Fundamentals of fiber waveguide modes

Transcription

1 SMR Winter College on Fibre Optics, Fibre Lasers and Sensors 1-3 February 007 Fundamentals of fiber waveguide modes (second part) K. Thyagarajan Physics Department IIT Delhi New Delhi, India

2 Fundamentals of fiber waveguide modes K Thyagarajan Physics Department IIT Delhi New Delhi, India ktrajan@physics.iitd.ernet.in Winter college on Fiber Optics, Fiber Lasers and Sensors February 1-3, 007, ICTP, Trieste Additional material Leaky modes Consider the following refractive index profile of a planar waveguide n c n f n(x) 0 a b For effective indices lying between n f and n s, the fields are decaying as x -> ± For effective indices lying between n s and n d, the fields are decaying as x -> - but oscillatory as x -> + Such modes correspond to leaky modes and energy in these modes leaks from the core n d n s x 1

3 Bending loss When a waveguide is bent, then it loses power The loss of power can be explained in terms of leakage of power from the core Consider a planar waveguide bent into the form of a circle ρ φ ξ Wave equation Wave equation is given by Using a cylindrical coordinate system, we can write for the mode iβρφ ψ = R( r) e Wave equation becomes 1 r d dr dr r dr ω ψ + n ψ = 0 c + ω n c β ρ ( r) R r ( r) = 0

4 Equivalent straight waveguide u( r) If we substitute R ( r) =, ξ = r ρ r Wave equation becomes d u ω ~ ( ) + ( ) = 0 n ξ β u ξ dξ c Where equivalent refractive index profile is ~ β n ( ξ ) = n ( r) + ω c 1 ρ + c ( ρ + ξ ) 4ω ( ρ + ξ ) Equivalent straight waveguide The wave equation for the bent waveguide resembles a straight planar waveguide with an effective refractive index profile n ~ ( ξ ) n(ξ) n eff For large value of ξ, the refractive index becomes greater than in the core, waveguide becomes leaky This is the basis for loss due to bend Modal field also shifts its peak away from the center ξ 3

5 Bend loss in optical fibers Bend loss depends on how tightly confined the mode is Higher NA fibers have lower bend loss Bend loss has become very important due to applications of fibers in fiber to the home (FTTH) etc. New fiber designs to reduce bend induced loss Fibers referred to as G.65D Example: AllWave FLEX ZWP fiber from OFS, USA Reducing bend loss Having high n in a step index fiber Trench in the refractive index profile Hole assisted fiber Large n Trench Hole assisted Ref: Himeno et al J. Lightwave Tech

6 Reduced bend loss fibers Loss < 0.05 db Ref: Sakabe et al SEI Tech. Rev. 005 Depressed clad fibers For tailoring dispersion characteristics, depressed clad fiber designs are used n(r) In these designs if the effective index of the mode falls below the outer cladding index, then the mode becomes leaky Even LP 01 mode has a finite cut off due to leakage Some recent interesting applications in optical fiber amplifiers, fiber gratings r 5

7 n (r) Leaky fiber n 1 n eff (Leaky mode with a small leakage loss) n 3 n a b As λ increases, n eff reduces Above a design λ, mode becomes leaky Leakage loss spectral variation can be controlled by adjusting profile parameters r Typical Leakage loss spectrum 40 Loss (db/m) b-a=1 µm b-a=13 µm b-a=14 µm Wavelength (nm) Applications: Tailoring transmission spectrum of LPGs Realizing S-band EDFA by suppressing C-band ASE Inherently gain flattened Raman fiber amplifiers 6

8 Leaky fiber discrete Raman amplifier Leakage loss used to flatten the gain of the amplifier Net Gain (1.1 ± 1.4) db L = 13.6 km P p (1450 nm) = 30 mw λs : 150 nm nm P s (in): -10 dbm Ref: Charu Kakkar and K Thyagarajan, Optics Comm. (005) Dispersion properties Designed fiber has slowly varying, negative dispersion coefficient of 83 to 84 ps/km.nm 13.6 km of designed fiber + 70 km of SMF 8 7

9 Segmented clad fiber (SCF) n 1 n 4 n 3 n n (r) a b c At λ c1 : n eff = n 4 At λ c : n eff = n 3 r λ < λ c1 : Perfectly guided mode λ c1 < λ < λ c : Leaky mode, with leakage loss spectrum governed by separation c-a λ > λ c : Leaky mode, with leakage loss spectrum governed by separation b-a Greater design freedom in leakage loss spectrum Ref: Charu Kakkar and K Thyagarajan, J. Lightwave Tech. (005) Leakage loss spectrum of SCF Leakage loss (db/m) 1 Average slope (db/m)/nm (db/m)/nm Wavelength (nm) High tunability of leakage loss spectrum Can be used to extend erbium doped fiber amplifier into shorter wavelength band 8

10 SCF based S-band EDFA Normally EDFAs operate in C-band Possible to extend operation to S-band using distributed loss Pump power: 15 mw, L = 9.3 m Gain ripple: ~ ± 0.9 db λ c1 =1497 nm, λ c =154 nm Wave guiding with metals Metals are characterized by negative dielectric constant at optical frequencies It is possible to guide light waves at a single interface between a metal and a dielectric Surface plasmon mode They also posses an imaginary part Imaginary part leads to loss Metal clad waveguides used as polarizers, sensors, 9

11 Surface plasmon polariton Consider an interface between a metal and a dielectric x Metal (Κ m ) Dielectric (Κ d ) For a guided mode the field should decay exponentially as x -> ± TE modes should satisfy continuity of E y and de y /dx at x = 0 This is not possible Only TM solutions exist z K: dielectric constant TM modes at the interface TM modes are characterized by H y, E x and E z H y and 1/n dh y /dx have to be continuous at x = 0 The field H y is given by where γ H = Ae m y x ; γ = Ae d x ; x > 0 x < 0 γ m = β ω K m; c γ d = β ω K d c 10

12 Eigenvalue equation Applying boundary conditions we obtain γ γ m K = m d Kd Since γ m and γ d need to be positive, the equation can be satisfied only if K m and K d have opposite signs Metal-dielectric interface can support such modes Surface plasmon polaritons Propagation constant The effective index of the SPP mode is neff KmK = d Km + Kd For the mode to exist K d < K m 11

13 Example Interface between silver (K m = -19) and glass (K d =.5) n eff = at 633 nm Note n eff is greater than the dielectric refractive index Depth of penetration into metal ~ 0 nm Depth of penetration into dielectric ~ 0.18 µm Metal clad waveguides Metal clad polarizers Metal cladding on a waveguide can lead to a coupling of guided mode to the SPP mode Since SPP modes are lossy, the TM mode is lost from the waveguide Plasmonics: deals with creating a subwavelength infrastructure for guiding electromagnetic waves for applications in photonics, imaging, biological sensing etc. Ref: Maier, IEEE Sel. Topics in Quant. Electron. Nov/Dec

14 Bragg reflection waveguides Using Bragg reflection from a multilayer stack for guidance Total internal reflection from the left boundary & Bragg reflection from right boundary Bragg structure Core Substrate n g n 1 n n c Modal field profile Modal field profile of a typical BRW 13

15 Other Bragg reflection waveguides n(x) n Λ = d 1 +d n 1 n n 1 n c n s Low index core asymmetric BRW x n c x Low index core symmetric BRW Highly dispersive Since the effective index is below the refractive indices of multilayer stack, the modes are all leaky Since periodic structures have strong dispersion, BRWs can be used in nonlinear interactions, dispersion compensation etc. Bragg fibers, photonic bandgap fibers use similar effects for guidance 14