## Signal-to-noise ratio in direct-detection mid-infrared Random-Modulation Continuous-Wave lidar in the presence of colored additive noise

Optics Express, Vol. 9, Issue 8, pp. 389-399 (2001)

http://dx.doi.org/10.1364/OE.9.000389

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### Abstract

We have derived the signal-to-noise ratio in direct-detection Random-Modulation Continuous-Wave
(RM-CW) lidar in the presence of colored additive noise. In contrast to a known formula derived for
the photon shot-noise regime, which may adequately describe experimental conditions in the
near-infrared, our result is applicable mainly at longer, mid-infrared wavelengths. Unlike the
former formula, our result is explicitly dependent on the pseudorandom code (PRC) used for
modulation. Three known modulation codes, the M-, A1-, and A2-sequence are compared and shown to
have practically equivalent signal and noise properties (provided that clutter inherent in the A1-
and A2-sequence is neglected), except that the M-sequence has a near-zero-frequency noise pickup
that degrades its performance in real measurement systems. This difference provides an alternative
explanation of a better performance of the A1-/A2-sequence in a previous experiment [

© Optical Society of America

## 1. Introduction

1 . N. Takeuchi , N. Sugimoto , H. Baba , and K. Sakurai , “ Random modulation cw lidar ,”
Appl. Opt. ** 22 ** , 1382 – 1386 ( 1983
). [CrossRef] [PubMed]

5 . Y. Emery and C. Flesia , “ Use of the A1- and the A2-sequences to modulate
continuous-wave pseudorandom noise lidar ,” Appl. Opt. ** 37 ** , 2238 – 2241 ( 1998
). [CrossRef]

6 . C. M. Gittins , E. T. Wetjen , C. Gmachl , F. Capasso , A. L. Hutchinson , D. L. Sivco , J. N. Baillargeon , and A. Y. Cho , “ Quantitative gas sensing by backscatter-absorption
measurements of a pseudorandom code modulated λ ~8-µm quantum cascade laser
,” Optics Lett. ** 25 ** , 1162 – 1164 ( 2000
). [CrossRef]

## 2. Signal-to-noise analysis in direct-detection mid-infrared RM-CW lidar

1 . N. Takeuchi , N. Sugimoto , H. Baba , and K. Sakurai , “ Random modulation cw lidar ,”
Appl. Opt. ** 22 ** , 1382 – 1386 ( 1983
). [CrossRef] [PubMed]

4 . J. L. Machol , “ Comparison of the pseudorandom noise code and pulsed
direct-detection lidars for atmospheric probing ,” Appl. Opt. ** 36 ** , 6021 – 6023 ( 1997
). [CrossRef] [PubMed]

_{i}, z

_{i}, and n

_{i}are discrete-time counterparts of the respective continuous-time quantities, and G

_{i}is the discrete-time counterpart of g(t)·Δt, where

_{0}·a(t) is the light power emitted into the atmosphere, where

_{0}is the laser output power when a=1, and a(t) is a (dimensionless) modulation waveform;

_{r}– receiver’s aperture area;

_{r}– differential backscattering coefficient;

_{r}(R) – transmission coefficient to the distance R:

_{i}is the detector noise.

_{d}. This factor converts the light power into the detector output signal, which – depending on the type of detector and amplifier used – can be voltage or current.

_{aá}(j) is the (normalized) crosscorrelation function defined as

_{j}/Δt is then equal to the atmospheric response function g from the distance R=(c·j·Δt)/2. From G

_{j}, β

_{r}(R) and its derivative parameters of the sensed medium can be determined. We are only concerned with the derivation of the S/N ratio in the measurement of G

_{j}.

### 2.1 The signal

3 . Ch. Nagasawa , M. Abo , H. Yamamoto , and O. Uchino , “ Random modulation cw lidar using new random sequence
,” Appl. Opt. ** 29 ** , 1466 – 1470 ( 1990
). [CrossRef] [PubMed]

*P*′

_{0}for

*P*

_{0}to account for possible losses in light power between the telescope and the detector, and N′ denotes the actual length of a given sequence.

### 2.2 The noise

_{a′}(τ) is the normalized autocorrelation function of the demodulation sequence, and η(f) is the positive-frequency noise power spectral density.

_{a′}(τ) is periodic, its power spectral density can be represented as a Fourier series:

_{0}=1/T

_{0}=1/(N·Δt), and c

_{n}are the Fourier coefficients of R

_{a′}(τ):

^{2}can be found from the impulse response function

_{0}=(T/T

_{0})=k»1, that is, if the measurement/averaging is carried out over a large number of periods. The distribution sinc

^{2}fT (of width ~1/T) can then be approximated by the δ(f) distribution:

### 2.3 The signal-to-noise ratio

_{j}is

_{d}and η in terms of commonly used infrared detector specifications, we note that

_{d}is detector’s area. Equation (21) then becomes

_{n}is significant (that is, from DC to ~1/Δt), we can simplify the above expression noting that

### 2.4 Comparison to the photon shot-noise regime

*et al*. [1

1 . N. Takeuchi , N. Sugimoto , H. Baba , and K. Sakurai , “ Random modulation cw lidar ,”
Appl. Opt. ** 22 ** , 1382 – 1386 ( 1983
). [CrossRef] [PubMed]

2 . N. Takeuchi , H. Baba , K. Sakurai , and T. Ueno , “ Diode-laser random-modulation cw lidar
,” Appl. Opt. ** 25 ** , 63 – 67 ( 1986
). [CrossRef] [PubMed]

*b̄*is the background radiation power;

_{Q}/hν is the conversion constant from light power to photoelectron number,

_{Q}is the detector’s quantum efficiency,

*lP*

_{0}

*Ḡ*

*b̄*, that is, background photon shot-noise-limited detection:

### 2.5 Performance comparison of the M-, A1-, and A2-sequence in the presence of colored noise

#### 2.5.1 The M-sequence

_{c}, called the chip length, equals Δt, and the fundamental frequency (n=1) is f

_{0}=1/(NT

_{c}).

#### 2.5.2 The A1- and A2-sequence

3 . Ch. Nagasawa , M. Abo , H. Yamamoto , and O. Uchino , “ Random modulation cw lidar using new random sequence
,” Appl. Opt. ** 29 ** , 1466 – 1470 ( 1990
). [CrossRef] [PubMed]

_{0}=1/(2NT

_{c}).

#### 2.5.3 Sequence parameters and performance comparison

_{j}).

5 . Y. Emery and C. Flesia , “ Use of the A1- and the A2-sequences to modulate
continuous-wave pseudorandom noise lidar ,” Appl. Opt. ** 37 ** , 2238 – 2241 ( 1998
). [CrossRef]

_{0}). Furthermore, this average is properly normalized, since – for all demodulation sequences – their normalized autocorrelation is equal to one at τ=0 (Eq. (24)). Therefore, the noise performance of the various demodulation sequences can be qualitatively compared by plotting the Fourier coefficients c

_{n}for each sequence (see Fig.2).

*et al*.[3

3 . Ch. Nagasawa , M. Abo , H. Yamamoto , and O. Uchino , “ Random modulation cw lidar using new random sequence
,” Appl. Opt. ** 29 ** , 1466 – 1470 ( 1990
). [CrossRef] [PubMed]

^{2}fT (T is the averaging time) rather than (1/T)δ(f). As a result, the line spectra described by the Fourier coefficients cn are in general windowed, frequency-broadened to ~1/T. Therefore, in practical systems, our concern is the pickup of near-zero-frequency noise (down to ~1/T) rather than “DC” noise. [In fact, our framework of stochastic noise analysis is not valid for frequencies lower or comparable to 1/T, although the description of DC noise pickup is qualitatively correct.] A semi-quantitative analysis shows that the S/N ratio is an order of magnitude greater in the A1- or A2-sequences compared to the M-sequence in typical experimental conditions (sequence length N=1000; chip length T

_{c}=30ns; integration time T=3s; and 1/f noise spectral density). The ~5-times greater S/N ratio in the A2-sequence compared to the M-sequence that was observed in an experiment carried out by Nagasawa

*et al*. [3

** 29 ** , 1466 – 1470 ( 1990
). [CrossRef] [PubMed]

#### 2.5.4 Effect of imbalance on overall performance

_{0}=(1/N)

^{2}, in agreement with

#### 2.6 Random modulation on a sinusoidal carrier

_{c}, covering the region of highest noise density in practical systems. To avoid this spectral coincidence, we could employ additional modulation with a sinusoidal carrier, which would shift the modulation spectrum and noise pickup to higher frequencies where the noise density is usually much lower. This would, however, degrade the signal properties of the entire system, as we show below.

_{m}»1/T

_{c}is the carrier frequency, such that the maximum instantaneous laser output power remains the same. The demodulation waveform is

_{m}.

_{m}τ+

_{m}-

_{m}/f

_{0}in the denominator (demodulated noise).

## 3. Summary and conclusions

## References and links

1 . | N. Takeuchi , N. Sugimoto , H. Baba , and K. Sakurai , “ Random modulation cw lidar ,”
Appl. Opt. |

2 . | N. Takeuchi , H. Baba , K. Sakurai , and T. Ueno , “ Diode-laser random-modulation cw lidar
,” Appl. Opt. |

3 . | Ch. Nagasawa , M. Abo , H. Yamamoto , and O. Uchino , “ Random modulation cw lidar using new random sequence
,” Appl. Opt. |

4 . | J. L. Machol , “ Comparison of the pseudorandom noise code and pulsed
direct-detection lidars for atmospheric probing ,” Appl. Opt. |

5 . | Y. Emery and C. Flesia , “ Use of the A1- and the A2-sequences to modulate
continuous-wave pseudorandom noise lidar ,” Appl. Opt. |

6 . | C. M. Gittins , E. T. Wetjen , C. Gmachl , F. Capasso , A. L. Hutchinson , D. L. Sivco , J. N. Baillargeon , and A. Y. Cho , “ Quantitative gas sensing by backscatter-absorption
measurements of a pseudorandom code modulated λ ~8-µm quantum cascade laser
,” Optics Lett. |

7 . | A. B. Carlson , |

8 . | S. Haykin , |

**OCIS Codes**

(010.3640) Atmospheric and oceanic optics : Lidar

(120.0280) Instrumentation, measurement, and metrology : Remote sensing and sensors

(280.0280) Remote sensing and sensors : Remote sensing and sensors

(290.0290) Scattering : Scattering

(300.0300) Spectroscopy : Spectroscopy

**ToC Category:**

Research Papers

**History**

Original Manuscript: May 30, 2001

Published: October 8, 2001

**Citation**

Adam Rybaltowski and Allen Taflove, "Signal-to-noise ratio in direct-detection mid-infrared Random-Modulation Continuous-Wave lidar in the presence of colored additive noise," Opt. Express **9**, 389-399 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-9-8-389

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### References

- N. Takeuchi, N. Sugimoto, H. Baba, and K. Sakurai, "Random modulation cw lidar," Appl. Opt. 22, 1382-1386 (1983). [CrossRef] [PubMed]
- N. Takeuchi, H. Baba, K. Sakurai, and T. Ueno, "Diode-laser random-modulation cw lidar," Appl. Opt. 25, 63-67 (1986). [CrossRef] [PubMed]
- Ch. Nagasawa, M. Abo, H. Yamamoto, and O. Uchino, "Random modulation cw lidar using new random sequence," Appl. Opt. 29, 1466-1470 (1990). [CrossRef] [PubMed]
- J. L. Machol, "Comparison of the pseudorandom noise code and pulsed direct-detection lidars for atmospheric probing," Appl. Opt. 36, 6021-6023 (1997). [CrossRef] [PubMed]
- Y. Emery and C. Flesia, "Use of the A1- and the A2-sequences to modulate continuous-wave pseudorandom noise lidar," Appl. Opt. 37, 2238-2241 (1998). [CrossRef]
- C. M. Gittins, E. T. Wetjen, C. Gmachl, F. Capasso, A. L. Hutchinson, D. L. Sivco, J. N. Baillargeon, and A. Y. Cho, "Quantitative gas sensing by backscatter-absorption measurements of a pseudorandom code modulated ~8-�m quantum cascade laser," Opt. Lett. 25, 1162-1164 (2000). [CrossRef]
- A. B. Carlson, Communication systems. An introduction to signals and noise in electrical engineering (McGraw-Hill, 1986).
- S. Haykin, Digital communications (John Wiley & Sons, 1988).

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