in ohic losses. Gold nanodots: Lattice constant 500 n Height 50 n Edge sizes n (at the center, decreasing to n at both sides) The waveguide is terinated at both ends by irrors consisting of a copressed lattice. Designed for the guiding of the light at a wavelength of.6 μ PHY430 Chapter Three 96

97 Plasonic Waveguides Arrays of Metal Nanoparticles This structure has been shown to have a decay length of about 50 μ, whereas finite-difference tie-doain (FDTD) siulations predict a /e energy attenuation length of 30 μ. PHY430 Chapter Three 97

98 Plasonic Waveguides Grooves in Metal Fils Surface plasons are bound to and propagate along the botto of V-shaped grooves illed in gold fils. λ 600 n d..3 μ θ ~5 Y-splitter Mach-Zehnder interferoeter S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, T. W. Ebbesen, Nature 006, 440, 508. PHY430 Chapter Three 98

99 Plasonic Waveguides Grooves in Metal Fils Waveguide-ring resonator PHY430 Chapter Three 99

100 Reading Materials S. O. Kasap, Optoelectronics and Photonics: Principles and Practices, Prentice Hall, Upper Saddle River, NJ 07458, 00, Chapter, Dielectric Waveguides and Optical Fibers. PHY430 Chapter Three 00