A digitally generated ultrafine optical frequency
comb for spectral measurements with 0.01-pm
resolution and 0.7-s response time
Supplementary Information
S1. Working principle of digital generation of ultrafine optical frequency combs
Figure S1 elucidates the procedures of UFOFC generation. The computer generates a
data train Dinput F  and conducts the inverted fast Fourier transform (IFFT). Here F
refers to the domain used in the data train. The post-IFFT data then undergo digitalto-analog conversion (DAC) to change into the electrical waveform E EO t  . This
waveform is applied to an electric-to-optical (E/O) modulator to modulate the narrowlinewidth continuous-wave laser into the optical waveform EUFOFC t  . The laser has
initially a single frequency f0. After the E/O modulation, it is changed into a
frequency comb.
Computer
EEO t 
01010110
Dinput F 
t
DAC
IFFT
Narrow-linewidth
tunable laser
Frequency comb
f0
f
E/O
EUFOFC t 
Figure S1. Generation of the ultrafine optical frequency comb using the digital data
process by the inverted fast Fourier transform (IFFT).
To find the proper expression of Dinput F  for a targeted frequency comb
EUFOFC t  , we can derive retrospectively. For simplicity, here we neglect the influence
of conversions between digital and analog. Assume the targeted frequency comb can
be expressed in the frequency domain as
S1
UFOFC  f    Eq expj q    f  f q 
N
（S1a）
q1
here Eq, q and fq are the amplitude, phase and frequency of the qth comb line,
respectively; and it has f q  f 0  qf , where f0 is the carrier frequency of the laser
source, f is the constant comb spacing, q is an integer with q = 1, 2, …, N, and N is
the number of comb lines.
In the time domain, the corresponding waveform of the frequency comb can
written as
EUFOFC t  
1
2
exp j 2  f 0  qf t  j q 
N
E
q1
q
（S1b）
Therefore, the electrical waveform E EO t  that is applied to the E/O modulator should
have the form of
E EO t  
1
2
N
E
q 1
q
exp j 2 qf t  j q 
（S2）
Compared with the UFOFC electrical expression EUFOFC t  , the signal E EO t  takes
away the carrier frequency f0 of the laser source.
To get the electrical signal E EO t  after the IFFT process, the input signal from
the computer should satisfy
Dinput F   FFT E EO  
1
2
N
E
q 1
q
exp j q  FFT exp  j 2 qf t
  E q exp j q   F  2qf 
N
（S3）
q 1
  E q exp j q   F  Fq 
N
q 1
here Fq  2qf . Based on Equation (S3), the data train generated by the computer is
essentially a comb in the data domain F, which has the value E q exp j q  at each
data point F  Fq . By comparing Equations (S1a) and (S3), one can see that they
have very similar form of expression. The only difference is that Equation (S1a) is in
the optical frequency domain, whereas Equation (S3) is in the data domain. Such a
similarity significantly simplies the generation of data train in the computer.
S2
S2. Additional experimental data of the generated ultrafine optical frequency
comb
Additional UFOFC data in the time and the frequency domain are shown below.
Figure S2 shows the overview and the close-up of time-domain waveforms of
the generated UFOFC. In the enlarged view over the time range of 10 s, the
waveform is periodic with the period of 0.7 s. Within one period, there are many
peaks, with the peak-to-average power ratio (PAPR) of 3.7. This is different from the
typically optically-generated frequency combs (e.g., by the femto-second lasers),
which has very short pulses separated periodically by relative long intervals
(determined by the repetition rate) and thus very high PAPR ( repetition period/pulse
width, typically > 103). The reason is because the pseudo-random relationship among
the phases of frequency comb lines we choose intentionally (see Materials and
Methods). Without the randomization of phase, saying if the phases of all the comb
lines are the same, the PAPR of the UFOFC would reach 3.2 106. From the
experiment’s point of view, a low PAPR is desirable as it eases the requirements to
the intensity modulator and the photodetector. This is the rationale for the choice of
pseudo-random phase relationship and is also one distinctive feature of the UFOFC as
Normalized power
compared to the optically-generated frequency combs.
1
0.8
0.6
0.4
0.2
0
0
10
20
30
40
50
60
70
80
90
100
26
27
28
29
30
Normalized power
Time (μs)
1
0.8
0.6
0.4
0.2
0
20
21
22
23
24
25
Time (μs)
Figure S2. Time-domain waveform of the generated frequency comb in the range of
100 s, and the enlarged view over the time range of 10 s.
S3
Figure S3(a) plots the overlapped power spectra of the UFOFC of 100
measurements, and Figure S3 (b) and (c) show the enlarged views. The sampling
interval is 1 s. It is seen in Figure S3 (c) that the frequency peaks present almost no
shift in either the position or intensity, and only the foot parts show slight variation.
This well shows the frequency and intensity stability of the generated UFOFC.
Frequency Comb
Frequency Comb
-90
(a)
Power [dBm]
Power [dBm]
-90
-100
-110
-120
25
50
Frequency [MHz]
75
(b)
-100
-110
-120
-1.7
Frequency Comb
0
Frequency [MHz]
1.7
-90
Power [dBm]
(c)
-100
-110
-120
-0.12
0
Frequency [MHz]
0.12
Figure S3. Overlapped power spectra of 100 measurements of the generated ultrafine
optical frequency comb. The measurement interval is 1 s.
S3. Derivation of equations of phase shift and peak wavelength shift in the
unbalanced Mach-Zehnder interferometer
In the sensing unit of Mach-Zehnder interferometer (MZI) as shown in Figure 4, the
two arms have unequal arm lengths. In fact, the sensing arm is longer than the
reference arm.
The optical path difference OPD consists of two parts,
OPD  nL  OPDi 0
(S4)
S4
here n is the refractive index of the sample, and L is the physical length of the sample
along the optical axis. The first term nL in Equation (S4) is the contribution of liquid
sample and the second term OPDi0 includes all the other parts (fibers and collimators
of the sensing arm, the reference arm, etc.). When the temperature of the sample is
changed, the first term on the right side of Equation (S5) varies whereas the second
term remains a constant.
The total phase difference due to the optical path length can be expressed as

2

OPD 
2

nL 
2

OPDi 0
(S5)
here  is the wavelength.
(A) Phase shift of the UFOFC method
For the UFOFC method that measures the phase, the phase shift UFOFC varies with
the temperature change T by the relationship
UFOFC 
2L

n
(S6)
Therefore, the thermo-optic coefficient (TOC) can be calculated by
dn
dn dUFOFC
 dUFOFC


dT dUFOFC dT
2L dT
(S7)
here dUFOFC dT can be obtained from the slope of measured curve UFOFC versus T
(see Figure 6).
(B) Spectral shift of the OSA method
For the OSA-based method, according to Equation (S5), the peak wavelength m of
the mth mode should satisfy
m 
2
m
nL 
2
m
OPLi 0  m2
(S8)
here m is an integer.
When the sample is subjected to a temperature change T, the peak is shifted by
, but the total phase should remain to be m2. Therefore,
m

T  m   0
T

(S9)
S5
According to Equations (S7) and (S8), there are
m 2L dn

T
m 0 dT
(S10a)
m
2
2
  2 n0 L  OPDi 0    2 OPD0

m 0
m 0
(S10b)
here n0, m0 and OPD0 are all the initial values before the temperature change.
Substitute Equations (S10a) and (S10b) into Equation (S9), it gives
dn OPD0 dm

dT
m 0 L dT
(S11a)
and thus
m 
m 0 L
OPD0
n
(S11b)
here dm dT can be obtained from the slope of measured curve m versus T using the
OSA (see Figure 6).
From the measured transmission spectra of the MZI sensing unit shown in
Figure 5, the period is F = 5.4961 GHz. Then, the initial optical path difference is
calculated to be
OPD0 
c
2.9979  108

 0.05455 m
F 5.4961  109
(S12)
here c =3.0  108 m/s is the speed of light in vacuum. In this example, the cuvette is
filled with ethanol.
S4. Calculation of resolutions and UFOFC response time
Table 1 lists the resolutions of the OSA method and the UFOFC method to different
parameters. Below shows how these values are calculated.
In the OSA method, the OSA (Yokogawa AQ6370B) has the best spectral
resolution OSA = 0.002 nm. The other parameters are m0 = 1551.220 nm, L = 5.01
mm and OPD0 = 0.05455 m. From Equation (S11b) (or Equation (3)), the resolution
of the refractive index change nOSA can be estimated as
nOSA 
OPD0
0.05455
OSA 
 0.002  10 9   1.4  10 5 (S13)
9
3
m 0 L
1551.220  10  5.01  10
S6
and the corresponding resolution of the temperature change T is
TOSA 
nOSA
TOCethanol

1.4  105
 0.033 C
4.28  10 4
(S14)
here TOCethanol is the thermo-optic coefficient of ethanol at 25 oC. It takes the value of
-4.2810-4
o -1
C
as measured by us using the Fresnel reflection method (see
Supplementary section S6).
In the UFOFC method, the comb spacing fUFOFC = 1.46 MHz determines the
resolution of the spectral shift UFOFC.
UFOFC 
2
c
fUFOFC
1550.129  10   1.46  10   0.01 pm

3.0  10
9 2
6
8
(S15)
For the resolution of phase change UFOFC, it can be read from Figure 5 that a
frequency change of F corresponds to a phase change of 2, therefore it has
UFOFC 
2
2
fUFOFC 
 1.46  106  0.0017 rad
F
5.4961 109


(S16)
From Equation (S6), it has
nUFOFC 

1550.129  109
UFOFC 
 0.0017  8.4  108
3
2L
2  5.01  10
(S17)
and then,
TUFOFC 
nUFOFC
TOC ethanol

8.4  10 8
 2.0  10 4 C
4
4.277  10
(S18)
The ratio of resolutions between the OSA method and the UFOFC method is
OSA
nOSA
TOSA


 167
UFOFC nUFOFC TUFOFC
(S19)
This shows clearly that the UFOFC method is 167 time finer than the OSA method in
terms of the resolutions of wavelength shift, refractive index change and temperature
variation.
The dynamic response of the UFOFC method can be estimated as below. To
calculate the phase shift in the UFOFC method, it needs 8192 sampling points. As the
sampling rate of the AWG is 12 GSamples/s, the time for one data generation cycle is
8192
 0.7 μs
12  109
(S20)
When the coherent receiver and the data calculation in the computer are fast enough,
the time for one measurement cycle is just 0.7 s.
S7
S5. Estimation of thermo-optic coefficients of ethanol
For the OSA-based method, the slope of linear fit in Figure 6 is -0.071 ± 0.032 nm/oC.
From Equation (S11a), it has
dn OPL0 dm

dT m 0 L dT
0.05455
  0.07106  0.03187   10 9
9
3
1551.22  10  5.01  10
 4.99  2.24   10 4 C -1

(S21a)
After the deduction of the influence of TOC of the quartz cuvette (see Supplementary
section S7), the TOC of ethanol is corrected to be
dn
 4.99  2.24  10 4 - 0.0494  10 4
dT
 4.9  2.2  10 4 C -1
(S21b)
2.2
 45%
4.9
(S21c)
The error percentage
For the UFOFC method, the data points are fitted to a straight line using the
least square fitting method (see Figure 6). The slope is -8.885 ± 0.018 rad/oC. From
Equation (S7), it gets
dn
 dUFOFC 1550.129  109


  8.885  0.018
dT 2L dT
2  5.01  10 3
 4.377  0.009   10 4 C-1
(S22a)
Similarly, the deduction of the influence of quartz cuvette corrects the TOC of ethanol
to be (see Supplementary section S7)
dn
 4.377  0.009  10 4 - 0.0494  10 4
dT
 4.328  0.009  10 4 C -1
(S22b)
0.009
 0 .2 %
4.328
(S22c)
45%
 225
0.2%
(S23)
The error percentage
The ratio of the error percentages is
This shows that the UFOFC is more accurate than the OSA method by a factor of 225.
S8
LED
Optical circulator
1
3
Optical
power meter
2
Thermometer
Reflection at
fiber end
nf
Liquid samples
n1
Water bath
Figure S4. Schematic of the Fresnel reflection method for refractive index
measurement.
S6. Measurement of thermo-optic coefficient of ethanol using the Fresnel
reflection method
As a reference of the TOC value, we conducted another experiment using the Fresnel
reflection method [S1]. The working principle is illustrated in Figure S4. A broadband
light (central wavelength 1550 nm, bandwidth 20 nm) from a light emitting diode
(LED) enters port 1 of an optical circulator and goes out from port 2, which is
connected to a single-mode optical fiber (Corning SMF-28e). The other end of the
fiber is cleaved and immerged into the liquid sample, whose temperature is controlled
by a water bath and is monitored by a thermometer. When the light hits the
fiber/liquid interface (see the inset of Figure S4), it undergoes a Fresnel reflection
under normal incidence. The reflected light goes back to port 2 of the optical
circulator and then passes to port 3, whose power is measured by an optical power
meter.
The Fresnel reflection at the fiber/liquid interface is given by
 n f  n1 

R
n n 
f
1


2
(S24)
here nf is the refractive index of optical fiber and n1 is that of liquid sample.
To offset the influence of fiber connection and temperature-dependence of
fiber refractive index, we first used the air as the sample and measured the reflected
power at different temperatures, and then used these data as the reference. The
S9
processed refractive index of ethanol is plotted in Figure S5 over the temperature
range of 15 – 56 oC. Every data point represents 5 measurements, and the error bars
indicate the standard deviation of 5 measurements. From the slope of linear fit, the
TOC of ethanol is (-4.28 ± 0.08)  10-4 oC-1.
Refractive index n
1.360
Measured by Fresnel reflection
1.355
Slope: (-4.277 ± 0.080)x10-4 oC-1
1.350
1.345
1.340
20
30
40
50
60
Temperature T [oC]
Figure S5. Refractive index of ethanol as a function of temperature as measured by
the Fresnel reflection method.
S7. Correction of thermo-optic coefficient after considering the influence of
quartz cuvette
(A) Thermo-optic coefficient of quartz
From the Sellmeier equation in Eq. (3.29) of Ref. [S2], the TOC of optical materials is
2n
dn
 GR  HR 2
dT
(S25)
here n is the refractive index，dn/dT is the TOC，G and H are coefficients，the
parameter R is defined as
R
2
2  2g
(S26)
here  is the wavelength，g denotes the wavelength that corresponds to the bandgap
by the relationship g  1.24 E g , Eg is the bandgap in the unit of eV, and g in m.
S10
Based on the data in pp. 136 of Ref. [S2], the parameters of quartz (-SiO2) at
room temperature are listed in Table S1. According to Equation (S25), the TOCs at
1550 nm are calculated to be -5.598610-6 oC-1 and -6.774910-6 oC-1 for the ordinary
ray and the extraordinary ray, respectively. On average, the TOC of quartz at room
temperature and 1550 nm is dn/dT = -6.186710-6 oC-1.
Table S1 Parameters for thermo-optic coefficient of quartz from Ref. [S2].
Ordinary ray
Extraordinary ray
Bandgap Eg (eV)
8.9
8.9
Refractive index n
1.515
1.520
G (oC-1)
-61.1840  10-6
-70.1182  10-6
H (oC-1)
43.9990  10-6
49.2985  10-6
Lq
Liquid
n
Optical beam
nq
Quartz sidewall
of cuvette
Liquid
L
Figure S6. Diagram of the optical beam passing through the quartz cuvette sidewalls
and the liquid sample.
(B) Correction to the TOC of liquid sample using the quartz’s TOC
It can be seen from Figure S6 that the optical path length (OPL) of the beam passing
through the quartz cuvette is given by
OPL  nL  nq ( 2 Lq )
(S27)
here n and L are the refractive index and the physical length of the liquid sample
along the optical axis, respectively, nq is the refractive index of quartz，Lq is the
thickness of quartz wall. Here the term 2Lq is to account for the two sidewalls of
cuvette.
When the quartz’s TOC is not considered, all the changes of OPL are attributed
to the nominal refractive index change n of liquid sample, that is,
S11
OPL  nL
(S28a)
OPL  nL  2nq Lq
(S28b)
From Equation (S27), it has
Comparing Equations (S28a) and (S28b) yields
n  n 
2 Lq
nq
(S29)
dn dn 2 Lq dnq


dT dT
L dT
(S30)
L
Therefore, the correction to the TOC is given by
For the quartz cuvette used in experiment, it has L = 5.01 mm, Lq = 2 mm, and dnq/dT
= -6.1867  10-6 oC-1. Then it has
dn dn 2  2


  6.1867  10 6 
dT dT 5.01
dn

 0.0494  10 4
dT
(S31)
here the term 0.0494  10-4 oC-1 is the correction term.
For ethanol, the nominal TOC measured by the OSA method is
dn
 4.99  2.24   10 4 C-1 (see Supplementary section S5), after correction it
dT
becomes
dn
 4.94  2.24   10 4 C-1 . For the nominal TOC measured by the
dT
UFOFC method
dn
 4.377  0.009   10 4 C-1 , after correction it changes to
dT
dn
 4.328  0.009   10 4 C-1 (see also Supplementary section S5).
dT
Reference:
[S1] Shlyagin, M. G., Manuel, R. M. & Esteban, Ó. Optical-fiber self-referred
refractometer based on Fresnel reflection at the fiber tip. Sens. Actuators B 178,
263–269 (2013).
[S2] Ghosh, G. Handbook of Thermo – Optic Coefficients of Optical Materials with
Applications (Academic, 1998).
S12