Lecture General concepts Digital modulation in general Optical modulation Direct modulation External modulation Modulation formats Differential detection Coherent detection Fiber Optical Communication Lecture, Slide 1
Encoding of optical signals We can write the electrical field corresponding to a monochromatic optical wave as e is a unit vector in the direction of E E z, t) eˆ A cos( k z ω t A 0 is the (real) amplitude k 0 is the wave number ω 0 is the angular frequency φ is the phase This is equal to It is often written simply Data can be encoded using Polarization, e E Amplitude, A 0 (Power is proportional to A 0 ) (Angular) frequency, ω 0 Phase, φ z, t) eˆ Re[ A e i(k 0z 0 ( Or by a combination of these 0 ω t) Fiber Optical Communication Lecture, Slide ] ( 0 0 0 ) E( z, t) eˆ A e 0 0 0 i(k zω t)
Modulation Using analog modulation, we talk about Amplitude modulation (AM) Frequency modulation (FM) Phase modulation (PM) Using digital modulation, the names are Amplitude-shift keying (ASK) Frequency-shift keying (FSK) Phase-shift keying (PSK) Simplest is ASK modulation in two levels This is also called on-off keying (OOK) Traditionally the most common format (for optical communication systems) Applying modulation to a monochromatic wave broadens the spectrum to a width comparable to the bit rate (symbol rate) Roughly: A 10 Gbit/s OOK signal can pass through a 10 GHz filter Fiber Optical Communication Lecture, Slide 3
Analog and digital signals (1..1) A physical signal is interpreted as being analog or digital A digital signal will be interpreted as one of the members from a finite set Digital modulation can be Pulse-position modulation (PPM) Pulse-duration modulation Pulse-code modulation (PCM) PCM is most common Required bit rate is B ( f )log ( M ) Δf is bandwidth of the analog signal M is number of quantization levels We see that B >> Δf What is gained from digital modulation? Fiber Optical Communication Lecture, Slide 4
Advantages over analog: Handles noise better Robust to distortion Can be regenerated Can use error correction (FEC) Disadvantages: Requires higher bandwidth Requires more electronics Digital optical modulation Difficult to implement with analog components Fiber Optical Communication Lecture, Slide 5
Our task: Carry a sequence of digital information over a channel (fiber + amplifiers) using an analog optical waveform Signal is often written as a k is the kth symbol p(t) is the pulse shape T is the duration of the symbol Traditional classification of optical pulse shapes Nonreturn-to-zero (NRZ) Return-to-zero (RZ) (Modern systems sometimes use digital pulse shaping) Possible modulation formats On-off Keying (OOK)... Signal modulation...and more modern ones, see following slides u( t) ak p( t kt ) Note: can have complex values (ASK + PSK = QAM) k Fiber Optical Communication Lecture, Slide 6
Optical modulation The modulation of data onto an optical carrier can be done in two ways Using direct modulation, the light source is modulated by the electrical signal This is simple and requires less hardware Hard to modulate amplitude without changing other parameters, like phase Cannot do PSK Using external modulation, a specifically made modulator is used Light source is not disturbed Modulators can have very high performance Laser electrical data electrical data Laser or LED modulator optical output optical output External modulation has better performance and is more flexible Unfortunately, it is also more expensive Fiber Optical Communication Lecture, Slide 7
External modulators are used for high performance There are two main types of external modulators A Mach-Zehnder modulator (MZM) Is made from Lithium niobate (LiNbO 3 ) Has large bandwidth Has higher V mod than EAM Has high extinction ratio (ER) An electro-absorption modulator (EAM) Is made from semiconductor material (InP) Has restricted bandwidth Has smaller V mod and ER External modulators Can be integrated with a laser Right: EAM + DFB (laser) Fiber Optical Communication Lecture, Slide 8
Nonreturn-to-zero modulation (1..3) Using nonreturn-to-zero modulation, the amplitude stays high between two consecutive ones The book illustrates with square pulses In reality, the pulse shape is given by the electrical drive signals/system response Shape is more smooth, can have overshoot etc. The square pulse has a spectrum sin( f ) FT[ E]( f ) TBsinc( TB f ),sinc( f ) f The power spectral density is plotted in the figure A δ-function at DC First nulls at f = ±B A realistic signal has a bandwidth B Lowpass filter is B/ (0.6 0.7 B) non-return-to-zero (NRZ)-signal return-to-zero (RZ)-signal 0 1 0 1 1 1 0 Fiber Optical Communication Lecture, Slide 9 pulse duration T time time bit period (bit-rate, B = 1/T f/b
Return-to-zero modulation Using return-to-zero modulation, the signal consists of pulses that are 0 1 0 1 1 1 0 more narrow than the bit slot The book illustrates with square pulses The ratio τ/t = τ/t B is called duty cycle In reality, a pulse carver is used or the pulses are generated by a pulse source The RZ spectrum is typically broader than the NRZ spectrum The RZ (OOK) spectrum contains a δ-function at f = B This is extracted (using a bandpass filter) to do clock recovery Clock recovery is much harder for NRZ non-return-to-zero (NRZ)-signal return-to-zero (RZ)-signal Pulse duration pulse duration /T = 0.5 T time time bit period (bit-rate, B = 1/T) Fiber Optical Communication Lecture, Slide 10 f/b
Eye diagrams for NRZ and RZ The eye diagram is a superposition of all bits on top of each other A way to provide visual feedback to monitor the performance An ideal OOK NRZ eye is seen in figure The bit slot is between the X-shaped curve intersections The upper and lower levels have no amplitude fluctuations All bits are in perfect synchronization Examples of measured eyes at 40 Gbit/s are seen in the figures NRZ eye is similar to the ideal one but there is noise Notice that transitions are smooth RZ eyes show the pulse shape Power of lower level is not exactly zero Fiber Optical Communication Lecture, Slide 11
Constellation diagrams (1..3, 10.1.1) A constellation diagram shows the symbols in the complex plane Using traditional OOK The amplitude is zero or a constant value The phase is unknown Drifting with time due to limited monochromaticity = finite linewidth Drifting faster for an LED than for a laser Using PSK, the data is encoded in the phase Binary PSK (BPSK) is seen in the figure There are two alternatives Phase can be tracked using hardware or software We can use relative phase between two consecutive symbols to carry data In this case: differential BPSK Q (Im) Q (Im) I (Re) I (Re) Fiber Optical Communication Lecture, Slide 1
Non-binary modulation formats The last ten years or so, non-binary modulation formats have become used in optical communication systems Can carry more than one bit per symbol Amplitude modulation in several levels is called pulse amplitude modulation (PAM) Proakis: Bandpass digital PAM is also called ASK Figure shows 4-PAM, carries two bits per symbol Phase is not important Symbols could have been written as circles Phase modulation in several levels is called phase-shift keying (PSK) Figure shows 4-PSK = quaternary PSK (QPSK) Carries two bits per symbol Symbol rate = Number of symbols transmitted/second Bit rate = Symbol rate [baud] number of bits per symbol Q (Im) Q (Im) I (Re) I (Re) Fiber Optical Communication Lecture, Slide 13
Quadrature amplitude modulation (QAM) Simultaneous modulation of the amplitude and the phase is called quadrature amplitude modulation (QAM) Figures show rectangular 16-QAM and 64-QAM Carries 4 and 6 bits per symbol In general: Modulation in M symbols carries log (M) bits per symbol Have been used in wireless systems a long time Difficult to use when symbol rates typically > 10 Gbaud Has not been used for a long time in optical systems For a given bit rate, these formats Have lower bandwidth requirements for the electronics Have higher spectral efficiency (more narrow spectrum) Can be difficult to implement High linearity requirements, typically requires DSP Are more susceptible to noise, require higher SNR Q (Im) Q (Im) I (Re) I (Re) Fiber Optical Communication Lecture, Slide 14
Modulator structures Lithium niobate modulators can be made in different ways A phase modulator changes the field according to The voltage V π changes the phase π Typically V π 5 V E out /E in is often called transfer function (although not in frequency domain) A Mach-Zehnder modulator (MZM) is an interferometer E E out in T 1 V1 ( t) V ( t) V exp i exp i 1 V V P P out Setting the voltage bias to V b = V π /......T changes from 1 to 0 as... in E E out E V t i ) out ( exp E in V V 1( t) V cos V...V 1 is changed from V π /4 to V π /4 in waveguide Electrical contact, applied voltage V(t) voltage V 1 (t) voltage V (t) = V 1 (t) + V b Fiber Optical Communication Lecture, Slide 15 b b
RZ data generation RZ data is typically generated by turning NRZ into RZ First, NRZ is generated using phase and/or intensity modulators Then, RZ pulses are cut out from the optical signal A pulse carver can be made using a MZM driven by a sinusoidal clock Clock is synchronous with the data By selecting the driving conditions, different pulse shapes are generated The RZ 50% pulse shape Select V b = V π / and V 1 = V π /4 cos(πbt) T( t) cos cos ( Bt) The full width at half maximum (FWHM) is T B / = 50% T B The clock frequency is B Bias point is at T = 1/ T t/t B Fiber Optical Communication Lecture, Slide 16
RZ data generation The RZ 33% pulse shape Select V b = 0 and V 1 = V π / cos(πbt) T( t) cos cos( Bt) T The FWHM is 33% T B The clock frequency is B/ Bias point is at T = 1 The RZ 67% pulse shape Select V b = V π and V 1 = V π / sin(πbt) T( t) cos sin( Bt) 1 T t/t B The FWHM is 67% T B The clock frequency is B/ Bias point is at T = 0 The RZ 67% has a more narrow spectrum than RZ 50% t/t B Fiber Optical Communication Lecture, Slide 17
Figure shows measured spectra at 4.7 Gbit/s OOK or DBPSK is used in all cases Spectra change for QAM Comments: NRZ is most narrow CS = carrier suppressed, phase shifting every second bit, no DC DB = duobinary, reduces spectral width by modifying the phases AMI = alternate mark inversion, Gives some reduction of nonlin. AP-RZ = alternate-phase RZ, reduces nonlin. by phase shifting DPSK = differential binary PSK Spectrum examples Fiber Optical Communication Lecture, Slide 18
Spectral efficiency The spectral efficiency (SE) is the throughput (bit rate) divided by the occupied spectral width The SE is increased by using a multilevel modulation format Often the word capacity is (mis-)used to denote system bit rate Example: At the same bit rate, the spectrum of a DQPSK signal is more narrow than that for a DBPSK signal As already seen, NRZ gives a more narrow spectrum than RZ Fiber Optical Communication Lecture, Slide 19
Differential detection When using differential detection, information is encoded in the phase change from symbol to symbol The phase changes are converted to intensity changes in the receiver Delay-line interferometers are frequently used for this Simplest example is differential binary PSK (DBPSK a.k.a. DPSK) This format will have a higher SNR for a given transmitted power Fiber Optical Communication Lecture, Slide 0
Balanced detection of DBPSK Only one photodetector is needed to recover the signal This is called single-ended detection By using two identical detectors, sensitivity can be improved up to 3 db The amplitude is doubled by the current subtraction This is called balanced detection Think of the junctions as couplers See next slide Constructive port i a T B i a i b i b Destructive port Delay interferometer (DI) Fiber Optical Communication Lecture, Slide 1
The 3-dB coupler A 3-dB coupler is a key component for understanding modulators/receivers Two waveguides are coupled Half the power is transferred The transferred field is π / out of phase The fields E 1 and E are arbitrary input fields The output fields are then E1 ie ie1 E E3, E4 If E = 0, then the output powers are P 3 = P 4 = P 1 / If both E 1 0 and E 0, then there will be interference The physical device is often a Y-junction E 1 (t) E (t) E 3 (t) E 4 (t) Fiber Optical Communication Lecture, Slide
Coherent detection Traditional receivers detect optical power phase information is lost Coherent detection can recover the phase data Uses a local oscillator (LO) laser in the receiver The simplest possible solution is seen in figure Uses free-space optics, mirror is partially reflecting The entering optical field is E det ( t) Asig ( t)exp( isig ( t) i0t) ALO exp( ilo( t) ilot) The detected current is proportional to E det idet ( t) Asig ( t) ALO Asig ( t) ALO cos ( sig( t) LO ( t)) ( 0 LO The term A LO is just a constant The term A sig can be neglected if A sig << A LO This is the case in practice i det ( t) Asig ( t) ALO cos ( sig( t) LO ( t)) ( 0 LO ) t) ) t) Fiber Optical Communication Lecture, Slide 3
Coherent detection If the signal and LO frequencies are equal and if φ LO = 0, we get idet ( t) Asig ( t)cos sig( t) Receiver is sensitive to both amplitude and phase The trouble is the assumptions made above The signal and LO lasers are not phase locked to each other The problem can be handled using hardware or software Hardware: Use a phase-locked loop (PLL) Drawback: Hard to construct, will need highly coherent lasers Software: Track the phase and frequency in digital signal processing (DSP) Drawback: Hard to do at high symbol rates Nevertheless, people are doing just that Fiber Optical Communication Lecture, Slide 4
Lecture Optical fibers as waveguides Maxwell s equations The wave equation Fiber modes Phase velocity, group velocity Dispersion Fiber Optical Communication Lecture, Slide 5
Maxwell s equations in an optical fiber (..1) H is the magnetic field [A/m] Ampère s circuital law, no current in fiber H D t E is the electric field [V/m] Faraday s law of induction E B t D is the electric flux density [C/m ] D 0 No free electric charges in fiber B is the magnetic flux density [Vs/m ] B 0 No free magnetic charges Fiber Optical Communication Lecture, Slide 6
The constitutive relations Relate the fields to the properties of the material P is the polarization field density The electric permittivity, ε 0 8.85 10 1 [As/Vm] D E P 0 The magnetization is zero in a fiber The magnetic permeability μ 0 1.6 10 6 [Vs/Am] B H 0 The permittivity and the permeability are related to the speed of light c 1/ 0 0 Fiber Optical Communication Lecture, Slide 7
The text-book uses the definition Fourier transform definition ~ s ( ) s ( t)exp( i t) dt Many different definitions are used in different fields of science With this definition t i s( t) 1 ~ s ( )exp( it) d A wave traveling in the positive z-direction is described by exp( it) s( t) ~ s ( ) i E( z, t) A0 cos( z 0t 0) Re[ A0e exp( iz i0 β is a propagation constant 0 t )] Fiber Optical Communication Lecture, Slide 8
Derivation of the wave equation We derive the wave equation E 0 t By using the Fourier transform of E we have ~ χ is the susceptibility The wave equation becomes H 0 0E P 0 it it E Er, te dt P Pt e d E ~ ~ E 1 ~ E c c ε is the relative dielectric constant (dimensionless) This is often written as ε r (not in the course book, though) ~ t c 1 0 ~ ~ t E E ~ t P Fiber Optical Communication Lecture, Slide 9
Derivation of the wave equation The dielectric constant is related to the refractive index n and the loss α c n i The frequency dependence of n is referred to as material dispersion We use the vector rule E It is assumed that the medium is homogeneous Finally the wave equation reads (using k 0 = ω/c and neglecting losses) ~ ~ ~ E E E ~ ~ E n ~ k E 0 0 Fiber Optical Communication Lecture, Slide 30
Optical fiber modes (..) Modes are solutions to the wave equation Satisfy the boundary conditions Do not change spatial distribution as they propagate Assume that the cladding extends to infinity The wave equation for E z in cylindrical coordinates (ρ, φ, z) 1 Ez 1 Ez Ez n k0 E 0 z z n = n 1 (ω) in the core (r < a) and n = n (ω) in the cladding (r > a) The same equation is obtained for the H z -component Maxwell s equations give the remaining four E- and H-field components The boundary conditions at z = 0 determines whether E z or H z are excited E z = 0 transverse electric (TE) mode H z = 0 transverse magnetic (TM) mode E z, or H z 0 hybrid mode, EH or HE Fiber Optical Communication Lecture, Slide 31
Solving the wave equation for the modes We use the method of separation of variables and obtain Z E z,, z F Zz z exp iz exp im where β is the propagation constant (to be determined) and m is an integer The equation for F is d F 1 df m 0 0 n k F d d This is Bessel s equation and its solution are Bessel functions Fiber Optical Communication Lecture, Slide 3
Initially given information: The propagation constant The fiber geometry, i.e., the core radius a, the core index n 1, and the cladding index n The operating wavelength/frequency, i.e. k 0 = π/λ ω/c Current status: We have the solution for E z with two unknown constants (amplitude in the core and the cladding) The equation and solution for H z is analogous The integer m is chosen by us Then we get the solution! The boundary conditions give an equation for β mn (ω) A number of solutions for each m-value Maxwell s equations give the other field components Each mode has a specific, wavelength-dependent propagation constant Fiber Optical Communication Lecture, Slide 33
For m = 0 either H z or E z are zero The mode family These are called TE 0n and TM 0n modes The other modes are hybrid modes; EH mn or HE mn The lowest order mode is HE 11, which exists for all wavelengths A Comsol simulation of HE 11 is seen to the right E x and E y are shown by the arrows E z is shown by the color The longitudinal (z) component is much smaller than the transverse Essentially linearly polarized Fiber Optical Communication Lecture, Slide 34
Further examples More calculated modes are shown below E z is not correct in third figure, top row Does not decrease properly with ρ Numerical artifact of an outer boundary Fiber Optical Communication Lecture, Slide 35
The effective index (mode index) The propagation constant must lie in the interval k 0 n < β < k 0 n 1 We define a mode index (or effective index) as n / k 0 The value is between the core and cladding index n n n 1 The effective index gives a measure of the mode confinement HE 11 mode well confined mode n weakly confined mode a -a intensity n 1 n n 1 n n intensity Fiber Optical Communication Lecture, Slide 36
V-parameter and effective propagation constant The normalized frequency is defined V a n1 n an1 c The normalized propagation constant is n n b n n All modes except HE 11 are cut-off for V <.405 For large V, approximately V / modes are guided 1 Fiber Optical Communication Lecture, Slide 37
The single-mode condition: Examples (..3) A multimode fiber with a = 5 μm and Δ = 0.005 has a value of V = 18 at λ = 1.3 mm. It then supports 18 / = 16 modes Single-mode fibers are often designed to have a cut-off at λ = 1. μm. By taking n 1 = 1.45, Δ = 0.004, we find that the required core radius a = 4 μm Fiber Optical Communication Lecture, Slide 38
The HE 11 mode and the LP modes The z-components of the E/H fields for the HE 11 mode are quite small (for small Δ), and the E x or E y component is dominating This mode is essentially linearly polarized The polarization state depends on how the E-field vector is directed Thus, the single mode consists of two degenerate polarization modes Two linearly polarized modes, depending on whether E x or E y is excited The HE 11 mode is often referred to as the LP 01 mode LP means linearly polarized Fiber Optical Communication Lecture, Slide 39
Propagating waves, phase velocity Each frequency component ω of the light propagates according to ~ ~ jz E( z, ) E(0, ) e Shows that the power spectrum, which is proportional to E(ω), will not change during transmission The propagation constant, β, is a function of ω This is called the dispersion relation We Taylor expand β(ω) to get 1 0 1, 0, m d m d m 0 A monochromatic wave propagates with the phase velocity v p 0 0 Fiber Optical Communication Lecture, Slide 40
Phase velocity The propagating field is E( z, t) E cos( 0z 0 0 t ) The phase velocity is v p 0 0 Fiber Optical Communication Lecture, Slide 41
A pulse E(t) at z = 0 will propagate as i.e. it moves with the group velocity Group velocity E( z, t) v g d d d E z d 0 t = k 0 n mode y = k 0 n 1 0 Fiber Optical Communication Lecture, Slide 4
Group velocity E( z, t) v g d E z d t Fiber Optical Communication Lecture, Slide 43
Group velocity dispersion (GVD) (.3) If the group velocity is different for different frequency components, the medium is dispersive Single-mode fibers have two contributions to the GVD: The dependence of n 1, on ω This is called material dispersion The mode behavior, which makes β depend on ω This is called waveguide dispersion In multimode fibers, the different velocities of the different modes is the main source of dispersion This is called modal dispersion Fiber Optical Communication Lecture, Slide 44
Pulse broadening by chromatic dispersion Study a discrete spectrum! (Signal will be periodic) Fiber Optical Communication Lecture, Slide 45
Inter-symbol interference The pulse distortion from dispersion leads to intersymbol interference (ISI) Neighboring pulses will broaden and overlap Dispersion limits the bit-rate! The information capacity of an optical fiber is often quantified by the bit-rate distance product fiber type multi-mode, step-index multi-mode, graded-index single mode, step-index maximum BL 0 Mbit/s km.5 Gbit/s km >.5 Gbit/s km Fiber Optical Communication Lecture, Slide 46