1 chapter five – optical frequency analysis contents 1. basic concepts 2. linewidth and chirp 3....

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

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

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 time-delayed portion of itself depends on the coherence time of the wave with respect to the optical delay

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

Basic Concepts

Coherent Time

The coherent time is defined for spectra with Lorentzian lineshapes as,

The coherent length is simply,

where vg is the group velocity of light: vg = c/ng, where ng is the group velocity index of refraction

1ct p n=

c g cL v t=

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]

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

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

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

Small Signal Chirp

Large Signal Chirp

( ) ( )2 log 1/11

4 sp effp rt

Pn a n

p t tD = +

max 2eff mmfa

æ ö¶ ÷çD = ÷ç ÷çè ø¶

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]

Linewidth and Chirp

The variable and constants are defined in the table below

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,

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

( ) ( ) ( )( )2 s sj f ts sE t P t e pn f+=

( ) ( ) ( )( )2 LO LOj f tLO LOE t P t e pn f+=

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

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

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

where is the detector responsivity, R given by

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)

( ) ( )T s LOE E t E t= +

( ) ( )2

Ti t R E t=

[ ]/dqR A Whhn

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):

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

( ) ( ) ( ) ( )( )2 cos 2s LO s LO IFi t R P t P P t P f t tp fé ù= + + +Dê úë û

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 Mach-Zehnder and Michelson interferometers, as well as Fabry-Perot interferometers

A Mach-Zehnder interferometer is shown in the figure below

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,

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

( ) ( ) ( ) ( )( )1 2 1 2 0 0 02 cos 2 ,i t R P t P P t P tpn t f té ù= + + +Dê úë û

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 optical-phase 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

At the quadrature point, the previous equation becomes,

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

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

( ) ( ) ( ) ( )( )1 2 1 2 02 sin ,i t R P t P P t P tf té ù= + + Dê úë û

( ) ( ) ( ) ( )1 2 1 2 02 ,i t R P t P P t P tf té ù= + + Dê úë û

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

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

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

Thus, each frequency component is translated to a low-frequency interference term described by

( ) ( ) ( ) ( )( )( ) 2 cos 2s LO s LO s LOi t R P t P P t P t tp n né ù= + + - +DFê úë û

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

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

( ) ( ) ( )[ ] ( ){ }2 2i d LO sS f R S f S Sn n= + Ä -

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

Delayed Self-Heterodyne

Delayed Self-Homodyne

Coherent Discriminator Method

Comparison of Techniques

Optical Spectral Measurement of a Modulated Laser

Heterodyne Method

Gated-Delayed Self-Homodyne

Laser Chirp Measurement

Frequency Modulation Measurement

1 chapter five – optical frequency analysis contents 1. basic concepts 2. linewidth and chirp 3....

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