## Correlation analysis of spectral fluctuations in inhomogeneously broadened spectra

Optics Express, Vol. 8, Issue 9, pp. 509-516 (2001)

http://dx.doi.org/10.1364/OE.8.000509

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### Abstract

It is shown that the lineshapes of inhomogeneously broadened spectra, due to the statistical nature of their formation, exhibit spectral fluctuations. Formulas are obtained that allow one, based on correlation analysis of different realizations of the inhomogeneously broadened line, to reconstruct its homogeneous lineshape and to evaluate the number of centers involved in its formation. The magnitude of these spectral fluctuations is estimated and it is shown that the proposed method can be efficiently used in practice.

© Optical Society of America

## 1 Introduction

2. A. Zrenner, L. V. Butov, M. Hagn, and G. Abstreiter,“Quantum dots formed by interface fluctuations in AlAs/GaAs coupled quantum well structures,”G. Böhm and G. Weimann. Phys. Rev. Lett. **72**, 3382 (1994). [CrossRef] [PubMed]

3. K. Brunner, G. Abstreiter, G. Böhm, G. Tränkle, and G. Weimann, “Sharp-line photoluminescence and two photon absorption of zero-dimensional biexcitons in a GaAs/AlGaAs structure,” Phys. Rev. Lett. **73**, 1138 (1994). [CrossRef] [PubMed]

4. H. F. Hess, E. Betzig, T. D. Harris, L. N. Pfeiffer, and K. W. West, Science264, 1740 (1994). [CrossRef] [PubMed]

5. D. Gammon, E. S. Snow, B. V. Shanobrook, D. S. Katzer, and D. Park, “Fine Structure Splitting in the Optical Spectra of Single GaAs Quantum Dots,” Phys. Rev. Lett. **76**, 3005 (1996). [CrossRef] [PubMed]

6. Q. Wu, R. D. Grober, D. Gammon, and D. S. Katzer, “Imaging Spectroscopy of Two-Dimensional Excitons in a Narrow GaAs/AlGaAs Quantum Well,” Phys. Rev. Lett. **83**, 2652 (1999). [CrossRef]

## 2 Mathematical procedure

*N*emitters, with the

*i*-th emitter being characterized by a homogeneously broadened line

*a*(

*ω*-

*ω*), where

_{i}*ω*(

_{i}*i*=1, …,

*N*) are mutually independent random quantities with the distribution function

*ρ*(

*ω*) (∫

*ρ*(

*ω*)

*dω*=1). We assume, for simplicity, that ∫

*a*(

*ω*)

*dω*=1. Then, each realization of the inhomogeneously broadened line will be described by the random function

*A*(

*ω*):

*ρ*(

*ω*) is nonzero is much larger than that for the function

*a*(

*ω*). Then, for the inhomogeneously broadened line averaged over realizations <

*A*(

*ω*)>, we have:

*N*≫1 and expressing

*ρ*(

*ω*) through the averaged profile of the inhomogeneously broadened line, we obtain:

*A*(

*ω*) (left-hand side of the equation). Suppose we have a set of

*N*realizations of the inhomogeneously broadened line, i.e.,

_{R}*N*functions

_{R}*A*(

_{r}*ω*),

*r*=1, …,

*N*. Then, the expression for auto-convolution of the homogeneously broadened line can be written in the form:

_{R}*ω*-

*ω′*and represents a function with a narrow peak (right-hand side), whose width is twice as large as that of the homogeneous line.

*N*emitters was obtained using formula (1) by generating

*N*random numbers

*ω*(

_{i}*i*=1, …,

*N*), with the Gaussian distribution function

*N*times, we obtained

_{R}*N*independent realizations of the inhomogeneously broadened line

_{R}*A*(

_{r}*ω*),

*r*=1, …,

*N*. Then we calculated the correlation function of the inhomogeneously broadened line and checked formula (4). Note that the frequency increment used for numerical generation of the arrays

_{R}*A*(

_{r}*ω*) had to be essentially smaller than the homogeneous linewidth

*δ*. The results thus obtained for different numbers of realizations

*N*, with the number of emitters

_{R}*N*=10

^{4}and the inhomogeneous/homogeneous linewidth ratio

*α*=20, are shown in Fig.1 The dashed line shows auto-convolution of the homogeneously broadened line. As seen from Fig.1, the results of computer simulation agree well with predictions of our analysis.

## 3 Magnitude of the effect

*δ*is much smaller than the inhomogeneous linewidth, formula (5) can be approximately represented in the form

*d*will evidently drop with increasing

*N*and, for sufficiently large

*N*, may ‘sink’ below the detected noise. It should be emphasized that the spectral noise of the inhomogeneously broadened line under study can be considered here, in the framework of this approach, as a statistical

*signal*, which

*should be the same*for identical realizations of the ensemble of emitters. The possibility of measuring the spectral fluctuations is determined by the ratio of their amplitude (

*amplitude of signal*) to amplitude of real irreproducible noise of measurements. The latter can be associated not only with usual noise of detection (shot noise of the photocurrent, noise of electronics, etc.), but also with various kinds of spurious spectral modulation (caused, e.g., by interferometric effects on parasitic interferometers in the optical channel).

*n*of the spectral data array (expressed in the number of photoelectrons) needed to be able to neglect the noise of detection will be given by

*d*is the variance of spectral fluctuations of the line profile, given by Eq.(6). In other words, the method under consideration can be used when the experimental numbers of the spectral data array

*n*essentially exceed (say by one or two orders of magnitude) the number of emitters falling into the interval equal to the homogeneous linewidth. In this case, the signal/noise ratio in the auto-convolution of the homogeneous line, obtained experimentally, will be determined, as before, by Eq. (7). Using this criterion, one can easily estimate that with the up-to-date methods of detection and accumulation of optical signals (with CCD detector arrays and automatic scanning of the optical excitation spot over the sample), this requirement can be fulfilled in a wide number of cases, which indicates feasibility of the proposed experimental approach. Note that to meet the requirement (8) one can increase the signal accumulation time and decrease the excitation spot size. As was already pointed out, spectral resolution for measurements of this kind should essentially exceed the measured homogeneous linewidth or, which is the same, the characteristic correlation interval of the spectral fluctuations.

## 4 Conclusion

## References and links

1. | J. Shah, |

2. | A. Zrenner, L. V. Butov, M. Hagn, and G. Abstreiter,“Quantum dots formed by interface fluctuations in AlAs/GaAs coupled quantum well structures,”G. Böhm and G. Weimann. Phys. Rev. Lett. |

3. | K. Brunner, G. Abstreiter, G. Böhm, G. Tränkle, and G. Weimann, “Sharp-line photoluminescence and two photon absorption of zero-dimensional biexcitons in a GaAs/AlGaAs structure,” Phys. Rev. Lett. |

4. | H. F. Hess, E. Betzig, T. D. Harris, L. N. Pfeiffer, and K. W. West, Science264, 1740 (1994). [CrossRef] [PubMed] |

5. | D. Gammon, E. S. Snow, B. V. Shanobrook, D. S. Katzer, and D. Park, “Fine Structure Splitting in the Optical Spectra of Single GaAs Quantum Dots,” Phys. Rev. Lett. |

6. | Q. Wu, R. D. Grober, D. Gammon, and D. S. Katzer, “Imaging Spectroscopy of Two-Dimensional Excitons in a Narrow GaAs/AlGaAs Quantum Well,” Phys. Rev. Lett. |

7. | A. van der Ziel, |

**OCIS Codes**

(240.6490) Optics at surfaces : Spectroscopy, surface

(300.6280) Spectroscopy : Spectroscopy, fluorescence and luminescence

**ToC Category:**

Research Papers

**History**

Original Manuscript: March 16, 2001

Published: April 23, 2001

**Citation**

Valerii Zapasskii and G. Kozlov, "Correlation analysis of spectral fluctuations in inhomogeneously broadened spectra," Opt. Express **8**, 509-516 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-8-9-509

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### References

- J. Shah, Ultrafast Spectroscopy of Semiconductors and Semiconductor Nanostructures, Springer series in Solid-State Sciences, (Springer-Verlag, Heidelberg, 1996).
- A. Zrenner, L. V. Butov, M. Hagn, G. Abstreiter,"Quantum dots formed by interface fluctuations in AlAs/GaAs coupled quantum well structures," G.B�hm, and G. Weimann. Phys. Rev. Lett. 72, 3382 (1994). [CrossRef] [PubMed]
- K. Brunner, G. Abstreiter, G. B�hm, G. Tr�nkle, and G. Weimann, "Sharp-line photoluminescence and two photon absorption of zero-dimensional biexcitons in a GaAs/AlGaAs structure," Phys. Rev. Lett. 73, 1138 (1994). [CrossRef] [PubMed]
- H. F. Hess, E. Betzig, T. D. Harris, L. N. Pfeiffer, and K. W. West, Science 264, 1740 (1994). [CrossRef] [PubMed]
- D. Gammon, E. S. Snow, B. V. Shanobrook, D. S. Katzer, and D. Park, "Fine Structure Splitting in the Optical Spectra of Single GaAs Quantum Dots," Phys. Rev. Lett. 76, 3005 (1996). [CrossRef] [PubMed]
- Q. Wu, R. D. Grober, D. Gammon, and D. S. Katzer, "Imaging Spectroscopy of Two-Dimensional Excitons in a Narrow GaAs/AlGaAs Quantum Well," Phys. Rev. Lett. 83, 2652 (1999). [CrossRef]
- A. van der Ziel, Noise in Measurements (John Wiley & Sons, New York, 1976).

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