A generic algorithm to numerically model the linewidth broadening of laser

International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN

0976 – 6464(Print), ISSN 0976 – 6472(Online) Volume 4, Issue 4, July-August (2013), © IAEME

26

A GENERIC ALGORITHM TO NUMERICALLY MODEL THE LINEWIDTH

BROADENING OF LASER DUE TO PHASE NOISE

Yogesh Prasad K R1, Srinivas T

2, Ramana D V

3

1Department of Electrical Communications Engineering, Indian Institute of Science, Bangalore-

560012, India 2Department of Electrical Communications Engineering, Indian Institute of Science, Bangalore-

560012, India 3Communications Systems Group, ISRO Satellite Centre, Vimanapura Post, Bangalore-

560017, India

ABSTRACT

A generic algorithm to numerically model the linewidth broadening of laser due to phase

noise is discussed. The application of optical heterodyning to generate microwave signal has been

considered to describe the utility of the algorithm in system level simulations. Optical heterodyning

experiment has been conducted to validate the simulation results obtained by using the algorithm.

Practical results have been found to agree closely with the simulated results.

Keywords: Laser lineshape, Laser linewidth, Laser phase noise, Microwave generation, Optical

Heterodyning

I. INTRODUCTION

Generation of microwave signals using optical techniques has been an active area of research.

When two optical signals are heterodyned by passing them through a photo-detector, which is a

square-law device, the resulting signal contains a term that corresponds to the frequency difference

between the two optical sources. By choosing the optical frequencies such that their difference lies in

the microwave region, a microwave signal can be generated by optical heterodyning. Spectral width

and phase noise profile of the microwave signal are determined by the characteristics of the optical

signals.

Laser output is not strictly monochromatic and hence cannot be modeled as a line spectrum.

Development of a numerical model for laser output based on a priori knowledge of its spectral

characteristics can prove to be very useful to study the effects of linewidth broadening in laser

applications. A generic method to numerically model laser linewidth, based on phase noise profile

defined at discrete frequency offsets, is presented in this paper. This model is applied to study the

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ISSN 0976 – 6464(Print)

ISSN 0976 – 6472(Online)

Volume 4, Issue 4, July-August, 2013, pp. 26-35

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effect of laser linewidth on the microwave signal generated by optical heterodyning. Optical

heterodyning experiment has been carried out and the results obtained have been shown to agree

closely with the simulation results generated using the proposed model.

II. EFFECT OF PHASE NOISE ON LASER LINEWIDTH

In the presence of phase noise, the output of a laser can be expressed as:

A(t) = E cos[2πfct+φn(t)] (1)

= E {cos[2πfct]cos[φn(t)] – sin[2πfct]sin[φn(t)]} (2)

A(t) ≅ E {cos[2πfct] – sin[2πfct][φ n(t)]} when φn(t) is small (3)

The above expression shows that phase noise ‘φn(t)’ mixes with the laser frequency ‘fc’ and

produces side bands around it. As a result, the spectrum of a laser exhibits finite non-zero linewidth.

III. ALGORITHM TO NUMERICALLY MODEL LASER LINEWIDTH BROADENING

DUE TO PHASE NOISE

Optical line shape and associated linewidth of a laser or its power spectral density of

frequency noise can be used to describe the spectral properties of a laser. The relationship between

frequency noise spectral density and laser linewidth has been reported in many papers. Theoretical

formulas linking the frequency noise spectral density to the laser line shape have been given by

Elliott and co-workers [1]. It was proposed that line shape obtained in the case of a rectangular noise

spectrum of finite bandwidth is Gaussian when the ratio of the frequency deviation to the noise

bandwidth is large and Lorentzian when small. The ideal case of a pure white frequency noise

spectrum has been extensively studied [2], as it can be fully solved analytically leading to the well-

known Lorentzian line shape described by the Schawlow–Townes–Henry linewidth [3,4]. However,

as the real noise spectrum of a laser is much more complicated it leads to a non-analytical line shape

that can be determined only numerically. Flicker noise, which generally affects lasers at low

frequency, has been widely studied in the literature [5-8]. Flicker noise produces spectral broadening

of the laser linewidth compared to the Schawlow–Townes–Henry limit.

It can be seen that the studies carried out so far aimed at creating a model for laser line shape

based on certain approximations or assumptions. For a given application, if a priori knowledge of

phase noise profile of the laser under consideration is available or if the laser can be characterized for

phase noise and these values be used for obtaining laser line shape then it would result in a more

accurate model. Development of such a model for laser output can prove to be very useful to study

the effects of linewidth broadening in laser applications. A generic method to numerically model

laser linewidth, based on phase noise profile defined at discrete frequency offsets, is proposed.

Phase noise values at discrete offsets from the laser frequency form the inputs for the

algorithm. An N-point array of discrete time domain points representing the laser output, whose FFT

satisfies the given phase noise profile, forms the output. A plot of phase noise values versus

frequency offsets from laser frequency, considered for demonstration, is shown in Fig. 1. The first

task is to obtain envelop of SSB phase noise spectrum, beginning at zero offset from carrier

frequency and ending at Nyquist frequency (Fs/2) of the discrete-time system, based on the input

phase noise data. Sampling frequency (Fs) must be chosen such that it is greater than twice the

maximum offset at which phase noise value is available in order to avoid aliasing. Available phase

noise data is interpolated on to M equally spaced points in the interval [0 Fs/2] including bounds as

shown in Fig. 2. This results in partitions of size, say, ∆f. Since the phase noise values are specified



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in units of dBc/Hz, multiplication of interpolated values with ∆f gives the integrated phase noise

power over the frequency band ∆f. Since amplitude varies as square root of power, the distribution of

phase noise amplitude over the interval [0 Fs/2] can be readily derived. This is stored as an M-point

vector Nph[n], where n varies from 1 to M. This vector represents the distribution of SSB phase noise

amplitudes that satisfies the given phase noise profile.

A complex vector NAWGN[n] of length M, representing random fluctuations in magnitude and

phase of a sampled AWGN signal in frequency domain, is now created as shown in Fig. 3. This can

be done by utilizing standard built-in MATLAB function. This noise, which by definition is

distributed uniformly over the interval [0 Fs/2], is tailored into the desired phase noise profile by

performing an element-to-element multiplication of NAWGN[n] and Nph[n]. SSB phase noise spectrum

over the interval [0 Fs/2], obtained by this process is shown in Fig. 4. Negative spectrum over the

interval (Fs/2 Fs), bounds not included, is derived from positive SSB spectrum based on symmetry.

DSB spectrum of phase noise about the carrier frequency is thus obtained as a vector of 2M-2

elements. Inverse FFT yields complex phase noise signal in time domain. This array when multiplied

element-to-element with 2M-2 samples of an ideal complex sinusoidal carrier signal, shown in Fig.

5, yields noisy carrier satisfying the given phase noise profile as shown in Fig. 6. Fig. 7 shows the

spectrum of the noisy carrier signal when viewed over a narrow frequency span.

Fig. 1. Phase noise data fed as input to the algorithm



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Fig. 2. Phase noise profile generated by interpolation

Fig. 3. Spectrum of AWGN signal



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Fig. 4. AWGN spectrum tailored to phase noise profile

Fig. 5. Spectrum of an ideal laser



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Fig. 6. Spectrum of laser modeled using phase noise data (viewed on a wide span)

Fig. 7. Spectrum of laser modeled using phase noise data (viewed on a narrow span)

IV. SIMULATION OF OPTICAL HETERODYNING USING THE PROPOSED MODEL

To demonstrate the utility of the algorithm, the application of optical heterodyning for

generation of microwave signal is considered. Using the model based on the proposed algorithm, the

effect of laser linewidth on the microwave signal generated by optical heterodyning is simulated. Fig.

8 shows the spectrum of the signal obtained by combining two laser signals that are to be

heterodyned. When this signal is heterodyned in a photodetector, spectrum of the resulting signal at



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difference frequency is as shown in Fig. 9. When the linewidth of one of the lasers is increased to

simulate poor phase noise and coherency, the spectrum of the resulting microwave signal would be

as shown in Fig. 10.

Fig. 8. Spectrum of the signal fed to photodetector for heterodyning

Fig. 9. Spectrum of the heterodyned output at difference frequency



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Fig. 10 Spectrum of heterodyned output when linewidth of laser is broad (150 MHz) due to poor

phase noise and coherency

IV. OPTICAL HETERODYNING EXPERIMENT TO VERIFY THE PROPOSED MODEL:

Optical heterodyning experiment was conducted using Tunable Laser Sources from Agilent

operating at 1550 nm. Wavelength of the laser modules is settable with a resolution of 0.1 picometer.

A wideband photodetector having a bandwidth of 18 GHz was used as the heterodyning element.

The photodetector current is dropped across a built-in 50 ohm load to provide the microwave output.

This output was monitored using RF Spectrum Analyzer. A typical spectrum of microwave signals

generated by optical heterodyning is shown in Fig. 11. The wavelength offset between the two lasers

and the generated microwave signal frequency is indicated in the figure.

The frequency of the microwave signal agrees with the theoretically determined frequency

based on the wavelength offset. Also, the line shape and width obtained by the experiment agree

closely with the simulated result shown in Fig. 9 thus validating the proposed numerical model.

The coherence length of one of the lasers involved in heterodyning was reduced to increase the

linewidth of the laser. The spectrum of the resulting signal is shown in Fig. 12. The spectral

characteristics of this signal were found to be in agreement with the simulated result shown in Fig.

10.



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Fig. 11. Spectrum of heterodyned output for λ1=1550 nm and λ2=1549.998 nm corresponding to

microwave signal of frequency ≅ 250 MHz

Fig. 12. Spectrum of heterodyned output for λ1=1550 nm and λ2=1549.996 nm corresponding to

microwave signal of frequency ≅ 500 MHz. Linewidth of laser used is 150 MHz



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V. CONCLUSION

A generic algorithm to numerically model the phase noise induced linewidth widening of

laser is proposed. The model has been used to simulate the generation of microwave signal by optical

heterodyning. An optical heterodyning experiment was conducted to evaluate the simulation result.

The linewidth/shape of the microwave signal obtained practically was found to be in close agreement

with that of the simulated signal. The model is ideal for system level simulations using applications

such as MATLAB. The model being based on the actual phase noise profile of characterized laser,

will be more accurate than the approximations generally used.

REFERENCES

[1] D. S. Elliott, R. Roy, and S. J. Smith, “Extracavity laser bandshape and bandwidth modification,”

Phys. Rev. A 26, 12–18 (1982).

[2] P. B. Gallion and G. Debarge, “Quantum phase noise and field correlation in single frequency

semiconductor laser systems,” IEEE J. Quantum Electron. 20, 343–350 (1984).

[3] A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112, 1940–1949

(1958).

[4] C. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18,

259–264 (1982).

[5] G. M. Stéphan, T. T. Tam, S. Blin, P. Besnard, and M. Têtu, “Laser line shape and spectral

density of frequency noise,” Phys. Rev. A 71, 043809 (2005).

[6] J.-P. Tourrenc, “Caractérisation et modélisation du bruit d’amplitude optique, du bruit de

fréquence et de la largeur de raie de VCSELs monomode,” Ph.D. dissertation (Université de

Montpellier II, 2005).

[7] L.B.Mercer, “1/f frequency noise effects on self-heterodyne linewidth measurements,”

J. LightwaveTechnol.9, 485–493(1991).

[8] K. Kikuchi, “Effect of llf-type fm noise on semiconductor-laser linewidth residual in high-power

limit,” IEEE J. Quantum Electron. 25, 684–688 (1989).

[9] Prof. B.N. Biswas, S. Chatterjee and S Pal, “Laser Induced Microwave Oscillator”, International

journal of Electronics and Communication Engineering &Technology (IJECET), Volume 3,

Issue 1, 2012, pp. 211 - 219, ISSN Print: 0976- 6464, ISSN Online: 0976 –6472.

A generic algorithm to numerically model the linewidth broadening of laser

Technology

laser phase noise

linewidth broadening

laser output

effect of laser linewidth

laser lineshape

laser applications

laser line shape

laser frequency fc