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