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Chapter Five – Optical Frequency Analysis Contents 1. Basic Concepts 2. Linewidth and Chirp 3. Interference between Two Optical Fields 4. Laser Linewidth Characterization 5. Optical Spectral Measurement of a Modulated Laser 6. Laser Chirp Measurement 7. Frequency Modulation Measurement 1 Basic Concepts Measurement Assumptions The laser under test are assumed to operate in a single-longitudinal mode (SLM) All the resonant frequencies of the laser cavity are suppressed with the exception of a single mode An illustration of a laser spectrum (no modulation) is shown in the figure below The laser line-shape typically has a Lorentzian-shaped central peak, small sidebands caused by relaxation oscillations, and small sidemodes (cavity frequencies) located further away 2 Basic Concepts Measurement Assumptions Optical mixing or interference plays a key role in the measurement methods presented here To obtain efficient interference, the following conditions are required between the interfering beams – polarization alignment, and spatial overlap All of the fibre in the circuits discussed in this chapter are singlemode Singlemode fibre insures good spatial overlap of the optical waves that are combined in the measurement setups Polarization state controller are often placed in the measurement circuits to permit polarization alignment Coherent Time The coherent time, τc of a laser is a measure of the spectral purity of the laser frequency over time In two-path interferometer, the degree to which an optical wave interferes with a timedelayed portion of itself depends on the coherence time of the wave with respect to the optical delay 3 Basic Concepts Coherent Time Coherent time is reduced by random events, such as spontaneous emission in the laser cavity, which alter the phase or frequency of the laser output field The figures illustrate the concept of coherence time In the upper figure, the coherence time is longer since the phase is predictable during the interval of time T1 – T2 In the lower figure, random phase changes cause an uncertainty in the phase relation between time T1 and time T2 4 Basic Concepts Coherent Time The coherent time is defined for spectra with Lorentzian lineshapes as, tc = 1 pD n The coherent length is simply, Lc = vg t c where vg is the group velocity of light: vg = c/ng, where ng is the group velocity index of refraction Question Consider a laser with a linewidth of 10 MHz. Calculate the coherence time and coherent length of the laser. Take ng = 1.476. [Ans. 32 ns, 6.5 m] 5 Linewidth and Chirp Dominant causes of spectral broadening in single-longitudinal mode lasers – phase noise and frequency chirp Random phase noise Created when spontaneous-emission, originating in the laser cavity gain media, changes the phase of the free-running laser frequency This process is magnified by physical effects within the laser cavity The magnification is quantified by the laser’s effective amplitude-phase coupling factor αeff αeff represents the link between power changes in the laser cavity to phase changes of the emitted light The result is a broadening of the laser spectral linewidth Relaxation oscillations Causes subsidiary peaks centred around the central mode of the laser These peaks generally lie within 20 GHz of the carrier and are much smaller in amplitude than the main peak 6 Linewidth and Chirp Laser frequency chirp results in significant spectral broadening when the laser injection current is modulated The magnitude of the chirp is proportional to the amplitude-phase coupling factor, αeff Material and structural properties of the laser contribute to the value of αeff, hence the resulting chirp The sweeping of the optical phase is due to the presence of frequency modulation or chirp on the optical carrier Laser chirp (without intensity modulation) is illustrated in the figure below 7 Linewidth and Chirp Some relations for estimating the laser linewidth, the relaxation resonance frequency, and the chirp of a semiconductor laser are given here Static linewidth Dn = log (1/ R ) 1 n sp (1 + a eff 2 )h n 4p P t p t rt Small Signal Chirp D nmax = a eff mfm 2 Large Signal Chirp D nc = a eff æ1 ¶ P ö ÷ çç ÷ è ø 4p P ¶ t ÷ Question A laser has an effective amplitude-phase coupling factor, αeff = 5. If the laser, undergoing 2.5 Gb/s intensity modulation has an intensity slope of 5 mW/30 ps at 4 mW average output, estimate the transient frequency chirp, Δνc. [Ans. 16. 6 GHz] 8 Linewidth and Chirp The variable and constants are defined in the table below 9 Interference between Two Optical Fields Heterodyne: Interference between Two Fields Consider the two optical fields incident on the photodetector after passing through the combiner as shown in the figure below, E s (t ) = E LO (t ) = Ps (t )e j (2pns f + f s t ) () PLO (t )e j (2pnLO f + f LO t ) () The two fields are scaled such that their magnitudes squared are optical powers (i.e. P(t) = |E(t)|2). The optical field frequencies and phases are designated by ν and φ(t). If either field were separately detected on a photodetector, the resulting photocurrent would follow only the power variations, P(t) and all phase information would be lost The optical phase, φ(t) takes into account any laser-phase noise or optical frequency modulation The total phase 2πνt + φ(t), of each optical field, changes at a rate much too fast for electronic instrumentation to respond 10 Interference between Two Optical Fields Heterodyne: Interference between Two Fields The optical spectrum corresponding to the two fields is shown in the figure below Here, the local oscillator has constant power and the signal laser has a small intensity modulation index of m To obtain the correct spectral display, the local oscillator frequency is set to a lower optical frequency than the signal under study 11 Interference between Two Optical Fields Heterodyne: Interference between Two Fields The optical combiner delivers the spatially overlapped optical fields to the photodetector where interference is detected The total field ET(t) at the photodetector is E T = E s (t ) + E LO (t ) Since power is detected (i.e. P(t) = |ET(t)|2), and not the optical field itself, photodetection is nonlinear with respect to the optical field. This fortunate situation allow us to detect interference between fields The photocurrent generated in the detector is proportional to the squared magnitude of the field i (t ) = R ET (t ) 2 where is the detector responsivity, R given by R= hdq [A / W ] hn where ηd (0 < ηd ≤ 1) is the detector quantum efficiency, a measure of the conversion efficiency of incident photons into electrical charge. The parameter q and h are electronic charge (1.6021 x 10-19 C) and photon energy (h = 6.6256 x 10-34 J, ν = c/λ) 12 Interference between Two Optical Fields Heterodyne: Interference between Two Fields Substituting both equations on slide 10 and the first equation on slide 12, into the second equation on slide 12, we obtain, using fIF = νs – νLO and Δφ = φs(t) – φLO(t): i (t ) = R éêPs (t ) + PLO + 2 Ps (t )PLO cos (2p fIF t + D f (t ))ù ú ë û The first two terms correspond to the direct intensity detection of Es(t) and ELO(t) The third term - the actual optical frequency is gone and only the difference frequency is left Thus the heterodyne method is able to shift spectral information from high frequencies to frequencies that can be measured with electronics In the heterodyne method, the local oscillator serves as a reference, with known frequency, amplitude and phase characteristics Thus the signal spectrum, including both intensity and frequency contributions can be obtained 13 Interference between Two Optical Fields Self-Homodyne: Interference between a Field and a Delayed Replica This is the case where one of the interfering optical fields is a delayed version of the other This condition can be created by a variety of two-path optical circuits such as the MachZehnder and Michelson interferometers, as well as Fabry-Perot interferometers A Mach-Zehnder interferometer is shown in the figure below 14 Interference between Two Optical Fields Self-Homodyne: Interference between a Field and a Delayed Replica The input field is split and routed along two paths with unequal lengths Time τ0 is the differential time delay between the two fields traversing the two arms of the interferometer The photocurrent generated at the detector is found in a similar way as with the heterodyne case, i (t ) = R éêP1 (t ) + P2 + 2 P1 (t )P2 cos (2pn0t 0 + D f (t , t 0 ))ù ú ë û where P1(t) and P2(t) are the powers delivered to the photodetector from each interferometer path The average phase-setting of the interferometer is given by 2πν0τ0 and Δφ(t, τ0) = φ(t) – φ(t – τ0) is the time-varying phase difference caused by phase or frequency modulation of the input signal, and the interferometer delay τ0 The interferometer free-spectral range (FSR) is defined as the change in optical frequency, to obtain a phase shift of 2π between the two combining fields – i.e. the frequency difference between the two peaks shown in the previous figure The FSR is the reciprocal of the net interferometer differential delay τ0 15 Interference between Two Optical Fields Self-Homodyne: Interference between a Field and a Delayed Replica Assuming Δφ(t, τ0) is small, varying the interferometer delay or the average optical frequency can cause the photocurrent to swing from minimum to maximum Limitations to the minimum and maximum current swings can be caused by a lack of polarization alignment between the fields, mismatch between path losses through the interferometer, or the limited coherence time of the optical source If the average phase 2πν0τ0 is equal to π/2, or more generally, equal to π(2n+1/2), n = 0, 1, 2, …, the interferometer is biased at quadrature, it can linearly transform small opticalphase excursion into photocurrent variations This is because the cosine characteristic varies linearly for small changes, about the quadrature point Thus the interferometer can function as a frequency discriminator as long as operation is confined to the approximately linear part of the interferometer transfer characteristic 16 Interference between Two Optical Fields Self-Homodyne: Interference between a Field and a Delayed Replica At the quadrature point, the previous equation becomes, i (t ) = R éêP1 (t ) + P2 + 2 P1 (t )P2 sin (D f (t , t 0 ))ù ú ë û If is small such that the approximation sin(Δφ(t, τ0)) ≈ Δφ(t, τ0) is valid, then the discriminator acts as a linear transducer converting phase or frequency modulation into power variations that can be measured with a photodetector i (t ) = R éêP1 (t ) + P2 + 2 P1 (t )P2 D f (t , t 0 )ù ú ë û The first two terms correspond to simple direct detection, the third term is the useful interference signal In the application of the interferometer as a discriminator to measure laser-phase noise, time-domain chirp, and FM response, the interferometer delay must be smaller than the source coherence time to maintain good interferometer contrast, which is a measure of the difference between Imax and Imin 17 Laser Linewidth Characterization This section – linewidth characterization of freerunning (unmodulated) single mode lasers Grating-based OSA – don’t offer the measurement resolution required for laser linewidth measurement The alternative methods – optical heterodyne method, the delayed self-heterodyne method, the delayed self-homodyne method, and an optical discriminator technique These methods – capable of obtaining the extremely high resolution required for laser linewidth measurement Heterodyne Using a Local Oscillator Capable of charaterizing nonsymmetrical spectral lineshapes Provides linewidth data and measures the optical power spectrum of an unknown optical signal Offers exceptional sensitivity and resolution Key component – stable, narrow linewidth reference laser 18 Laser Linewidth Characterization Heterodyne Using a Local Oscillator The setup for optical heterodyne is illustrated in the figure below The reference laser (local oscillator) is tuned appropriately and then its optical frequency is fixed during the measurement – possible because of the wide analysis bandwidth offered by ESA Alternative way – to have a narrow bandwidth electrical detection and a swept local oscillator – sets stringent requirements on the tuning fidelity of the local oscillator 19 Laser Linewidth Characterization Heterodyne Using a Local Oscillator Light form the local oscillator (LO) is combined with the signal laser under test LO – e.g. grating-tuned external cavity diode laser Polarization state converters are placed in the LO path to align the polarization state of the LO to that of the signal under test Coupler – combines the two fields, delivering half the total power to each output port One port leads to a photodetector which detect the interference beat tone, converting it to and electrical tone The LO laser frequency must be tuned close to the signal laser frequency to allow the mixing product to fall within the bandwidth of typical detection electronics Course alignment of the LO wavelength – performed using an OSA or a wavelength meter The LO frequency – tuned to a frequency just lower than the average frequency of the laser under study This creates a heterodyne beat tone between the LO and each of the frequency components in the signal spectrum as indicated in the figure on the next slide 20 Laser Linewidth Characterization Heterodyne Using a Local Oscillator Thus, each frequency component is translated to a low-frequency interference term described by i(t ) = R éêPs (t ) + PLO + 2 Ps (t )PLO cos (2p (ns - nLO )t + DF (t ))ù ú ë û 21 Laser Linewidth Characterization Heterodyne Using a Local Oscillator If the LO phase noise is small with respect to the laser under test - the beat tone will be broadened primarily by the phase noise of the laser under study The beat frequencies due to signal phase noise are measured using an ESA Heterodyne Power Spectrum ESA display is proportional to the power spectrum of the photodetector current which contains products of optical heterodyne mixing as well as direct detection terms S i (f ) = R 2 {S d (f ) + 2[S LO (n )]Ä S s (- n )} Sd(f) is the ordinary direct detection that could be measured with just a photodetector and ESA The second term – the useful heterodyne mixing product which is the convolution of the LO spectrum SLO(ν) with the signal spectrum Ss(ν) The convolution originates from the multiplication of the time-varying LO field with the signal field in the photodetector Multiplication in the time domain is equivalent to convolution in the frequency domain The lineshape of the laser, including any asymmetries, is replicated at a low frequency set by the optical frequency difference between the two lasers 22 Laser Linewidth Characterization Heterodyne Using a Local Oscillator Heterodyne Power Spectrum The convolution given in the previous slide is illustrated in the figure below The net result is a translation of the test-laser lineshape to the average difference frequency between the LO and the test laser As the LO linewidth broadens, its linewidth can dominate the photocurrent spectrum and decrease the frequency resolution of the heterodyne measurement 23 Laser Linewidth Characterization Heterodyne Using a Local Oscillator 24 Laser Linewidth Characterization Heterodyne Using a Local Oscillator 25 Laser Linewidth Characterization Heterodyne Using a Local Oscillator 26 Laser Linewidth Characterization Heterodyne Using a Local Oscillator 27 Laser Linewidth Characterization Heterodyne Using a Local Oscillator 28 Laser Linewidth Characterization Heterodyne Using a Local Oscillator 29 Laser Linewidth Characterization Heterodyne Using a Local Oscillator 30 Laser Linewidth Characterization Heterodyne Using a Local Oscillator 31 Laser Linewidth Characterization Heterodyne Using a Local Oscillator 32 Laser Linewidth Characterization Delayed Self-Heterodyne 33 Laser Linewidth Characterization Delayed Self-Heterodyne 34 Laser Linewidth Characterization Delayed Self-Heterodyne 35 Laser Linewidth Characterization Delayed Self-Heterodyne 36 Laser Linewidth Characterization Delayed Self-Heterodyne 37 Laser Linewidth Characterization Delayed Self-Heterodyne 38 Laser Linewidth Characterization Delayed Self-Heterodyne 39 Laser Linewidth Characterization Delayed Self-Heterodyne 40 Laser Linewidth Characterization Delayed Self-Heterodyne 41 Laser Linewidth Characterization Delayed Self-Homodyne 42 Laser Linewidth Characterization Delayed Self-Homodyne 43 Laser Linewidth Characterization Delayed Self-Homodyne 44 Laser Linewidth Characterization Delayed Self-Homodyne 45 Laser Linewidth Characterization Delayed Self-Homodyne 46 Laser Linewidth Characterization Delayed Self-Homodyne 47 Laser Linewidth Characterization Coherent Discriminator Method 48 Laser Linewidth Characterization Coherent Discriminator Method 49 Laser Linewidth Characterization Coherent Discriminator Method 50 Laser Linewidth Characterization Coherent Discriminator Method 51 Laser Linewidth Characterization Coherent Discriminator Method 52 Laser Linewidth Characterization Coherent Discriminator Method 53 Laser Linewidth Characterization Coherent Discriminator Method 54 Laser Linewidth Characterization Coherent Discriminator Method 55 Laser Linewidth Characterization Coherent Discriminator Method 56 Laser Linewidth Characterization Coherent Discriminator Method 57 Laser Linewidth Characterization Coherent Discriminator Method 58 Laser Linewidth Characterization Coherent Discriminator Method 59 Laser Linewidth Characterization Coherent Discriminator Method 60 Laser Linewidth Characterization Comparison of Techniques 61 Laser Linewidth Characterization Comparison of Techniques 62 Laser Linewidth Characterization Comparison of Techniques 63 Laser Linewidth Characterization Comparison of Techniques 64 Optical Spectral Measurement of a Modulated Laser 65 Optical Spectral Measurement of a Modulated Laser 66 Optical Spectral Measurement of a Modulated Laser Heterodyne Method 67 Optical Spectral Measurement of a Modulated Laser Heterodyne Method 68 Optical Spectral Measurement of a Modulated Laser Heterodyne Method 69 Optical Spectral Measurement of a Modulated Laser Heterodyne Method 70 Optical Spectral Measurement of a Modulated Laser Heterodyne Method 71 Optical Spectral Measurement of a Modulated Laser Gated-Delayed Self-Homodyne 72 Optical Spectral Measurement of a Modulated Laser Gated-Delayed Self-Homodyne 73 Optical Spectral Measurement of a Modulated Laser Gated-Delayed Self-Homodyne 74 Optical Spectral Measurement of a Modulated Laser Gated-Delayed Self-Homodyne 75 Optical Spectral Measurement of a Modulated Laser Gated-Delayed Self-Homodyne 76 Optical Spectral Measurement of a Modulated Laser Gated-Delayed Self-Homodyne 77 Optical Spectral Measurement of a Modulated Laser Gated-Delayed Self-Homodyne 78 Optical Spectral Measurement of a Modulated Laser Gated-Delayed Self-Homodyne 79 Laser Chirp Measurement 80 Laser Chirp Measurement 81 Laser Chirp Measurement 82 Laser Chirp Measurement 83 Laser Chirp Measurement 84 Laser Chirp Measurement 85 Laser Chirp Measurement 86 Laser Chirp Measurement 87 Laser Chirp Measurement 88 Laser Chirp Measurement 89 Laser Chirp Measurement 90 Frequency Modulation Measurement 91 Frequency Modulation Measurement 92 Frequency Modulation Measurement 93 Frequency Modulation Measurement 94 Frequency Modulation Measurement 95 Frequency Modulation Measurement 96 Frequency Modulation Measurement 97 Frequency Modulation Measurement 98 Frequency Modulation Measurement 99