Open AccessArticle

Tunable Optical Frequency Comb Generated Using Periodic Windows in a Laser and Its Application for Distance Measurement

Zhuqiu Chen

Can Fang

Yuxi Ruan

Yanguang Yu

Qinghua Guo

Jun Tong

and

Jiangtao Xi

School of Electrical, Computer and Telecommunications Engineering, Faculty of Engineering and Information Sciences, University of Wollongong, Northfields Ave, Wollongong, NSW 2522, Australia

Author to whom correspondence should be addressed.

Sensors 2023, 23(21), 8872; https://doi.org/10.3390/s23218872

Submission received: 9 September 2023 / Revised: 27 October 2023 / Accepted: 30 October 2023 / Published: 31 October 2023

(This article belongs to the Special Issue Radar Remote Sensing and Applications)

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Abstract

A novel method for the generation of an optical frequency comb (OFC) is presented. The proposed approach uses a laser diode with optical feedback and operating at a specific nonlinear dynamic state named periodic window. In this case, the laser spectrum exhibits a feature with a series of discrete, equally spaced frequency components, and the repetition rate can be flexibly adjusted by varying the system parameters (e.g., external cavity length), which can provide many potential applications. As an application example, a dual-OFC system for distance measurement is presented. The results demonstrate the system’s ability to achieve target distance detection, underscoring its potential for real-world applications in this field.

Keywords:

optical feedback; laser dynamics; laser diode; optical frequency comb; distance measurement

1. Introduction

In recent decades, the optical frequency comb (OFC) has seen significant advancements, providing a wide range of stabilized optical wavelengths that can be simultaneously utilized. By emitting a broad spectrum of uniformly spaced ultrashort pulses with stable and narrow lines, these OFCs effectively function as continuous-wave (CW) lasers evenly distributed across the frequency domain [1,2]. Capitalizing on these inherent advantages, OFCs have found applications in various fields, such as atomic clock comparisons [3], long distance quantum key distribution [4,5], calibration of astronomical spectrographs [6], ultralow-noise microwave generation [7], distance measurement and laser ranging [1,8], as well as many applications in medical areas [9,10]. In the context of distance measurement, several OFC-based techniques have been proposed. These include time-of-flight (TOF) distance measurements, which are achieved by the optical cross-correlation method [11]; the synthetic wavelength interferometry method, which generates harmonics of an OFC’s repetition rate through different optical modes and enables a connection between the peak-finding method and single-wavelength heterodyne interferometry [12]; the multiple-wavelength interferometry method, which is based on multichannel phase detection to determine the target position [13]; and others.

However, in most of these ranging applications, single-frequency combs are employed, but they do not fully exploit all the teeth of the comb. This limitation arises because it is often challenging to distinguish adjacent comb lines with megahertz to gigahertz mode spacing [14]. Additionally, the process of adjusting the repetition rate to align the reference pulses and target pulses, which is essential for single-comb ranging, introduces a significant complexity to the systems [15,16]. To address these limitations, dual-comb interferometry is utilized, employing two optical frequency combs with slightly different repetition rates. The beams from these two OFCs interfere on a photodetector, generating a microwave frequency comb (MFC) in the beat spectra. This enables the acquisition of intensity and phase information encoded in the individual comb modes through standard high-speed electronics, facilitating a fast Fourier transform analysis [17]. This principle considerably simplifies the receiver setup and has been demonstrated to be effective for absolute distance measurements, aiming to achieve improved ranging resolution, high update rate, and a low signal-to-noise ratio [18,19].

Regarding the generation source of the dual-comb signal, various methods have been developed. In one approach [20], mode-locked lasers (MLLs) with slightly different repetition rates are utilized for distance measurement. This results in a comblike spectrum through the interaction between the gain medium (erbium-doped fiber) and the laser resonator cavity. However, these systems are costly and complex, requiring additional devices like Kalman filters to mitigate timing jitter. Another method [21] involves the generation of two optical comb signals through electro-optic modulation for achieving ranging. This approach increases system complexity as each electro-optic modulation comb generation scheme consists of an intensity modulator followed by a phase modulator. Additionally, in [22], two femtosecond fiber lasers are used for achieving absolute distance measurement, involving the synchronization of multiple laser modes to produce a train of optical pulses with a fixed repetition rate. Nonetheless, it should be noted that femtosecond lasers can be relatively expensive, potentially leading to increased system costs.

Furthermore, an essential approach to optical frequency comb (OFC) generation involves leveraging laser dynamics triggered by external perturbations. The dynamics of a laser diode (LD) with external perturbation can be highly intricate, with interactions between relaxation oscillation and external feedback frequencies leading to nonlinear instabilities. The method of optical frequency comb generation grounded in laser nonlinear dynamics offers versatile advantages by enabling a range of nonlinear dynamics, and by manipulating controllable parameters such as optical feedback strength, the LD can exhibit various dynamic states, including steady state, P1 oscillation, quasi-periodic oscillation, and chaos, providing flexibility for diverse applications [23,24]. Additionally, it allows a precise control over the comb properties, including repetition rate and comb teeth relative amplitudes, enabling tailored comb solutions for various needs [25,26,27]. Optical injection is a technique that introduces an external perturbation to the LD by injecting light from another LD. In previous studies [25,26], an OFC was generated by optically injecting an LD, leading to nonlinear dynamics in the period-one (P1) oscillation state. The repetition rate of the OFC could be tuned using electro-optic modulation. Additionally, in another study [27], an OFC was generated by injecting a suitable OFC into an LD, resulting in a rich variety of nonlinear dynamics that were characterized by a comblike behavior. The number of comb lines and relative amplitudes of these lines could be adjusted by controlling injection parameters, such as injection strength and detuning frequency. Optical injection enables an adjustable repetition rate for flexibility, while providing stable, low-noise microwave signals, ensuring precision and stability in OFC generation. However, it is essential to recognize that the optical injection method introduces complexities, necessitating the use of at least two LDs, which, in turn, results in higher costs and implementation difficulties.

The compact system configuration of an LD with external optical feedback (EOF) makes it a highly suitable candidate for investigating the properties of LD nonlinear dynamics. This is achieved by introducing external perturbation through the reflection and partial reinjection of emitted light back into the laser cavity. Its practicality and efficiency in providing such perturbations are widely acknowledged [28,29,30].

In this study, we investigate the use of external optical feedback (EOF) to induce laser dynamics for the generation of optical frequency combs (OFCs) with tunable repetition rates. Unlike conventional methods, our approach involves operating a laser diode (LD) within periodic windows, resulting in an OFC source offering a wide tunable repetition rate range of up to 18.07 MHz by adjusting the external cavity length. This approach simplifies the setup significantly, as it relies on a single laser diode with optical feedback, eliminating the need for complex configurations involving mode-locked lasers, electro-optic modulators, and optical amplifiers. Moreover, it offers cost-effectiveness compared to traditional dual-comb systems that rely on expensive mode-locked lasers. By utilizing specific periodic window regions, our method produces stable and low-noise optical frequency combs, addressing common stability challenges encountered in laser systems. Additionally, our technique provides precise control over comb parameters, such as repetition rate and the number of comb teeth, through adjustments in the external cavity length and other operating conditions—a level of controllability challenging to achieve in most conventional comb generation methods. These attributes highlight the potential of our approach for practical applications in various fields, including distance measurement.

2. Periodic Window and OFC Generation

2.1. Dynamic States of an LD with EOF

Figure 1 illustrates the schematic diagram of the LD with EOF used in our study. The system comprises an LD and its external cavity. A reflection mirror positioned on a piezoelectric transducer (PZT) is used to form the cavity. The injection current ratio to the LD is precisely controlled by the LD controller. A variable attenuator (VA) is inserted into the cavity to adjust the EOF strength. The PZT controller governs the amplitude and frequency of the PZT for a precise adjustment of the external cavity length.

To study the laser dynamic states, one portion of the light is directed to an optical spectrum analyzer (OSA) for spectrum analysis. The other portion is captured by a photodiode (PD) and displayed on an oscilloscope (OSC) for a time domain analysis. An optical isolator (OI) is used to prevent backscattering of the light from beam splitter 2 (

B S_{2}

The nonlinear dynamics of a laser diode (LD) with EOF are described by the widely recognized Lang and Kobayashi (L-K) equations, as shown by Equations (1)–(3) [31].

E (t)

represents the amplitude of the electric field,

ϕ (t)

corresponds to the electric field phase, and

N (t)

denotes the carrier density. In this paper,

E^{2} (t)

is treated as the laser intensity.

\begin{matrix} \frac{d E (t)}{d t} = & \frac{1}{2} \{G [N (t), E (t)] - \frac{1}{τ_{p}}\} E (t) + \frac{κ}{τ_{i n}} \cdot E (t - τ) \cdot cos [ω_{0} τ + ϕ (t) - ϕ (t - τ)] \end{matrix}

(1)

\begin{matrix} \frac{d ϕ (t)}{d t} = & \frac{1}{2} α \{G [N (t), E (t)] - \frac{1}{τ_{p}}\} - \frac{κ}{τ_{i n}} \cdot \frac{E (t - τ)}{E (t)} \cdot sin [ω_{0} τ + ϕ (t) - ϕ (t - τ)] \end{matrix}

(2)

\begin{matrix} \frac{d N (t)}{d t} = \frac{J}{e V} - \frac{N (t)}{τ_{s}} - G [N (t), E (t)] E^{2} (t) \end{matrix}

(3)

where

G [N (t), E (t)] = G_{N} [N (t) - N_{0}] [1 - ε Γ E^{2} (t)]

is the modal gain per unit time. The physical meanings and values of parameters used in Equations (1)–(3) are shown in Table A1 in Appendix A [32].

The parameters relevant to the LD used for our study on OFC generation are presented in Table A1 (called internal parameters). The other parameters including injection current, optical feedback strength, and external cavity length are also shown in Table A1 (called controllable parameters). The external parameters are controllable and serve to govern the dynamic states of the LD. An LD with EOF may operate at different states including steady state (S), P1 oscillation state, quasi-periodic (QP) oscillation state, periodic windows state, and chaos. Regarding periodic window, it is the region embedded within chaos and demonstrating a quasiperiodic behavior [23,33,34].

Note that period-N (denoted by PN) in the following means there are N peaks within a period of the laser intensity in time domain. Each state can be identified based on the waveform of

E^{2} (t)

. The transition route from a steady state to chaos, influenced by one or more controllable variables, is known as the bifurcation diagram [23,24]. To construct the bifurcation diagram, we sampled the local maxima

E^{2} {(t)}_{M a x}

(or minima

E^{2} {(t)}_{M i n}

) of the waveform

E^{2} (t)

for a given set of LD controllable parameters. This allowed us to gain valuable insights into the complex dynamics and behavior exhibited by the LD system under different cavity conditions.

Based on our study, a specific set of controllable parameters were chosen as:

α = 2.0

L = 5.2 cm

and

J = 1.2 J_{th}

. We firstly explored the impact of the EOF strength (

κ

) on the dynamic states by varying

κ

over the range (0, 0.06). The corresponding bifurcation diagram was obtained and is depicted on Figure 2.

From Figure 2, we observe six dynamic states. The steady state corresponds to

0 < κ < 0.0147

. As

κ

increases beyond this point, an undamped relaxation oscillation occurs after the Hopf bifurcation, leading the system to enter the period-1 (P1) oscillation state within the range of

0.0147 < κ < 0.0281

. Subsequently, a quasi-periodic (QP) state emerges corresponding to

0.0281 < κ < 0.0318

. The first chaotic region appears in the range of

0.0318 < κ < 0.0399

and the second chaotic region with

0.0456 < κ < 0.0600

. Within the two chaotic regions, we observe a periodic window starting at

κ = 0.0399

and ending at

κ = 0.0456

. This periodic window further exhibits a P3 state followed by a P6 state.

2.2. OFC Generation and Its Tunability

Let us examine the characteristics of the laser intensity

E^{2} (t)

within this periodic window. We selected two operating points with

κ = 0.041

and

κ = 0.042

, respectively, and plotted their corresponding time series (i.e., laser intensity

E^{2} (t)

) in Figure 3. Figure 3a shows that the laser intensity exhibits a periodic structure with three peaks within each period of 1.25 ns. Figure 3b shows a periodic structure with six peaks within each period of 2.48 ns. After extensive simulation tests, as a result, this periodic window combined a P3 state along with a P6 state with

κ = 0.0416

as their boundary line.

We then looked at the spectrum feature when the LD operating at the periodical window. Figure 4 illustrates the laser spectrum at

κ = 0.041

within the P3 region. Clearly it shows a series of discrete, equally spaced frequency components, demonstrating the generation of an OFC in this operating condition. The OFC generated exhibits a repetition rate of

f_{r e p} = 0.8

GHz. The total bandwidth spans from 193.365 THz to 193.464 THz. Within the −10 dB region, there are 66 comb teeth. Hence, the introduced LD with external optical feedback (EOF) system can be regarded as an optical frequency comb (OFC) generation source. In the subsequent sections, we refer to it as OFC-EOF.

Next, we show that the repetition rate of the generated OFC can be adjusted by varying the cavity length L. By gradually increasing L with increments

Δ L = 15.50

nm,

23.25

nm,

31.00

nm and

38.75

nm, respectively, the repetition rate changes from 810.5 MHz to 818.4 MHz, as depicted in Figure 5.

To further determine the tunable range of the repetition rate denoted as

Δ f_{r e p}

, extensive simulations were conducted to generate an OFC under different external cavity lengths L within the range of

0 \leq Δ L \leq 37.2

nm. In these simulations,

α

and J were fixed at

α = 2.0

and

J = 1.2 J_{th}

, and

κ

was fixed at

κ = 0.041

As shown in Figure 6, the repetition rate of the OFC increased from 800.67 MHz to 818.74 MHz as L increased from 5.2 cm to 5.2 cm + 37.2 nm.

3. A Dual-OFC System for Distance Measurement

The simulation analysis revealed that the proposed LD system, operating within a specific periodic window region, could generate an optical frequency comb (OFC) source with a tunable repetition rate spanning from 800.67 MHz to 818.74 MHz. This tunability allowed the system to generate OFC signals with repetition rate differences of up to 18.07 MHz. Such a feature of OFC generation, coupled with its tunable range, holds the potential for various applications in optical and microwave measurements. As an application example, in this section, we present a dual-OFC system for achieve target distance detection. This application showcases the versatility and advantages of the proposed system in practical and precise distance measurement scenarios.

Figure 7 illustrates the schematic of the proposed dual-OFC system for distance measurement. The boxed part of the diagram represents the source of the dual-comb light, consisting of two independent OFC-EOF systems. These systems generate two distinct optical frequency combs, denoted as

OF C_{1}

and

OF C_{2}

, with a slight difference in their repetition rates, referred to as

f_{r e p_d i f f}

. Each OFC-EOF system incorporates a laser diode (LD) with an external feedback cavity, and the cavity lengths (

L_{1}

and

L_{2}

) are precisely adjusted using a piezoelectric transducer (PZT) with a difference of 1.55 nm. For a typical dual-OFC ranging system, generally, a single-mode DFB laser diode is used as the OFCs generation source, after amplification by EDFAs to provide an output power up to around 200–500 mW [16,22,35,36].

Optical isolators are used to prevent light backscattering, which could potentially interfere with the dynamic states of the LDs. The OFC generated by

OFC - EO F_{1}

and

OFC - EO F_{2}

are designated as the

OF C_{1}

beam and

OF C_{2}

beam, respectively. Upon exiting the system, the

OF C_{1}

beam is directed towards a beam splitter (BS) and subsequently divided into two beams: a target measurement beam denoted by

OF C_{1_mea}

(expressed by

E_{1_m e a} (t) = \sum_{n} E_{1 n} e^{j [2 π (n f_{1_r e p} + f_{1_1}) t - ϕ_{n}]}

) and a reference beam denoted by

OF C_{1_ref}

(expressed by

E_{1_r e f} (t) = \sum_{n} E_{1 n} e^{j [2 π (n f_{1_r e p} + f_{1_1}) t]}

), where

E_{1_m e a} (t)

and

E_{1_r e f} (t)

are electric fields of the reflected measurement beam and reference comb, respectively. The

OF C_{1_mea}

beam and

OF C_{1_ref}

beam containing multiple optical frequencies components (up to n) have light magnitudes

E_{1 n}

and

E_{2 n}

, respectively. These beams contain multiple optical frequency components with respective electric fields. The first comb tooth frequencies in

OF C_{1_mea}

and

OF C_{1_ref}

are

f_{1_1}

, and both OFCs have the same repetition rate denoted as

f_{1_r e p}

. The phase difference between each pair of frequency comb teeth in

OF C_{1_mea}

and

OF C_{1_ref}

is represented by

ϕ_{n}

The

OF C_{1_mea}

beam is directed to the target at a distance D and then reflected by the target surface. Simultaneously, the reference beam (

OF C_{1_ref}

) is reflected by a reflector, carrying distance information in its phase. The measurement beam (

OF C_{1_ref}

) and the reflected beam

OF C_{1_mea}

are recombined before passing through

CO L_{2}

. The interference between the two beams is then amplified by an erbium-doped fiber application amplifier (EDFA) to compensate for any power losses during propagation. Subsequently, the interference beam is combined with the

OF C_{2}

beam (expressed by

E_{2} (t) = \sum_{n} E_{2 n} e^{j [2 π (n f_{2_r e p} + f_{2_1}) t]}

) using a fiber coupler (FC) and detected by a photodiode (PD), yielding a photocurrent denoted as

I_{b e a t}

(expressed as

I_{b e a t} = {|E_{1_m e a} (t) + E_{1_r e f} (t) + E_{2} (t)|}^{2}

), where the first comb tooth frequency of

OF C_{2}

is denoted as

f_{2_1}

, and its repetition rate is

f_{2_r e p}

. There exists a slight difference (

f_{r e p_d i f f}

) between

f_{1_r e p}

and

f_{2_r e p}

, which can be controlled by adjusting the external cavity length

L_{1}

and

L_{2}

. It is assumed that the oscillations of the internal resonant cavities of the LDs in

OFC - EO F_{1}

and

OFC - EO F_{2}

are synchronized, ensuring the equality of

f_{1_1}

and

f_{2_1}

. After passing through a low-pass filter, we gain the output photocurrent denoted by

I_{b e a t_l o w}

, expressed by

I_{b e a t_l o w} = \sum_{n} E_{1 n} E_{2 n} cos (\frac{ϕ_{n}}{2}) cos (2 π n f_{r e p_d i f f} t - \frac{ϕ_{n}}{2})

, and it is recorded by the OSC for further signal processing to extract the distance.

By applying a fast Fourier transform (FFT) on

I_{b e a t_l o w}

, the spectrum of the resulting beat spectrum is obtained. The repetition rate of the beat signal is equal to the repetition rate difference between

OF C_{1_mea}

and

OF C_{1_ref}

. The distance information is then extracted from the phase of the respective frequency comb components. This configuration allows us to achieve target distance detection through the analysis of the beat spectrum.

The main process for retrieving the distance information can be found in [37]. Figure 8 is the processing diagram. As depicted in Figure 8, the process of distance measurement involves a sequence of six distinct steps. In the initial stage, the electric fields of

OF C_{1_mea}

OF C_{1_ref}

, and

OF C_{2}

are derived from the dual-OFC source. The mathematical representations of their respective electric fields are presented within the first block. Subsequently, in step 2, the

OF C_{2}

is subjected to a beating process with the

OF C_{1_mea}

and

OF C_{1_ref}

signals, which are reflected from the target and reflector, respectively. This interaction takes place on a PD. Moving on to step 3, the detected current signal denoted as

I_{b e a t}

originating from the photodiode passes through a low-pass filter. The primary purpose of this filtering step is to eliminate high-frequency components. It is noteworthy that

I_{b e a t_l o w}

in this context symbolizes the superimposed intensity of a beat signal characterized by a repetition rate of

f_{r e p_d i f f}

. Subsequently, the collected spectrum of the low-pass-filtered signal is acquired using an OSC. Each comb tooth’s power is extracted individually, and the application of the fast Fourier transform (FFT) serves to eliminate direct current (DC) terms. During steps 4 and 5, the phase (

ϕ_{n}

) pertaining to each comb tooth is determined by subjecting the low-pass-filtered signal to the Hilbert transform, where

τ_{D}

is the time delay between the

OF C_{1_mea}

beam and the

OF C_{1_ref}

beam, given by

τ_{D} = 2 D / c

, where D is the distance difference between target and reflector, and c is the speed of light. The Hilbert transform is an operation that converts a real-valued signal into a complex-valued signal while preserving the original frequency components. It is commonly used in signal processing to extract the instantaneous amplitude and phase information from a signal [38].

Then, we employ an unwrapping process to determine the fit line slope of the relationship between phases and the order of a comb tooth (up to n). Lastly, in the sixth step, we compute the measured distance

D_{m e a}

by equations provided in the sixth block. The measured distance can be calculated by the first equation in the step 6 block when

0 \leq |D_{p r e}| \leq 1 / 2 D_{n a m b}

, where

D_{n a m b}

is the range of nonambiguity denoted by

D_{n a m b} = c / 2 f_{1_r e p} = 187, 200 μ

m. We observe that when the

D_{p r e}

is in the range of

0 \leq |D_{p r e}| \leq 1 / 2 D_{n a m b}

, the

d ϕ_{n} / d n

(slope) can be used for calculating

D_{m e a}

, indicating a positive correlation between

D_{p r e}

and

d ϕ_{n} / d n

. However, as

D_{p r e}

increases, the

d ϕ_{n} / d n

becomes closer to

π

and then decreases when

D_{p r e}

is larger than

1 / 2 D_{n a m b}

, falling into the range of

1 / 2 D_{n a m b} \leq |D_{p r e}| \leq D_{n a m b}

. In this case,

D_{p r e}

and

d ϕ_{n} / d n

change to be negatively correlated and the expression of

D_{m e a}

is modified to the second equation in the step 6 block. When

D_{m e a}

exceeds

D_{n a m b}

, the relationship between

D_{p r e}

and the slope of unwrapped phases repeats for each

D_{n a m b}

. To measure distances larger than

D_{n a m b}

, the repetition rate needs to be tuned accordingly. For a specific

D_{p r e}

, different repetition rates can result in different

d ϕ_{n} / d n

’s. In this case,

D_{m e a}

can be calculated by the third equation in the step 6 block. Following the aforementioned processing diagram, we present an example with a predefined distance of

D_{p r e}

= 2,400,000.0

μ

m. In this case,

D_{p r e}

exceeds

D_{n a m b}

, necessitating the use of the third equation in the step 6 block for distance measurement.

OFC - EO F_{1}

and

OFC - EO F_{2}

were both fixed with

α = 2.0

J R a t i o = 1.2

and

κ = 0.041

, resulting in the generation of two OFC signals with repetition rates of

f_{1_r e p} =

800.67 MHz and

f_{2_r e p} = f_{1_r e p} +

1.1267 MHz, respectively. In this case, an

f_{r e p_d i f f}

of 1.1267 MHz was achieved.

Figure 9a shows the spectrum of the beat signal of the

OF C_{1_mea}

beam,

OF C_{1_ref}

beam, and

OF C_{2}

beam after low-pass filtering. It is observed as a 59-teeth beat spectrum with the first comb tooth frequency and repetition rate both at 1.1267 MHz, which is consistent with the expression of

I_{b e a t_l o w}

shown in the step 3 block. Subsequently, the power of each comb tooth was extracted. To extract the distance information carried by the phase of each comb tooth, we applied the Hilbert transform on the extracted power spectrum, as shown in Figure 9b. The phase of each comb tooth was wrapped within the range of

(- π, π)

. To obtain the derivative of phase with respect to n, an unwrapping process was applied. The results of the 59 points are shown in Figure 9c, and the corresponding fitting line is plotted in black. It can be observed that the obtained

d ϕ_{n} / d n

was 1.1992.

In this study, our OFC-EOF system can generate OFCs with variable repetition rates ranging from 800.67 MHz to 818.74 MHz. In the context of distance measurement, the repetition rates

f_{1_r e p}

and

f_{2_r e p}

are tuned by precisely adjusting the external cavity lengths

L_{1}

and

L_{2}

in increments of

Δ L = 38.75

nm. In Figure 9d, the correlation between the varied repetition rate

f_{1_r e p}

and the corresponding

d ϕ_{n} / d n

is illustrated. The distance can be obtained twice with the results of

D_{m e a}

= 2,400,032.5

μ

m and

D_{m e a}

= 2,400,070.3

μ

m, respectively.

4. Conclusions

In this study, we successfully demonstrated the generation of optical frequency combs (OFCs) with a tunable repetition rate using a laser diode (LD) with external optical feedback (EOF). By operating the LD within periodic windows, we were able to control the laser dynamics and achieve a tunable repetition rate for the OFCs. The bifurcation diagram provided insights into the different laser dynamic states, with a focus on the period-three window. We established the relationship between system parameters, such as the external cavity length and the tunable range of the repetition rate. Furthermore, we showcased the practicality of our proposed OFC generation method by implementing it in a dual-OFC system for distance measurement. The results showed a precise target distance detection, indicating the method’s potential for practical applications in fields such as remote sensing and precision metrology. Overall, this study contributes to the understanding of LD nonlinear dynamics and provides a novel approach for generating OFCs with tunable repetition rates. The tunability capabilities of the proposed method open up new possibilities for applications in various fields requiring precise frequency comb sources.

Author Contributions

Conceptualization, Z.C., C.F., and Y.Y.; methodology, Z.C., C.F., and Y.R.; software, Z.C., C.F., and Y.R.; formal analysis, Z.C. and C.F.; writing—original draft preparation, Z.C.; writing—review and editing, Y.Y., J.X., Q.G., and J.T.; supervision, Y.Y., J.X., and Q.G. All authors regularly discussed the progress during the entire work. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Data sharing does not apply to this article as no new data were created or analyzed in this study.

Conflicts of Interest

The authors declare no conflict of interest.

Appendix A

Table A1. Physical meaning and value of symbols in Lang and Kobayashi (L-K) equations.

	Symbol	Physical Meaning	Value
Internal Parameters	$G_{N}$	Model gain coefficient	$8.1 \times 10^{- 13}$ m $^{3}$ s $^{- 1}$
	$N_{0}$	Carrier density at transparency	$1.1 \times 10^{24}$ m $^{- 3}$
	$ε$	Nonlinear gain compression coefficient	$2.5 \times 10^{- 23}$ m $^{3}$
	$Γ$	Confinement factor	$0.3$
	$τ_{p}$	Photon lifetime	$2.0 \times 10^{- 12} s$
	$τ_{s}$	Carrier lifetime	$2.0 \times 10^{- 9} s$
	$τ_{i n}$	Internal cavity round-trip time	$8.0 \times 10^{- 12} s$
	e	Elementary charge	$1.6 \times 10^{- 19}$ C
	V	Volume of the active region	$1 \times 10^{- 16}$ m $^{3}$
	$ω_{0}$	Unperturbed optical angular frequency of a laser diode, $ω_{0} = 2 π c / λ_{0}$ , where c is the speed of light, and $λ_{0} = 1550 nm$ is the wavelength of the LD	$1.2 \times 10^{15}$
	$α$	Line-width enhancement factor	$2.0$
External (Controllable) Parameters	J	Injection current	$1.2 J_{th}$
	$κ$	Feedback strength	$0.041$
	L	External cavity length	5.2 cm
	$τ$	External cavity round trip time, $τ = 2 L / c$	34.6 ns

κ

—Feedback strength: This refers to the fraction of emitted power from the laser diode that is fed back into the cavity after reflection from the external mirror. It determines the amount of external optical feedback.

α

—The linewidth enhancement factor, denoted as “a” or “

α

”, is a parameter that characterizes the change in the linewidth of a laser diode as a function of the modulation frequency. It quantifies the degree to which the laser’s spectral linewidth is broadened when the laser is subjected to intensity modulation or phase modulation.

J—Injection current density: This represents the current pumped into the laser diode relative to the threshold current needed for lasing. It is quantified by the ratio

J_{th}

, where

J_{th}

is the threshold current density.

L—External cavity length: This is the round-trip length of the external cavity formed between the laser facet and the external reflector. Tuning L provides a means to adjust the repetition rate of the optical frequency comb.

References

Fortier, T.; Baumann, E. 20 years of developments in optical frequency comb technology and applications. Commun. Phys. 2019, 2, 153. [Google Scholar] [CrossRef]
Chang, L.; Liu, S.; Bowers, J.E. Integrated optical frequency comb technologies. Nat. Photonics 2022, 16, 95–108. [Google Scholar] [CrossRef]
Bodine, M.I.; Deschênes, J.D.; Khader, I.H.; Swann, W.C.; Leopardi, H.; Beloy, K.; Bothwell, T.; Brewer, S.M.; Bromley, S.L.; Chen, J.S.; et al. Optical atomic clock comparison through turbulent air. Phys. Rev. Res. 2020, 2, 033395. [Google Scholar] [CrossRef]
Wang, X.B.; Yu, Z.W.; Hu, X.L. Twin-field quantum key distribution with large misalignment error. Phys. Rev. A 2018, 98, 062323. [Google Scholar] [CrossRef]
Lucamarini, M.; Yuan, Z.L.; Dynes, J.F.; Shields, A.J. Overcoming the rate–distance limit of quantum key distribution without quantum repeaters. Nature 2018, 557, 400–403. [Google Scholar] [CrossRef] [PubMed]
Ycas, G.G.; Quinlan, F.; Diddams, S.A.; Osterman, S.; Mahadevan, S.; Redman, S.; Terrien, R.; Ramsey, L.; Bender, C.F.; Botzer, B.; et al. Demonstration of on-sky calibration of astronomical spectra using a 25 GHz near-IR laser frequency comb. Opt. Express 2012, 20, 6631–6643. [Google Scholar] [CrossRef]
Brochard, P.; Schilt, S.; Südmeyer, T. Ultra-low noise microwave generation with a free-running optical frequency comb transfer oscillator. Opt. Lett. 2018, 43, 4651–4654. [Google Scholar] [CrossRef]
Newbury, N.R. Searching for applications with a fine-tooth comb. Nat. Photonics 2011, 5, 186–188. [Google Scholar] [CrossRef]
Ji, X.; Yao, X.; Klenner, A.; Gan, Y.; Gaeta, A.L.; Hendon, C.P.; Lipson, M. Chip-based frequency comb sources for optical coherence tomography. Opt. Express 2019, 27, 19896–19905. [Google Scholar] [CrossRef]
Bajraszewski, T.; Wojtkowski, M.; Szkulmowski, M.; Szkulmowska, A.; Huber, R.; Kowalczyk, A. Improved spectral optical coherence tomography using optical frequency comb. Opt. Express 2008, 16, 4163–4176. [Google Scholar] [CrossRef]
Lee, J.; Lee, K.; Lee, S.; Kim, S.W.; Kim, Y.J. High precision laser ranging by time-of-flight measurement of femtosecond pulses. Meas. Sci. Technol. 2012, 23, 065203. [Google Scholar] [CrossRef]
Wu, G.; Takahashi, M.; Inaba, H.; Minoshima, K. Pulse-to-pulse alignment technique based on synthetic-wavelength interferometry of optical frequency combs for distance measurement. Opt. Lett. 2013, 38, 2140–2143. [Google Scholar] [CrossRef] [PubMed]
Wang, G.; Jang, Y.S.; Hyun, S.; Chun, B.J.; Kang, H.J.; Yan, S.; Kim, S.W.; Kim, Y.J. Absolute positioning by multi-wavelength interferometry referenced to the frequency comb of a femtosecond laser. Opt. Express 2015, 23, 9121–9129. [Google Scholar] [CrossRef] [PubMed]
Zhu, Z.; Wu, G. Dual-comb ranging. Engineering 2018, 4, 772–778. [Google Scholar] [CrossRef]
Cui, M.; Schouten, R.; Bhattacharya, N.; Berg, S. Experimental demonstration of distance measurement with a femtosecond frequency comb laser. J. Eur. Opt. Soc.-Rapid Publ. 2008, 3, 08003. [Google Scholar] [CrossRef]
Hu, D.; Wu, Z.; Cao, H.; Shi, Y.; Li, R.; Tian, H.; Song, Y.; Hu, M. Dual-comb absolute distance measurement of non-cooperative targets with a single free-running mode-locked fiber laser. Opt. Commun. 2021, 482, 126566. [Google Scholar] [CrossRef]
Lee, J.; Han, S.; Lee, K.; Bae, E.; Kim, S.; Lee, S.; Kim, S.W.; Kim, Y.J. Absolute distance measurement by dual-comb interferometry with adjustable synthetic wavelength. Meas. Sci. Technol. 2013, 24, 045201. [Google Scholar] [CrossRef]
Godbout, M.; Deschênes, J.D.; Genest, J. Spectrally resolved laser ranging with frequency combs. Opt. Express 2010, 18, 15981–15989. [Google Scholar] [CrossRef]
Liu, T.A.; Newbury, N.R.; Coddington, I. Sub-micron absolute distance measurements in sub-millisecond times with dual free-running femtosecond Er fiber-lasers. Opt. Express 2011, 19, 18501–18509. [Google Scholar] [CrossRef]
Wu, G.; Zhou, Q.; Shen, L.; Ni, K.; Zeng, X.; Li, Y. Experimental optimization of the repetition rate difference in dual-comb ranging system. Appl. Phys. Express 2014, 7, 106602. [Google Scholar] [CrossRef]
Martin, B.; Feneyrou, P.; Dolfi, D.; Martin, A. Performance and limitations of dual-comb based ranging systems. Opt. Express 2022, 30, 4005–4016. [Google Scholar] [CrossRef] [PubMed]
Zhang, H.; Wei, H.; Wu, X.; Yang, H.; Li, Y. Absolute distance measurement by dual-comb nonlinear asynchronous optical sampling. Opt. Express 2014, 22, 6597–6604. [Google Scholar] [CrossRef] [PubMed]
Uchida, A. Optical Communication with Chaotic Lasers: Applications of Nonlinear Dynamics and Synchronization; John Wiley & Sons: Hoboken, NJ, USA, 2012. [Google Scholar]
Ohtsubo, J.; Ohtsubo, J. Semiconductor Lasers and Theory; Springer: Berlin/Heidelberg, Germany, 2017. [Google Scholar]
Zhang, Y.; Liao, S.; Wang, G.; Yang, K.; Zhang, Z.; Zhang, S.; Liu, Y. Robust Optical Frequency Comb Generation by Using a Three-Stage Optical Nonlinear Dynamic. Front. Phys. 2021, 9, 711959. [Google Scholar] [CrossRef]
Zhang, Y.; Ge, M.; Zhang, Z.; Zhang Sr, S.; Liu, Y. Self-recirculating modulation-enhanced optical frequency comb generation pumped by optical injection. In Semiconductor Lasers and Applications IX; SPIE: Bellingham, DC, USA, 2019; Volume 11182, pp. 89–94. [Google Scholar]
Doumbia, Y.; Malica, T.; Wolfersberger, D.; Panajotov, K.; Sciamanna, M. Nonlinear dynamics of a laser diode with an injection of an optical frequency comb. Opt. Express 2020, 28, 30379–30390. [Google Scholar] [CrossRef]
Tager, A.A.; Petermann, K. High-frequency oscillations and self-mode locking in short external-cavity laser diodes. IEEE J. Quantum Electron. 1994, 30, 1553–1561. [Google Scholar] [CrossRef]
Yee, D.S.; Leem, Y.A.; Kim, S.T.; Park, K.H.; Kim, B.G. Self-pulsating amplified feedback laser based on a loss-coupled DFB laser. IEEE J. Quantum Electron. 2007, 43, 1095–1103. [Google Scholar] [CrossRef]
Yu, L.; Lu, D.; Pan, B.; Zhao, L.; Wu, J.; Xia, G.; Wu, Z.; Wang, W. Monolithically integrated amplified feedback lasers for high-quality microwave and broadband chaos generation. J. Light. Technol. 2014, 32, 3595–3601. [Google Scholar] [CrossRef]
Lang, R.; Kobayashi, K. External optical feedback effects on semiconductor injection laser properties. IEEE J. Quantum Electron. 1980, 16, 347–355. [Google Scholar] [CrossRef]
Nie, B.; Ruan, Y.; Yu, Y.; Guo, Q.; Xi, J.; Tong, J. Period-one microwave photonic sensing by a laser diode with optical feedback. J. Light. Technol. 2020, 38, 5423–5429. [Google Scholar] [CrossRef]
Maranhao, D.M.; Baptista, M.S.; Sartorelli, J.C.; Caldas, I.L. Experimental observation of a complex periodic window. Phys. Rev. E 2008, 77, 037202. [Google Scholar] [CrossRef]
Qin, J.; Wang, L.; Yuan, D.; Gao, P.; Zhang, B. Chaos and bifurcations in periodic windows observed in plasmas. Phys. Rev. Lett. 1989, 63, 163. [Google Scholar] [CrossRef] [PubMed]
Riemensberger, J.; Lukashchuk, A.; Karpov, M.; Weng, W.; Lucas, E.; Liu, J.; Kippenberg, T.J. Massively parallel coherent laser ranging using a soliton microcomb. Nature 2020, 581, 164–170. [Google Scholar] [CrossRef] [PubMed]
Zhang, Z.; Gu, C.; Sun, J.; Wang, C.; Gardiner, T.; Reid, D.T. Asynchronous midinfrared ultrafast optical parametric oscillator for dual-comb spectroscopy. Opt. Lett. 2012, 37, 187–189. [Google Scholar] [CrossRef] [PubMed]
Hei, K.; Anandarajah, K.; Martin, E.P.; Shi, G.; Anandarajah, P.M.; Bhattacharya, N. Absolute distance measurement with a gain-switched dual optical frequency comb. Opt. Express 2021, 29, 8108–8116. [Google Scholar] [CrossRef]
Feldman, M. Hilbert transform in vibration analysis. Mech. Syst. Signal Process. 2011, 25, 735–802. [Google Scholar] [CrossRef]

Figure 1. Schematic diagram for an LD with EOF. LD, laser diode; BS, beam splitter; VA, variable attenuator; RM, reflection mirror; PZT, piezoelectric transducer; OI, optical isolator; OSA, optical spectrum analyzer; PD, photodiode; OSC, oscilloscope; PC, computer.

Figure 2. Bifurcation diagram with a periodic window embedded in chaos.

Figure 3. Time series (a) P3 region. (b) P6 region.

Figure 4. Optical spectrum at

κ = 0.041

Figure 4. Optical spectrum at

κ = 0.041

Figure 5. Demonstration of the repetition rate changing with L.

Figure 6. The relationship between

Δ L

and

Δ f_{r e p}

Figure 6. The relationship between

Δ L

and

Δ f_{r e p}

Figure 7. Schematic setup of a dual-OFC system for distance measurement. OFC-EOF, external optical feedback-based OFC sources; OI, optical isolator; COL, collimator; FC, fiber coupler; EDFA, erbium-doped fiber application amplifier; PD, photodiodes; LF, low-pass filter; OSC, digital oscilloscopes; PC, computer for further processing.

Figure 8. Measurement principle and processing diagram for distance measurement.

Figure 9. (a) Beat spectrum of the

OF C_{1_mea}

beam,

OF C_{1_ref}

beam, and

OF C_{2}

beam after low-pass filtering. (b) Wrapped phase obtained by the Hilbert transform. (c) Unwrapped phase and the fit line. (d) The relationship between the varied

f_{1_r e p}

and

d ϕ_{n} / d n

when

D_{p r e}

= 2,400,000.0

μ

Figure 9. (a) Beat spectrum of the

OF C_{1_mea}

beam,

OF C_{1_ref}

beam, and

OF C_{2}

beam after low-pass filtering. (b) Wrapped phase obtained by the Hilbert transform. (c) Unwrapped phase and the fit line. (d) The relationship between the varied

f_{1_r e p}

and

d ϕ_{n} / d n

when

D_{p r e}

= 2,400,000.0

μ

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Chen, Z.; Fang, C.; Ruan, Y.; Yu, Y.; Guo, Q.; Tong, J.; Xi, J. Tunable Optical Frequency Comb Generated Using Periodic Windows in a Laser and Its Application for Distance Measurement. Sensors 2023, 23, 8872. https://doi.org/10.3390/s23218872

AMA Style

Chen Z, Fang C, Ruan Y, Yu Y, Guo Q, Tong J, Xi J. Tunable Optical Frequency Comb Generated Using Periodic Windows in a Laser and Its Application for Distance Measurement. Sensors. 2023; 23(21):8872. https://doi.org/10.3390/s23218872

Chicago/Turabian Style

Chen, Zhuqiu, Can Fang, Yuxi Ruan, Yanguang Yu, Qinghua Guo, Jun Tong, and Jiangtao Xi. 2023. "Tunable Optical Frequency Comb Generated Using Periodic Windows in a Laser and Its Application for Distance Measurement" Sensors 23, no. 21: 8872. https://doi.org/10.3390/s23218872

APA Style

Chen, Z., Fang, C., Ruan, Y., Yu, Y., Guo, Q., Tong, J., & Xi, J. (2023). Tunable Optical Frequency Comb Generated Using Periodic Windows in a Laser and Its Application for Distance Measurement. Sensors, 23(21), 8872. https://doi.org/10.3390/s23218872

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Tunable Optical Frequency Comb Generated Using Periodic Windows in a Laser and Its Application for Distance Measurement

Abstract

1. Introduction

2. Periodic Window and OFC Generation

2.1. Dynamic States of an LD with EOF

2.2. OFC Generation and Its Tunability

3. A Dual-OFC System for Distance Measurement

4. Conclusions

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Conflicts of Interest

Appendix A

References

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