**3. The application in laser noise measurement**

#### **3.1. Experimental setup**

Based on the 120° phase difference technology, the experimental setup of laser noise measurement is shown in **Figure 2** [18]. It consists of a commercially available 3 × 3 optical fiber coupler (OC), a circulator (C), two Faraday rotator mirrors (FRMs), three photodetectors (PDs), a data acquisition board (DAC) or a digital oscilloscope, and a computer. The 120° phase difference Michelson interferometer is composed of the 3 × 3 coupler and the FRMs. On one hand, the FRM will remove the polarization fading caused by external disturbance on the two beam fibers of the interferometer. On the other hand, the length or index of the fiber configuring the interferometric arms would change randomly because of temperature fluctuations, vibration, and other types of environmental disturbances; thus it induces low-frequency random-phase drifts in the interferometric signal. So in the proposed experimental setup, the complete interferometer is housed in an aluminum box enclosed in a polyurethane foam box for thermal and acoustic isolation. Meantime, the two fiber arms of the Michelson interforometer are placed closely in parallel to improve the stability against the perturbation.

( )

*t t*

D= - -

 j t

parameter one by one as said in Ref. [15]. Eq. (13) can be transformed as.

222 2 2 2 2 2 21 22 2 2 <sup>333</sup> <sup>3</sup> <sup>333</sup> 3 31 32

 hVx

 hVx

 hVx

= +-+ ¢ ¢

1 2 1 2 2 2 1 2 1 2 21 1 11

2 cos 2 sin

*n nn n n n nn n n n nn n nm mm nm m nn n*

*rc c rc c*

h

= =

V

x

q q q

jj

( )

hVx

hVx

hVx

– |*E*(*t* – *τ*)|2

**3.1. Experimental setup**


238 Optical Interferometry

form

( )

*ttt*

 jt

**3. The application in laser noise measurement**

 j

j

*X I t rc c*

( )( ) () ( )

*rc c n c p bb m*

= + æ ö <sup>=</sup> ç ÷ <sup>=</sup> ¢ è ø <sup>=</sup>

The parameters *ηn*, *ςn*, and *ξn* are constant for the setup, once the devices and structure are determined. They can be obtained by a broadband light source without measuring each

2 2 1 1 1 11 12 2 2 <sup>111</sup> <sup>1</sup> <sup>111</sup>

Eq. (16) shows the relation between the differential phase ∆*ϕ* and the detectors outputs (*I*1, *I*2, *I*3). We note that the second term on the right-hind side of Eq. (16) becomes zero or can be omitted under the following conditions: (a) the splitting ratios of the 3 × 3 coupler are uniform (i.e., *c*n1 = *c*n2 for all *n*), or (b) the intensity of the laser under test is constant or periodical (i.e.,

the differential phase Δ*ϕ*(*t*) accumulated in delay time *τ* can be obtained in the following simple

Based on the 120° phase difference technology, the experimental setup of laser noise measurement is shown in **Figure 2** [18]. It consists of a commercially available 3 × 3 optical fiber coupler (OC), a circulator (C), two Faraday rotator mirrors (FRMs), three photodetectors (PDs), a data acquisition board (DAC) or a digital oscilloscope, and a computer. The 120° phase difference Michelson interferometer is composed of the 3 × 3 coupler and the FRMs. On one hand, the FRM will remove the polarization fading caused by external disturbance on the two beam fibers of the interferometer. On the other hand, the length or index of the fiber configuring the interferometric arms would change randomly because of temperature fluctuations, vibration,

() () ( ) 2 2

æö æö ¢ ¢ D = - -= ç÷ ç÷ - ¢ ¢ èø èø

() 2

*I t rc c*


*rc c I t*

 qq

 q

 q

1,2,3 , 1,2

( ) ( ) ( )

2 2

=0), or (c) the extinction ratio of the Michelson interferometer. As a result,

1 1 ( ) ( ) arctan arctan

*Xt Xt*

( ) ( ) *Xt Xt*

( ) ( )

t

*Et Et*

(15)

(16)

(17)

**Figure 2.** Experimental setup used to measure the laser phase and frequency noise, and the output interference fringe of the PD1, PD2, PD3 (inset). LUT: laser under test, C: circulator, OC: optical fiber coupler, FRM: Faraday rotation mirror, PD: photodetector, DAC: data acquisition board [18].

The laser under test (LUT) injects the left port 1 of the 3 × 3 optical fiber coupler through a circulator and then splits into three parts by the coupler. Two of them interfere mutually in the coupler after being reflected by Faraday mirrors and with different delay times, and the third part of them is made reflection-free. Then the interference fringes are obtained from the left port 1, 2, and 3 of the coupler and read by a DAC or a digital oscilloscope.

In the experimental setup, a swept laser source with linewidth of about 2.5 kHz [19] is used as the broadband light source to show clearly the small free spectral range (FSR) of the MI. The measured interference fringe is shown in the inset figure of **Figure 1**. On the other hand, all parameters of the devices are considered in the differential phase fluctuation calculation process, so the possible errors from device defects such as imperfect splitting ratio or phase difference are removed, and the requirements for the device performance parameters are also relaxed. In our experimental setup, the final setup parameters are *τ* = 244 ns (corresponding FSR of the MI is 4.1 MHz), and

$$
\begin{pmatrix}
\eta\_1 & \boldsymbol{\varrho}\_1 & \boldsymbol{\xi}\_1 \\
\eta\_2 & \boldsymbol{\varrho}\_2 & \boldsymbol{\xi}\_2 \\
\eta\_3 & \boldsymbol{\varrho}\_3 & \boldsymbol{\xi}\_3
\end{pmatrix} = \begin{pmatrix}
0.1198 & 0.0002 & 0.1223 \\
\end{pmatrix} \tag{18}
$$

When the setup parameters are calibrated, the whole setup is in a state of plug-and-play.

**Figure 3.** Demodulation principle of the setup for triangle waveforms with different modulation periods (a) *T* ≠ 2*τ* and (b) *T* = 2*τ* [18].

**Figure 4.** Modulated and demodulated (a) triangle phase amplitude and (b) waveform at different modulated voltage amplitudes [18].

To verify the correction of the phase demodulation of the setup, a demodulated test for a preset modulated phase by a LiNbO3 phase modulator is demonstrated. A narrow linewidth laser that is phase-modulated with the LiNbO3 phase modulator is used as LUT. The triangle waveform is selected to the modulation waveform for holding the frequency components as such and much more. Hence, the fact of the interference at the coupler is a subtraction between two triangle waveforms with delay time *τ* as shown in **Figure 3**. The modulation period *T* needs to be twice of the delay time difference *τ*. Otherwise, there are some constants in the interference fringe as shown in Figure 3(a), accordingly the triangle wave cannot be demodulated. Once the triangle waveform is demodulated correctly as shown in Figure 3(b), the demodulated amplitude would be twice the input modulated amplitude. **Figure 4**. shows the demodulated triangle phase amplitude and their waveforms at different modulated voltage amplitudes. The results confirmed the correction of the differential phase demodulation both in terms of waveform and amplitude. **Figure 5** shows the time-domain interference fringes of the MI and the corresponding demodulated phase waveform at a fixed modulated voltage. The red curve and the black curve represent the results of two independent tests at different time, respectively. Despite the interference fringe changes at different time due to the environment variation, the demodulated phase would not change and hold the consistence with the input phase modulation, so the consistency and robustness of the proposed setup is verified.

**Figure 5.** Output voltages of (b) PD1, (c) PD2, (d) PD3, and (a) corresponding demodulated phase waveforms at a fixed modulated voltage *V*m = 3 V. The red and black lines represent the first and second test, respectively [18].

#### **3.2. Laser noise theory**

**Figure 3.** Demodulation principle of the setup for triangle waveforms with different modulation periods (a) *T* ≠ 2*τ* and

**Figure 4.** Modulated and demodulated (a) triangle phase amplitude and (b) waveform at different modulated voltage

To verify the correction of the phase demodulation of the setup, a demodulated test for a preset modulated phase by a LiNbO3 phase modulator is demonstrated. A narrow linewidth laser that is phase-modulated with the LiNbO3 phase modulator is used as LUT. The triangle waveform is selected to the modulation waveform for holding the frequency components as such and much more. Hence, the fact of the interference at the coupler is a subtraction between two triangle waveforms with delay time *τ* as shown in **Figure 3**. The modulation period *T* needs to be twice of the delay time difference *τ*. Otherwise, there are some constants in the interference fringe as shown in Figure 3(a), accordingly the triangle wave cannot be demodulated. Once the triangle waveform is demodulated correctly as shown in Figure 3(b), the demodulated amplitude would be twice the input modulated amplitude. **Figure 4**. shows the demodulated triangle phase amplitude and their waveforms at different modulated voltage amplitudes. The results confirmed the correction of the differential phase demodulation both

(b) *T* = 2*τ* [18].

240 Optical Interferometry

amplitudes [18].

Considering the relation between the delay phase *ϕ* and frequency *ν*;

$$
\varphi = 2\pi \frac{n l \nu}{c} = 2\pi \pi \nu \tag{19}
$$

the differential phase variation is from the laser frequency variation in the time interval *τ*, because the delay time difference *τ* of the MI is fixed and the random variation of the fiber is eliminated carefully by some techniques as described above. So from the differential phase fluctuation Δ*ϕ*(*t*), the laser frequency fluctuation in time interval *τ* defined as differential frequency fluctuation Δ*ν*(*t*) can be expressed as

$$
\Delta\nu(t) = \Delta\phi(t) / \left(2\pi\tau\right). \tag{20}
$$

Hence, the PSD of differential phase fluctuation Δ*ϕ*(*t*) and differential frequency fluctuation Δ*ν*(*t*) can be calculated, respectively, in the computer by PSD estimation method [20] and .

denoted as *S*Δ*<sup>ϕ</sup>*(*f*) and *S*Δ*<sup>ν</sup>*(*f*), respectively, where *f* is the Fourier frequency. Meantime, from the linearity of the Fourier transform, they have a fixed relation

$$S\_{\Lambda\nu}(f) = \left(\frac{1}{2\pi\tau}\right)^2 S\_{\Lambda\rho}(f) \,. \tag{21}$$

So far, the differential phase fluctuation Δ*ϕ*(*t*) accumulated in the delay time difference *τ* of the MI, corresponding differential frequency fluctuation Δ*ν*(*t*), and their PSD is calculated. But these values are not the instantaneous information of the LUT. Furthermore, considering the relation of differential phase fluctuation Δ*ϕ*(*t*) and instantaneous phase fluctuation *ϕ*(*t*) expressed in Eq. (17), the definition of the PSD [21], linearity, and time-shifting properties of Fourier transform, we derived strictly the PSD of laser instantaneous phase fluctuation and frequency fluctuation, which can be expressed as,

$$S\_{\phi}(f) = \frac{1}{4\left[\sin(\pi f \tau)\right]^2} S\_{\Lambda\phi}(f) \tag{22}$$

$$S\_{\nu}(f) = \frac{f^2}{4\left[\sin(\pi f \tau)\right]^2} S\_{\Lambda\rho}(f) = \frac{1}{\left[\text{sinc}(\pi f \tau)\right]^2} S\_{\Lambda\nu}(f) \tag{23}$$

From Eq. (22), the single-side-band phase noise can also be obtained with [22].

$$L(f) = \frac{1}{2} S\_{\phi}(f) = \frac{1}{8\left[\sin(\pi f \tau)\right]^2} S\_{\Lambda\phi}(f) \tag{24}$$

Eq. (22) means that, at the low Fourier frequency domain, the PSD of the laser instantaneous phase fluctuation *Sϕ*(*f*) would be larger than the PSD of the differential phase fluctuation *S*Δ*<sup>ϕ</sup>*(*f*), but at the high Fourier frequency domain, the former is smaller than the latter. Eq. (23) means that, however, the PSD of the laser instantaneous frequency fluctuation *Sν*(*f*) is larger than the PSD of the differential frequency fluctuation *S*Δ*<sup>ν</sup>*(*f)* at any positive Fourier frequency. On the other hand, it is observed that if the differential phase and frequency fluctuation are normalized in 1 m delay fiber (*τ* ~ 5 ns), then

$$S\_{\nu}(f) \approx S\_{\Lambda\nu}(f), \quad \text{for } f < \\$\,\text{MHz}.\tag{25}$$

Eq. (25) means that the PSD of the laser instantaneous frequency fluctuation *Sν*(*f*) would approximately equal to the PSD of the differential frequency fluctuation *S*Δ*<sup>ν</sup>*(*f*) at the Fourier frequency less than MHz level. In physics, the results can be so explained that the frequency is the differential of the phase and the delay of the MI is equivalent to the differential operation for the phase. The conclusions are very important that the characterization of differential phase and instantaneous phase (laser phase) should be carefully distinguished, but the instantaneous frequency can be replaced by the differential frequency sometimes in the practical engineering applications.

### **3.3. Laser noise measurement results**

denoted as *S*Δ*<sup>ϕ</sup>*(*f*) and *S*Δ*<sup>ν</sup>*(*f*), respectively, where *f* is the Fourier frequency. Meantime, from the

2

æ ö = . ç ÷ è ø p

So far, the differential phase fluctuation Δ*ϕ*(*t*) accumulated in the delay time difference *τ* of the MI, corresponding differential frequency fluctuation Δ*ν*(*t*), and their PSD is calculated. But these values are not the instantaneous information of the LUT. Furthermore, considering the relation of differential phase fluctuation Δ*ϕ*(*t*) and instantaneous phase fluctuation *ϕ*(*t*) expressed in Eq. (17), the definition of the PSD [21], linearity, and time-shifting properties of Fourier transform, we derived strictly the PSD of laser instantaneous phase fluctuation and

 j (21)

<sup>1</sup> () () <sup>2</sup> *Sf Sf*

tD D

[ ]

[ ] [ ]

1 1 () () ( ) <sup>2</sup> 8 sin( ) *Lf S f S f*

= = <sup>D</sup>

*Sf S f f* ( ) ( ), for 5 MHz.

 n

= = D D

From Eq. (22), the single-side-band phase noise can also be obtained with [22].

<sup>1</sup> ( ) ( ) ( ) 4 sin( ) sinc( ) *<sup>f</sup> S f S f S f f f*

t<sup>=</sup> <sup>D</sup>

<sup>1</sup> ( ) ( ) 4 sin( ) *S f S f f*

2

2 2

[ ]

Eq. (22) means that, at the low Fourier frequency domain, the PSD of the laser instantaneous phase fluctuation *Sϕ*(*f*) would be larger than the PSD of the differential phase fluctuation *S*Δ*<sup>ϕ</sup>*(*f*), but at the high Fourier frequency domain, the former is smaller than the latter. Eq. (23) means that, however, the PSD of the laser instantaneous frequency fluctuation *Sν*(*f*) is larger than the PSD of the differential frequency fluctuation *S*Δ*<sup>ν</sup>*(*f)* at any positive Fourier frequency. On the other hand, it is observed that if the differential phase and frequency fluctuation are

*f*

t

jn

2

 j

 t

p p (23)

» < D (25)

 j

<sup>p</sup> (22)

<sup>p</sup> (24)

linearity of the Fourier transform, they have a fixed relation

frequency fluctuation, which can be expressed as,

n

normalized in 1 m delay fiber (*τ* ~ 5 ns), then

n

.

242 Optical Interferometry

j

2

t

j

n

First, the phase and frequency noise of an external-cavity semiconductor laser (RIO OR-IONTM) [23] with a wavelength of 1551.7 nm and a linewidth of about 2 kHz are measured. The PSD of the differential phase fluctuation is normalized to 1 m delay fiber (*S*Δ*<sup>ϕ</sup>*(*f*) @ 1 m), the PSD of the differential frequency fluctuation is normalized to 1 m delay fiber (*S*Δ*<sup>ν</sup>*(*f*) @ 1 m), and the PSD of instantaneous frequency fluctuation *Sν*(*f*), the PSD of the instantaneous phase fluctuation *Sϕ*(*f*), and the laser phase noise *L*(*f*) are shown in **Figure 6**. The data are very close to that given in the product datasheets or the typical data given in the website [23]. The curves clearly demonstrate the relations between these physical quantities as described above. At the focused frequency range (<1 MHz), *S*Δ*<sup>ν</sup>*(*f*) @ 1 m approximately equals to *Sν*(*f*), *S*Δ*<sup>ϕ</sup>*(*f*) @ 1 m is much less than *Sϕ*(*f*), and laser phase noise *L*(*f*) = *Sϕ*(*f*)/2. So the usage of PSD of differential phase as the laser phase noise is not strictly correct and needs more careful definition and consideration.

**Figure 6.** PSD of the differential phase fluctuation (*S*Δ*ϕ*(*f*) @ 1 m), differential frequency fluctuation (*S*Δ*ν*(*f*) @ 1 m), instantaneous phase fluctuation *Sϕ*(*f*), instantaneous frequency fluctuation *Sν*(*f*) and phase noise *L*(*f*) of the RIO laser [18].

From the PSD of the frequency fluctuation *Sν*(*f*), the linewidth at different observation time can be calculated. **Figure 7** shows the linewidth calculated with the approximated model presented in Ref. [24] for different values of the integration bandwidth. The results indicate that the linewidth is very dependent on the integration bandwidth. Linewidth would increase with the increase of observation time (in other words, linewidth increases with the decrease of the lower limit of the integration bandwidth). It mainly results from the presence of the 1/*f* α type noise in the PSD of frequency fluctuation. In a high-frequency domain (>100 kHz), there is only white noise, and the minimum linewidth of about *δν*1 = 2 kHz is calculated. Meantime, the inset figure shows the linewidth of the same laser measured by the self-delay heterodyne (SDH) method [14] with heterodyne frequency of 80 MHz and optical fiber delay length of 45 km. The Lorentz fitted linewidth at −20 dB from the spectrum measured by the SDH method is about 51.7 kHz, so the laser linewidth is about *δν*<sup>2</sup> = 2.6 kHz, and the fitted linewidth would not vary with the observation time. It indicates that the nonwhite noise components are not revealed in the Lorentz fitted results of SDH method, but the values are also conservative for white noise components and about 30% larger than the values calculated by the PSD of frequency fluctuation. This is because the tail of the spectrum measured by SDH method is not taken into consideration in the Lorentz fitting process, resulting in the fitted value being larger than the real value. Therefore, PSD of frequency fluctuation is recommended to completely describe the frequency noise behavior, and a specified linewidth value should be reported with the corresponding integration bandwidth or observation time.

**Figure 7.** PSD of instantaneous frequency fluctuation of the RIO laser and the *β* – separation line given by *Sν*(*f*) = 8ln2*f*/π2 [24] (left axis), laser linewidth (FWHM) obtained by the method in Ref. [24] (right axis) and by the SDH method (inset) [18].

Second, the noise features of commercial distributed feedback (DFB) semiconductor laser are measured under different operating temperatures (24 and 22.7°C) and different operating currents (71, 91, 111, and 136 mA), shown in Figures 8 and 9. **Figure 8** shows that this DFB laser is more suitable for working at 24°C than at 22.7°C. **Figure 9** shows that the PSDs of frequency fluctuation *Sν*(*f*) decrease with the increase of operating currents from 71 to 136 mA. According to above results, if we measure the PSDs of frequency fluctuation *Sν*(*f*) of the laser under a wide range of operating temperatures and currents, the optimum operating temperature and current can be found out.

**Figure 8.** *S <sup>ν</sup>*(*f*) of the DFB laser at operating temperature of 22.7 and 24.0°C.

increase of observation time (in other words, linewidth increases with the decrease of the lower

in the PSD of frequency fluctuation. In a high-frequency domain (>100 kHz), there is only white noise, and the minimum linewidth of about *δν*1 = 2 kHz is calculated. Meantime, the inset figure shows the linewidth of the same laser measured by the self-delay heterodyne (SDH) method [14] with heterodyne frequency of 80 MHz and optical fiber delay length of 45 km. The Lorentz fitted linewidth at −20 dB from the spectrum measured by the SDH method is about 51.7 kHz, so the laser linewidth is about *δν*<sup>2</sup> = 2.6 kHz, and the fitted linewidth would not vary with the observation time. It indicates that the nonwhite noise components are not revealed in the Lorentz fitted results of SDH method, but the values are also conservative for white noise components and about 30% larger than the values calculated by the PSD of frequency fluctuation. This is because the tail of the spectrum measured by SDH method is not taken into consideration in the Lorentz fitting process, resulting in the fitted value being larger than the real value. Therefore, PSD of frequency fluctuation is recommended to completely describe the frequency noise behavior, and a specified linewidth value should be reported with the

**Figure 7.** PSD of instantaneous frequency fluctuation of the RIO laser and the *β* – separation line given by *Sν*(*f*) =

Second, the noise features of commercial distributed feedback (DFB) semiconductor laser are measured under different operating temperatures (24 and 22.7°C) and different operating currents (71, 91, 111, and 136 mA), shown in Figures 8 and 9. **Figure 8** shows that this DFB laser is more suitable for working at 24°C than at 22.7°C. **Figure 9** shows that the PSDs of frequency fluctuation *Sν*(*f*) decrease with the increase of operating currents from 71 to 136 mA. According to above results, if we measure the PSDs of frequency fluctuation *Sν*(*f*) of the laser

[24] (left axis), laser linewidth (FWHM) obtained by the method in Ref. [24] (right axis) and by the SDH meth-

α type noise

limit of the integration bandwidth). It mainly results from the presence of the 1/*f*

corresponding integration bandwidth or observation time.

8ln2*f*/π2

od (inset) [18].

244 Optical Interferometry

**Figure 9.** *S <sup>ν</sup>*(*f*) of the DFB laser at operating current of 71, 91, 111, 136 mA.
