## Sensitivity optimization of the one beam Z-scan technique and a Z-scan technique immune to nonlinear absorption |

Optics Express, Vol. 21, Issue 13, pp. 15350-15363 (2013)

http://dx.doi.org/10.1364/OE.21.015350

Acrobat PDF (1178 KB)

### Abstract

It is presented a criteria for selecting the optimum aperture radius for the one beam Z-scan technique (OBZT), based on the analysis of the transmittance of the aperture. It is also presented a modification to the OBZT by directly measuring the beam radius in the far field with a rotating disk, which allows to determine simultaneously the non-linear absorptive coefficient and non-linear refractive index, much less sensitive to wave front distortions caused by inhomogeneities of the sample with a negligible loss of signal to noise ratio. It is demonstrated its equivalence to the OBZT.

© 2013 OSA

## 1. Introduction

1. M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quant. Electron. **26**(4), 760–769 (1990) [CrossRef] .

*z*axis.

*z*axis) to measure on axis transmitted power (

*P*), second,

*P*is only a small fraction of the total laser power (

*P*

_{0}), compromising the signal to noise ratio (SNR), and perhaps the most important of all, when the sample exhibits nonlinear absorption, there is no way to know whether changes in

*P*due to changes in the refractive index or absorption when the sample is displaced, forcing to repeat the experiment without the aperture for measuring the nonlinear absorption and deduct it, as will be seen later.

2. J. Wang, B. Gu, Y. M. Xu, and H. T. Wang, “Enhanced sensitivity of Z-scan technique by use of flat-topped beam,” Appl. Phys. B **95**(4), 773778 (2009) [CrossRef] .

3. T. Xia, D. J. Hagan, M. Sheik Bahae, and E. Van Stryland, “Eclipsing Z-scan measurement of *λ*/10^{4} wave-front distortion,” Opt. Lett. **19**(5), 317–319 (1994) [CrossRef] [PubMed] .

4. G. Tsigaridas, M. Fakis, I. Polyzos, P. Persephonis, and V. Giannetas, “Z-scan technique through beam radius measurements,” Appl. Phys. B **76**(1), 83–86 (2003) [CrossRef] .

5. P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. Mckay, and R. G. Mcduff, “Single-beam Z-scan: Measurement techniques and analysis,” J. Nonlinear Opt. Phys. Mater **6**(3), 251–293 (1997) [CrossRef] .

6. I. A. Ryasnyansky and B. Palpant, “Theoretical Investigation of the off-axis z-scan technique for nonlinear optical refraction measurement,” Appl. Opt. **45**(12), 2773–2776 (2006) [CrossRef] [PubMed] .

## 2. Description of the chopper-width technique

7. R. L. McCally, “Measurement of Gaussian beam parameters,” Appl. Opt. **23**(14), 2227–2227 (1984) [CrossRef] [PubMed] .

8. P. B. Chapple, “Beam waist and M2 measurement using a finite slit,” Opt. Eng. **33**(7), 2461–2466 (1994) [CrossRef] .

9. P. J. Shayler, “Laser beam distribution in the focal region,” Appl. Opt. **17**(17), 2673–2674 (1978) [CrossRef] [PubMed] .

10. S. Nemoto, “Determination of waist parameters of a Gaussian beam,” Appl. Opt. **21**(21), 3859–3863 (1986) [CrossRef] .

11. J. M. Khosrofian and B. A. Garetz, “Measurement of a Gaussian laser beam diameter through the direct inversion of knife-edge data,” Appl. Opt. **22**(21), 3406–3410 (1983) [CrossRef] [PubMed] .

*ζ*) eclipses the beam transversely to its propagation axis (

*z*axis), the beam width (

*W*(

*ζ*)) is determined from the derivative plot of the transmitted power versus the knife-edge position. This method is simple but a bit tedious, slow and noise sensitive. These issues can be overcome with the technique described below.

*W*(

*ζ*), from the rise or fall time, of a electric signal [12

12. J. A. Arnaud, W. M. Hubbard, G. D. Mandeville, B. delaClaviére, E. A. Franke, and J. M. Franke, “Technique for fast measurement of Gaussian laser beam parameters,” Appl. Opt. **10**(12), 2775–2776 (1971) [CrossRef] [PubMed] .

*ζ*) [13

13. Y. Suzaki and A. Tachibana, “Measurement of the *μm* sized radius of Gaussian laser beam using the scanning knife-edge,” Appl. Opt. **14**(12), 2809–2810 (1975) [CrossRef] [PubMed] .

*W*(

*ζ*) measured in this technique is larger than the obtained with the knife-edge technique, due the curved trajectory to cut the beam, rather than perpendicular. This drawback can be overcome calibrating the chopper-width technique, this is presented in the appendix at the end of this work. Let us consider a Gaussian beam propagating along the

*z*axis impinging on a photoreceptor, the chopper is located at the

*ζ*position where the

*W*(

*ζ*) is to be measured, like the chopper has

*l*slots and rotates with constant angular frequency Ω

_{0}, as shown in Fig. 1. To each blade will take

*θ*subtended by the beam. If

*R*≫

*W*(

*ζ*),

*τ*is given by where

*R*is the distance from the axis of rotation to the beam axis, therefore, Eq. (1) provides a way to measure

*W*(

*ζ*) from the time

*τ*.

4. G. Tsigaridas, M. Fakis, I. Polyzos, P. Persephonis, and V. Giannetas, “Z-scan technique through beam radius measurements,” Appl. Phys. B **76**(1), 83–86 (2003) [CrossRef] .

14. A. Nag, A. Kumar De, and D. Goswami, “Two-photon cross-section measurements using an optical chopper: z-scan and two-photon fluorescence schemes,” J. Phys. B: At. Mol. Opt. Phys. **42**(6), 065103, (2009) [CrossRef] .

15. I. Bhattacharyya, S. Priyadarshi, and D. Goswami, “Molecular structure-property correlations from optical nonlinearity and thermal-relaxation dynamics,” Chem. Phys. Lett. **469**, 104–109, (2009) [CrossRef] [PubMed] .

*P*, however, here the chopper is used as meter width. Before experimentally demonstrate the feasibility and advantages of this approach, let us present its justification and the comparison of the sensitivities of chopper and OBZT.

## 3. Equivalence and sensitivities

*T*(

*z*)) of the aperture located at

*ζ*fixed, is defined as [1

1. M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quant. Electron. **26**(4), 760–769 (1990) [CrossRef] .

*z*is the position of the nonlinear sample referred to the minimum waist position (

*z*= 0) and the power transmitted by the aperture is given by

*ρ*

_{0}is the aperture radius,

*W*(

*z*) and

*W*(∞) =

*W*are the beam radius at the position of the aperture when the sample is located at

_{L}*z*and far away (∞) from the focus (linear regime) respectively,

*P*

_{0}is the optical power of the beam incident to the aperture. Henceforth, we shall assume the implicit dependence of

*W*on

*z*in order to simplify the notation.

*T*(∞) = 1, then, the effect of the presence of sample is measured from this value. Also, it is easy to realize that

### 3.1. The signal of the OBZT

*W*to

_{L}*W*, therefore, the beam width change (Δ

*W*) is this causes that the transmitted power (

*P*), changes by Δ

*P*, let us calculate the optimum value of

*ρ*

_{0}to detect (

*W*(using the criteria of

_{L}*P*. To reinforce this assertion, in Fig. 2, it is shown an experimental plot of Δ

*P*as function of the ratio

*ρ*

_{0}≅ 0.71

*W*, corroborating Eq. (9), the experimental details are given in section 4.

_{L}*W*≪

*W*), Eq. (10) can be approximated to Eq. (11) is important because allows to relate beam width changes (Δ

_{L}*W*) with changes in the transmitted power (Eq. (4)). Substituting Eq. (6), Eq. (9) and Eq. (11) into Eq. (4) gives Eq. (12) provides the equivalence of the chopper technique. The minus sign in Eq. (12), means that the chopper Z-scan curve is inverted with respect to the OBZT Z-scan, because when there is a positive increment of beam width (Δ

*W*> 0) the beam intensity and therefore the aperture transmittance diminishes (Δ

*T*< 0). In the following we shall work with

### 3.2. The signal for the chopper Z-scan

*τ*for eclipsing the beam width (

*W*) is given by Eq. (1), when

*W*changes due to the presence of the sample,

*τ*also changes by Δ

*τ*, given by: therefore, the minimum Δ

*τ*that noise allows to discriminate, determines, the minimum Δ

*W*that can be resolved with this technique.

### 3.3. Calculus of noise power for both techniques

*V*(which only depends on the sample position

_{ph}*z*) is:

*R*is the amplifier zero frequency gain in Ohms, ℜ is the responsivity of the photodiode in

_{L}*P*is the optical power incident to the photodiode in

*Watt*.

*P*, the photoreceptor is limited by thermal noise [16], therefore, the signal to noise ratio (

*k*is the Boltzmann constant,

*T*is the temperature in Kelvin and Δ

*f*is the detection bandwidth of the photoreceptor in

*Hz*given by [17] the expression in parentheses in Eq. (17) is the photo-current generated by the photodiode.

*P*

_{min}) detectable is to obtain an expression for the magnitude of the minimum change of width detectable (

*P*), of course, if other additive noise sources were present,

_{n}*P*would represent the total noise power, hence, the sensitivity of the OBZT is given by Eq. (21).

_{n}*P*

_{0}) impinges on the photodiode, therefore, a lower gain value of

*R*is needed, lowering the noise power due to this cause, however, the bandwidth requirement (Δ

_{L}*f*) is greater than the OBZT, somewhat compensating the previous advantage, as shown in the following subsection.

#### 3.3.1. Bandwidth need of the chopper z-scan

*P*(

_{ch}*t*) is a periodic function of the time

*t*, due to periodic eclipsing;

*P*(

_{ch}*t*) is given by: let us define the following ratio for Ω

_{0}

*t*≪ 1 the raising part of Eq. (22) can be approximated to the Fourier series of

*P*(

_{ch}*t*) using the approximation (Eq. (24)) is if the series is cut up to

*m*≥ 2

*β*(

*m*integer) harmonic, the error is less than 0.6%, therefore the chopper Z-scan bandwidth need (Δ

*f*) in

_{c}*Hz*is: where it has been used the fact that Ω

_{0}= 2

*πf*

_{0}and Ω

_{0}

*t*≪ 1, hence, the photoreceptor must satisfy the following requirement where Δ

*f*is given in Eq. (18).

*τ*was calculated in Eq. (1), therefore, the photoreceptor must at least be as fast as the eclipsing time, consequently, its bandwidth Δ

*f*must be the following is also meet Ω

_{c}_{0}= 2

*πf*

_{0}where

*f*

_{0}is the rotation frequency in

*Hz*, thus, Eq. (28) can be expressed as

#### 3.3.2. Minimum change of width detectable with the chopper Z-scan technique

*τ*is the time elapsed for

*V*(

_{ph}*t*) to vary from

*V*

_{th1}to

*V*

_{th2}as shown in Fig. 4, therefore,

*V*(

_{ph}*t*) is compared with the threshold voltages

*V*

_{th1}and

*V*

_{th2}, hence,

*V*(

_{ph}*t*) fluctuations due to noise (

*n*(

*t*)) result in uncertainty in the comparison process (

*jitter*), in this case the following relation is satisfied between the variance in the metric

*j*of the time of the threshold crossing [18] where

*t*is the expected time of the threshold crossing and

_{c}*V*(

_{ph}*t*) is the voltage delivered by the photoreceptor system. In order to compare both techniques let us assume that the same amplifier than the OBZT is used and Eq. (27) is satisfied, then,

*P*(

_{ch}*t*) is given by Eq. (22). For noise with zero mean,

*var*(

*n*(

*t*)) =

*P*[19

_{n}19. B. E. A. Saleh and M. C. Teich, *Fundamentals of Photonics* (John Wiley & Sons Inc, 1991) [CrossRef] .

*P*is the noise power of

_{n}*n*(

*t*), therefore, equation (30) can be expressed as from Eq. (1) and Eq. (32) we can establish the following relation Note that

*W*is minimized if

*V*

_{th1}and

*V*

_{th2}are set as close as possible to the maximum value of the profile, hence, if they are established where the profile drops to

*V*

_{th1}≅ 0.25

*V*(

_{ph}*t*) and

*V*

_{th2}≅ 0.75

*V*(

_{ph}*t*)), then from Eq. (1) using Eq. (31), Eq. (24) and Eq. (34) the derivative of

*V*is substituting Eq. (35) into Eq. (33) and taking into account that two comparisons are done (at

_{ph}*V*

_{th1}and

*V*

_{th2}), the minimum relative change of the beam width (

*τ*versus

*z*, and to use Eq. (37) derived from Eq. (1) and Eq. (13) to estimate the phase change where Δ

*τ*

_{p−v}is the peak-valley difference of the Z-scan curve, and

*τ*is the rising time of

_{L}*V*(

_{ph}*t*) in the linear regime.

## 4. Experimental results

*P*) passing through the aperture is function of its position (

*ζ*) along beam axis and its radius (

*ρ*

_{0}), i.e.

*P*(

*W*(

*ζ*),

*ρ*

_{0}), thus, the aperture was located at a distance (

*ζ*) where the beam width was (

*W*(

*ζ*) = 4.5

*mm*); then

*P*(

*W*(

*ζ*),

*ρ*

_{0}) was measured for ten increasing radius

*ρ*

_{0}, after, the aperture was displaced at (

*ζ*

_{1}) where the width was (

*W*(

*ζ*

_{1}) = 4

*mm*),

*P*(

*W*(

*ζ*

_{1}),

*ρ*

_{0}) was measured for the same values of

*ρ*

_{0}as before. In Fig. 2 it is shown a plot of Δ

*P*=

*P*(

*W*(

*ζ*

_{1}),

*ρ*

_{0})−

*P*(

*W*(

*ζ*),

*ρ*

_{0}) versus

*ρ*

_{0}for detecting changes in the transmitted power is

*μm*thick, the JDS 1145P laser, a 5 cm lens, a Thorlabs PIN photodiode SM1PD1A, a home-made trans-resistance amplifier with

*R*= 82

_{L}*k*Ω,

*C*= 10

_{L}*nF*and operational amplifier OP27, an oscilloscope model TDS1012C-EDU for monitoring

*V*, disposed according to the setup shown in Fig. 3(A), the distance from the output laser to the lens was 50cm, the distance of the lens to the aperture was 50cm; to perform the chopper technique the same elements were used arranged according to Fig. 3(B), replacing the aperture with a disk with 10 slots rotating at Ω

_{ph}_{0}= 18.85

### 4.1. Discussion of results

*μW*.

#### 4.1.1. OBZT

20. X. Liu, S. Guo, H. Wang, N. Ming, and L. Hou, “Investigation of the influence of finite aperture size on the Z-scan transmittance curve,” J. Nonlinear Opt. Phys. Mater. **10**(4), 431–439 (2001) [CrossRef] .

*V*=

*V*−

_{ph}*V*,

_{L}*V*is the voltage

_{L}*V*measured when the sample is far from the focus (linear regime). Again, the optimum radius turned out to be determined by the Eq. (9).

_{ph}#### 4.1.2. Chopper Z-scan

*V*(

_{ph}*t*) recorded with the oscilloscope, for illustrative purposes different curves were displaced to the left as the position of the sample was increased, from such figure it is clear how the chopper technique works: from the amplitude of

*V*(

_{ph}*t*) the nonlinear absorption is estimated and from the raising time (

*τ*) the refractive index is estimated because

*τ*depends only on the beam width (see Fig. 4).

*τ*vs

*z*(the position of the sample), also in the inset is shown the amplitude of the pulses (such curve represents the transmitance).

5. P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. Mckay, and R. G. Mcduff, “Single-beam Z-scan: Measurement techniques and analysis,” J. Nonlinear Opt. Phys. Mater **6**(3), 251–293 (1997) [CrossRef] .

*μW*; from Fig. 9 it is evident that the nonlinear absorption distorts the OBZT Z-scan curve [20

20. X. Liu, S. Guo, H. Wang, N. Ming, and L. Hou, “Investigation of the influence of finite aperture size on the Z-scan transmittance curve,” J. Nonlinear Opt. Phys. Mater. **10**(4), 431–439 (2001) [CrossRef] .

*P*(

*W*(∞),

*ρ*

_{0}) =

*P*, the Eq. (38) gives the following result: one might think that the OBZT is more sensitive than the chopper technique, however, it should not be confused for the result of Eq. (39), the optical power is fully utilized in the chopper technique, achieving a better signal to noise ratio (SNR) than the SNR of the OBZT, for example, for the opening radius

_{L}*ρ*

_{0}= 0.1

*W*, the optical power detected with the OBZT (Eq. (3)) technique is 50 times less than the maximum power (

_{L}*P*

_{0}) detected with the chopper technique, this is the reason of the use of a device that improves the SNR (lock-in). The OBZT increases its SNR as

*ρ*

_{0}is increased up to

## 5. Conclusion

*W*is the beam width (using the criteria of

_{L}## 6. Appendix

*W*) is larger than that of the knife (

_{ch}*W*), this difference is negligible if condition (

_{kn}*R*≫

*W*) is met, if not, the difference can be deducted if an expression for it is found, by measuring

*W*(

*z*) of a Gaussian beam, propagating along z-axis with both techniques at different positions (

*ζ*) under the same experimental conditions. In Fig. 11,

*W*is plotted versus

_{kn}*W*, the data fitting Eq. (41) allows to find the relation between both techniques. In Fig. 12 it is show the width measured of a the laser used in this paper, where it can be seen an excellent agreement between knife and calibrated chopper techniques.

_{ch}## Acknowledgments

## References and links

1. | M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quant. Electron. |

2. | J. Wang, B. Gu, Y. M. Xu, and H. T. Wang, “Enhanced sensitivity of Z-scan technique by use of flat-topped beam,” Appl. Phys. B |

3. | T. Xia, D. J. Hagan, M. Sheik Bahae, and E. Van Stryland, “Eclipsing Z-scan measurement of |

4. | G. Tsigaridas, M. Fakis, I. Polyzos, P. Persephonis, and V. Giannetas, “Z-scan technique through beam radius measurements,” Appl. Phys. B |

5. | P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. Mckay, and R. G. Mcduff, “Single-beam Z-scan: Measurement techniques and analysis,” J. Nonlinear Opt. Phys. Mater |

6. | I. A. Ryasnyansky and B. Palpant, “Theoretical Investigation of the off-axis z-scan technique for nonlinear optical refraction measurement,” Appl. Opt. |

7. | R. L. McCally, “Measurement of Gaussian beam parameters,” Appl. Opt. |

8. | P. B. Chapple, “Beam waist and M2 measurement using a finite slit,” Opt. Eng. |

9. | P. J. Shayler, “Laser beam distribution in the focal region,” Appl. Opt. |

10. | S. Nemoto, “Determination of waist parameters of a Gaussian beam,” Appl. Opt. |

11. | J. M. Khosrofian and B. A. Garetz, “Measurement of a Gaussian laser beam diameter through the direct inversion of knife-edge data,” Appl. Opt. |

12. | J. A. Arnaud, W. M. Hubbard, G. D. Mandeville, B. delaClaviére, E. A. Franke, and J. M. Franke, “Technique for fast measurement of Gaussian laser beam parameters,” Appl. Opt. |

13. | Y. Suzaki and A. Tachibana, “Measurement of the |

14. | A. Nag, A. Kumar De, and D. Goswami, “Two-photon cross-section measurements using an optical chopper: z-scan and two-photon fluorescence schemes,” J. Phys. B: At. Mol. Opt. Phys. |

15. | I. Bhattacharyya, S. Priyadarshi, and D. Goswami, “Molecular structure-property correlations from optical nonlinearity and thermal-relaxation dynamics,” Chem. Phys. Lett. |

16. | A. Rogalski and Z. Bielecki, “Detection of optical radiation,” Bull. Pol. Ac.: Tech. |

17. | C. D. Motchenbacher and J. A. Connelly, |

18. | J. Phillips and K. Kundert, “Noise in mixers, oscillators, samplers, and logic an introduction to cyclostationary noise,” Proceedings of the IEEE custom integrated circuits conference , 431–439, (2000). |

19. | B. E. A. Saleh and M. C. Teich, |

20. | X. Liu, S. Guo, H. Wang, N. Ming, and L. Hou, “Investigation of the influence of finite aperture size on the Z-scan transmittance curve,” J. Nonlinear Opt. Phys. Mater. |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(190.4720) Nonlinear optics : Optical nonlinearities of condensed matter

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: January 29, 2013

Revised Manuscript: March 4, 2013

Manuscript Accepted: March 11, 2013

Published: June 20, 2013

**Citation**

José A. Dávila Pintle, Edmundo Reynoso Lara, and Marcelo D. Iturbe Castillo, "Sensitivity optimization of the one beam Z-scan technique and a Z-scan technique immune to nonlinear absorption," Opt. Express **21**, 15350-15363 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-13-15350

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### References

- M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quant. Electron.26(4), 760–769 (1990). [CrossRef]
- J. Wang, B. Gu, Y. M. Xu, and H. T. Wang, “Enhanced sensitivity of Z-scan technique by use of flat-topped beam,” Appl. Phys. B95(4), 773778 (2009). [CrossRef]
- T. Xia, D. J. Hagan, M. Sheik Bahae, and E. Van Stryland, “Eclipsing Z-scan measurement of λ/104 wave-front distortion,” Opt. Lett.19(5), 317–319 (1994). [CrossRef] [PubMed]
- G. Tsigaridas, M. Fakis, I. Polyzos, P. Persephonis, and V. Giannetas, “Z-scan technique through beam radius measurements,” Appl. Phys. B76(1), 83–86 (2003). [CrossRef]
- P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. Mckay, and R. G. Mcduff, “Single-beam Z-scan: Measurement techniques and analysis,” J. Nonlinear Opt. Phys. Mater6(3), 251–293 (1997). [CrossRef]
- I. A. Ryasnyansky and B. Palpant, “Theoretical Investigation of the off-axis z-scan technique for nonlinear optical refraction measurement,” Appl. Opt.45(12), 2773–2776 (2006). [CrossRef] [PubMed]
- R. L. McCally, “Measurement of Gaussian beam parameters,” Appl. Opt.23(14), 2227–2227 (1984). [CrossRef] [PubMed]
- P. B. Chapple, “Beam waist and M2 measurement using a finite slit,” Opt. Eng.33(7), 2461–2466 (1994). [CrossRef]
- P. J. Shayler, “Laser beam distribution in the focal region,” Appl. Opt.17(17), 2673–2674 (1978). [CrossRef] [PubMed]
- S. Nemoto, “Determination of waist parameters of a Gaussian beam,” Appl. Opt.21(21), 3859–3863 (1986). [CrossRef]
- J. M. Khosrofian and B. A. Garetz, “Measurement of a Gaussian laser beam diameter through the direct inversion of knife-edge data,” Appl. Opt.22(21), 3406–3410 (1983). [CrossRef] [PubMed]
- J. A. Arnaud, W. M. Hubbard, G. D. Mandeville, B. delaClaviére, E. A. Franke, and J. M. Franke, “Technique for fast measurement of Gaussian laser beam parameters,” Appl. Opt.10(12), 2775–2776 (1971). [CrossRef] [PubMed]
- Y. Suzaki and A. Tachibana, “Measurement of the μm sized radius of Gaussian laser beam using the scanning knife-edge,” Appl. Opt.14(12), 2809–2810 (1975). [CrossRef] [PubMed]
- A. Nag, A. Kumar De, and D. Goswami, “Two-photon cross-section measurements using an optical chopper: z-scan and two-photon fluorescence schemes,” J. Phys. B: At. Mol. Opt. Phys.42(6), 065103, (2009). [CrossRef]
- I. Bhattacharyya, S. Priyadarshi, and D. Goswami, “Molecular structure-property correlations from optical nonlinearity and thermal-relaxation dynamics,” Chem. Phys. Lett.469, 104–109, (2009). [CrossRef] [PubMed]
- A. Rogalski and Z. Bielecki, “Detection of optical radiation,” Bull. Pol. Ac.: Tech.52(1), 43–66 (2004).
- C. D. Motchenbacher and J. A. Connelly, Low-noise Electronic System Design (John Wiley & Sons Inc, 1993).
- J. Phillips and K. Kundert, “Noise in mixers, oscillators, samplers, and logic an introduction to cyclostationary noise,” Proceedings of the IEEE custom integrated circuits conference, 431–439, (2000).
- B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley & Sons Inc, 1991). [CrossRef]
- X. Liu, S. Guo, H. Wang, N. Ming, and L. Hou, “Investigation of the influence of finite aperture size on the Z-scan transmittance curve,” J. Nonlinear Opt. Phys. Mater.10(4), 431–439 (2001). [CrossRef]

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