Novel method for ultrashort laser pulse-width measurement based on self-diffraction effect

doi:10.1364/OE.10.001099

Optics Express

Editor: Michael Duncan
Vol. 10, Iss. 20 — Oct. 7, 2002
pp: 1099–1104

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Novel method for ultrashort laser pulse-width measurement based on self-diffraction effect

Peng Xi, Changhe Zhou, Enwen Dai, and Liren Liu »View Author Affiliations

Optics Express, Vol. 10, Issue 20, pp. 1099-1104 (2002)
http://dx.doi.org/10.1364/OE.10.001099

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▸ Article Outline
– Abstract
1. Introduction
2. Talbot effect under ultrashort-pulse illumination
3. Conclusion
– References and links

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Abstract

Previous pulse-width measurement methods for ultrashort laser pulses have broadly employed nonlinear effects; thus any of these previous methods may experience problems relating to nonlinear effects. Here we present a new pulse-width measuring method based on the linear self-diffraction effect. Because the Talbot effect of a grating with ultrashort laser pulse illumination is different from that with continuous laser illumination, we are able to use this difference to obtain information about the pulse width. Three new techniques—the intensity integral technique, the intensity comparing ratio technique, and the two-dimensional structure technique— are introduced to make this method applicable. The method benefits from the simple structure of the Talbot effect and offers the possibility to extend the measurement of infrared and x-ray waves, for which currently used nonlinear methods do not work.

[Optical Society of America ]

1. Introduction

Previous ultrashort laser pulse-width measuring methods have employed nonlinear effects, and as a result the optical setups of the previous methods are always complex. For example, the earliest popular autocorrelation method [1

A. Brun, P. Georges, G. L. Saux, and F. Salin, “Single-shot characterization of ultrashort light pulses,” J. Phys. D 24, 1225–1233 (1991). [CrossRef]

] requires that three sensitive degrees of freedom be carefully adjusted: two spatial and one temporal. Also, to satisfy the phase-matching condition, the SHG crystal must be made very thin, leading to a weak signal. To yield the full intensity and phase of the pulse, frequency-resolved optical gating (FROG) [2

D. J. Kane and R. Trebino, “Single-shot measurement of intensity and phase of an arbitrary ultrashort pulse by using frequency-resolved optical gating,” Opt. Lett. 18, 823–825 (1993). [CrossRef] [PubMed]

] has been developed. Because an autocorrelator is employed in FROG, it inherits the drawbacks mentioned above. Other measuring apparatuses add more alignment degrees of freedom. An increase in system complexity leads to spending more time on adjusting and maintaining the alignment, decreasing the accuracy, and increasing the expense. The recently reported method of grating-eliminated no-nonsense observation of ultrafast incident laser light e-fields (GRENOUILLE) [3

P. O’Shea, M. Kimmel, X. Gu, and R. Trebino, “Highly simplified device for ultrashort-pulse measurement,” Opt. Lett. 26, 932–934 (2001). [CrossRef]

] combines several functions into just four devices. But when the complexity is diminished, the sensitive range is also lessened, to where 50–200 fs is needed for high accuracy. There are also problems relating to the nonlinear crystal: the detectable wavelength must be within the transparent area of the crystal, and the intensity of the laser must be carefully controlled to generate a sufficient nonlinear signal [4

D. N. Fittinghoff, J. L. Bowie, J. N. Sweetser, R. T. Jennings, M. A. Krumbügel, K. W. Delong, R. Trebino, and I. A. Walmsley, “Measurement of the intensity and phase of ultraweak, ultrashort laser pulses,” Opt. Lett. 21, 884–886 (1996). [CrossRef] [PubMed]

] but not too strong to cause laser damage. To overcome the drawbacks of nonlinear methods, some linear measuring methods are also introduced. But their resolution is usually inadequate for pulse widths below 100 fs. For example, the time-to-frequency conversion method has a resolution of 3 ps [5

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64, 270–272 (1994). [CrossRef]

], and the all-electronic method put forward by S. Prein et al. can achieve an accuracy of ~50 fs [6

S. Prein, S. Diddams, and J.-C. Diels, “Complete characterization of femtosecond pulses using an all-electronic detector,” Opt. Commun. 123, 567–573 (1996). [CrossRef]

].

The illumination of an ultrashort-pulsed laser beam is not monochromatic but includes a spectrum distribution. In this case the total field in an observation plane is a coherent superposition of the contribution from each frequency component. On this basis, Jiang et al. [7

Z. Jiang, R. Jacquemin, and W. Eberhardt, “Time dependence of Fresnel diffraction of ultrashort laser pulses by a circular aperture,” Appl. Opt. 36, 4358–4361 (1997). [CrossRef] [PubMed]

] and Gu and Gan [8

M. Gu and X. S. Gan, “Fresnel diffraction by circular and serrated apertures illuminated with an ultrashort pulsed-laser beam,” J. Opt. Soc. Am. A 13, 771–778 (1996). [CrossRef]

] studied the diffraction characteristics of different apertures illuminated with ultrashort laser pulses, and Wang et al. [9

H. Wang, C. Zhou, S. Zhao, P. Xi, and L. Liu, “The temporal Fresnel diffractive field of a grating illuminated by an ultrashort pulsed-laser beam,” J. Opt. A: Pure Appl. Opt. 3, 159–163 (2001). [CrossRef]

,10

H. Wang, C. Zhou, J. Li, and L. Liu, “Talbot effect of a grating under ultrashort pulsed-laser illumination,” Micro. Opt. Tech. Lett. 25, 184–187 (2000). [CrossRef]

] studied the Talbot effect under ultrashort-pulse illumination in both the temporal and the spatial domain. In this paper we study this subject further, and on this basis we propose a completely new method of ultrashort laser pulse measurement. Because ours is a linear diffraction measuring method, it avoids all the drawbacks relating to nonlinear effects and nonlinear crystals. We introduce three new techniques to improve this method for application: the intensity integral technique makes the detection more accurate, the intensity comparing ratio technique avoids the task of intensity calibration, and the two-dimensional structure technique enhances the accuracy and enlarges the range of application. Taking advantage of the Talbot effect, our method’s optical setup is believed to be the simplest of any current pulse-width measuring method. It should be noted that this method cannot obtain the full characteristics of a pulse with a square-law detector [11

V. Wong and I. A. Walmsley, “Linear filter analysis of methods for ultrashort-pulse-shape measurements,” J. Soc. Am. B 12, 1491–1499 (1995). [CrossRef]

]. Nevertheless, it gives us the possibility to detect pulses of any wavelength, especially for those wavelengths where nonlinear crystals are hard to find.

Fig. 1. Optical setup of pulse-width measurement based on the Talbot effect.

2. Talbot effect under ultrashort-pulse illumination

2.1 Theory

When a grating is illuminated with monochromic coherent light, exact images of the grating are shown in the periodic distances Z_T = 2nd ²/λ where λ is the wavelength, d is the period of the grating, and n is an integer. This effect is called the Talbot effect, and the distance is the nth Talbot distance [12

A. W. Lohmann and J. A. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt. 29, 4337–4340 (1990). [CrossRef] [PubMed]

]. The optical setup for ultrashort-pulse illumination is shown in Fig. 1. A grating is illuminated by the ultrashort laser, and a detector is placed at a certain distance z after the grating. Without loss of generality, the ultrashort laser pulse can be assumed to take a Gaussian shape in time [8–10

M. Gu and X. S. Gan, “Fresnel diffraction by circular and serrated apertures illuminated with an ultrashort pulsed-laser beam,” J. Opt. Soc. Am. A 13, 771–778 (1996). [CrossRef]

]

r (t, Δ τ) = \exp [i ω_{0} t - 4 \ln 2 {(\frac{t}{Δ τ})}^{2}],

(1)

where ω ₀ denotes the central frequency of the pulse and Δτ denotes the full width at half-maximum of the pulse [8–10

M. Gu and X. S. Gan, “Fresnel diffraction by circular and serrated apertures illuminated with an ultrashort pulsed-laser beam,” J. Opt. Soc. Am. A 13, 771–778 (1996). [CrossRef]

]. The distribution in the frequency domain R(ω,Δτ) is the Fourier transform of r(t), which can be expressed as

R (ω, Δ τ) = \frac{Δ τ}{4 \sqrt{π \ln 2}} \exp [\frac{- Δ τ^{2} {(ω - ω_{0})}^{2}}{8 \ln 2}] .

(2)

The diffraction intensity distribution of pulsed light waves in free space can be described by the Fresnel diffraction formula. Under paraxial approximation and for an incident illumination of a given frequency ω, it can be expressed as

U (x, z, ω) = \frac{\exp (i \frac{2 π}{λ} z)}{\sqrt{i λ z}} \int_{- \infty}^{+ \infty} U_{0} (x_{0}, ω) \exp [- \frac{i π {(x - x_{0})}^{2}}{λ z}] d x_{0},

(3)

where λ = 2πc/ω is the wavelength of the incident light. U ₀(x ₀,ω) and U(x,ω) are the amplitude distribution in the diffraction plane and in the observation plane, respectively. A rectangular grating is used to study the diffraction feature of the ultrashort-pulse beam. The grating can be expressed as

U_{0} (x) = \sum_{l} A_{l} \exp (i \frac{2 πlx}{d}) .

(4)

where d represents the period length and l and A_l represent the Fourier level and coefficient, respectively. Applying Eq. (4) to Eq. (3), we can have

U (x, z, ω) = \exp (i \frac{2 π}{λ} z) \times \sum_{l} A_{l} \exp (i \frac{2 πlx}{d}) \times \exp (\frac{i 2 π l^{2} z}{\frac{2 d^{2}}{λ}}),

(5)

where λ = 2πc/ω is the wavelength of the frequency parameter ω.

Fig. 2. (98.4KB) The intensity distribution detected at one Talbot distance with different pulse width (central wavelength 800 nm).

The illumination of an ultrashort pulse can be treated as a summation of coherent monochromic beams, with the central frequency ω ₀. In this sense, the diffraction pattern can be regarded as the summation of a series of monochromatic components. The amplitude distribution of ω in the frequency domain can be expressed as

G (x, z, ω, Δ τ) = R (ω, Δ τ) U (x, z, ω) .

(6)

The intensity distribution on the imaging plane can be expressed as [9

H. Wang, C. Zhou, S. Zhao, P. Xi, and L. Liu, “The temporal Fresnel diffractive field of a grating illuminated by an ultrashort pulsed-laser beam,” J. Opt. A: Pure Appl. Opt. 3, 159–163 (2001). [CrossRef]

,10

H. Wang, C. Zhou, J. Li, and L. Liu, “Talbot effect of a grating under ultrashort pulsed-laser illumination,” Micro. Opt. Tech. Lett. 25, 184–187 (2000). [CrossRef]

]

I (x, z, Δτ) = 2 π \int_{- \infty}^{+ \infty} {∣ G (x, z, ω, Δτ) ∣}^{2} dω .

(7)

Then for z = Z_T = 2nd ²/λ ₀ where λ ₀ =2πc/ω ₀ is the central wavelength and n is the Talbot number, we can have

I (x, z, Δτ) = \frac{Δ τ^{2}}{8 \ln 2} \int_{- \infty}^{+ \infty} \exp [- \frac{Δ τ^{2} {(ω - ω_{0})}^{2}}{8 \ln 2}]

\times \sum_{l,}^{+ \infty} \sum_{m = - \infty}^{+ \infty} A_{l} A_{m} \exp [i \frac{2 π (l - m) x}{d}] \times \exp [\frac{i 2 π (l^{2} - m^{2}) n ω_{0}}{ω}] dω .

(8)

Numerically solving Eq. (8), we can obtain the distribution of I(x,z,Δτ). Because a shorter pulse has a larger frequency range, a greater distortion of energy distribution occurs as compared with that of continuous-wave illumination. We can also increase the distortion by increasing the Talbot number n. The distortion is what we can use to detect the pulse width. A Ti:sapphire laser is usually used to generate the ultrashort pulse, which has a central wavelength of λ₀ = 800 nm. Figure 2 gives the intensity profile of pulses with different widths across three periods at the Talbot distance z ₀, here z ₀ = 2d ²/λ ₀.

Fig. 3. Talbot effect of pulses with different wavelengths at a pulse width of 100 fs. The detected distance is z = 2nd ²/λ ₀.

Since the basis of this method is linear diffraction, this method can also be used in infrared and x-ray pulse-width detection. Figure 3 illustrates the Talbot ultrafast effect of different wavelengths. From Figs. 2 and 3 we can obtain that I(0) and I(d/2) are always extreme points, because of symmetry. This gives us the extra advantage of being able to locate these two points precisely.

2.2 Techniques

We introduce three techniques to make this method applicable for practical use:

The first is the intensity integral technique. The casual error makes the detection of point intensity inaccurate. We define

P (h_{1}, h_{2}, Δτ) = \int_{h_{1} d}^{h_{2} d} I (x, z, Δτ) d x .

(9)

Because the casual error is averaged by the integration, a more reliable result can be obtained with this technique.

The second approach is the intensity comparing ratio technique. Let us define

S (Δ τ) = \frac{P (\frac{1}{4}, \frac{3}{4}, Δ τ)}{P (- \frac{1}{4}, \frac{1}{4}, Δ τ)} .

(10)

We can obtain the relationship between S(Δτ) and Δτ through numerical simulation. From the comparison, the common factor is eliminated, and thus the complex calibration work is avoided. Then we can obtain the pulse width by finding the corresponding point in the S(Δτ) ~ Δτ curve. The S(Δτ) ~ Δτ curves are shown in Fig. 4. An ideal S(Δτ) ~ Δτ curve should be monotonic; a linear shape with a large slope is preferred. From Fig. 4 we can see that although the trends of S(Δτ) ~ Δτ curves are linear, there are ripples in the S(Δτ) ~ Δτ curves when Δτ exceeds 20 fs. Thus this method is ideal for pulse widths of less than 20 fs with a high accuracy. The ripples can lead to a relative large error, e.g., a maximum of 9 fs (relative error 0.107) for M = 2 and 6 fs (relative error 0.085) for M = 3 within 20–100-fs detection. The error can be greatly reduced by use of the following two-dimensional structure technique. The relative FROG and GRENOUILLE errors are 0.031 and 0.013, respectively [3

P. O’Shea, M. Kimmel, X. Gu, and R. Trebino, “Highly simplified device for ultrashort-pulse measurement,” Opt. Lett. 26, 932–934 (2001). [CrossRef]

]. This method’s error tends to decrease with the decrease of the detected pulse width. In contrast, for nonlinear effects the error is usually increased with the decrease of pulse width. Note that in Eq. (10) the integral limits are not constrained; proper choice of integral limits can make the curve more linear.

Fig. 4. Relationship between the intensity ratio S(Δτ) and pulse-width Δτ is shown in S(Δτ) ~ Δτ curves. 1/M is the opening ratio of the corresponding grating.

The third is the two-dimensional structure technique. To detect the pulse width accurately, a two-dimensional grating with different opening ratios at x and y dimensions can be employed, as shown in Fig. 5. Then because x and y dimensions are orthogonal, we can have S_x (Δτ) and S_y (Δτ) at the same time. Thus two advantages can be obtained from this technique: (1) through finding the corresponding pulse width in two S(Δτ) ~ Δτ curves, a solitary pulse width can be decided, and thus a higher accuracy (less than 1-fs error within 1–100 fs) is achieved; (2) because the wavelength sensitivity range is related to the period of the grating, we can adopt different periods in the two coordinates to enlarge the sensitivity range. In other words, these three techniques work together to make this method more reliable.

In summary, since this method avoids the nonlinear effect, it leads to a simple structure and has a low energy requirement and no wavelength limitation. Compared with other linear pulse-width measuring methods, this method has a much higher accuracy, especially for pulse widths below 20 fs. The Gaussian wave-shape assumption of the ultrashort laser pulse is well accepted and can be chosen for other assumptions such as sech². Remarkably, in this method many freedoms such as the grating, the comparing area, and the detecting distance are not constrained; they can be chosen for adaptation for different pulses. For example, the grating can be any complex pattern to improve accuracy, and a phase grating can be used instead of the amplitude grating to enhance efficiency.

Fig. 5. Illustration of the two-dimensional grating (black area denotes transparent; white area denotes opaque). The opening ratio is as follows: vertical 1/2, horizontal 1/3.

3. Conclusion

Most traditional pulse-width measuring methods employ the nonlinear effect. The drawback of the nonlinear effect is that it requires a complex optical setup, which makes the system difficult to establish and maintain. Other problems relating to nonlinear crystals are also present, such as the wavelength sensitivity and intensity limits. Although some linear methods exist for this task, their accuracy is usually too low for an ultrashort pulse with a pulse width of less than 100 fs. By means of the Talbot effect under ultrashort-pulse illumination, we propose a novel method of measuring the pulse width of ultrashort laser pulses based on the linear diffraction effect. Thus our method avoids all the drawbacks mentioned above. Meanwhile, it can be easily realized with the simple structure of the Talbot effect. Unlike other methods, our method is more sensitive to shorter pulses, because shorter pulses have wider spectra. Three new techniques are presented to improve this method for practical application: the intensity integral technique can obtain more accurate intensity values, the intensity comparing ratio technique can avoid the difficulty task of intensity calibration, and the two-dimensional structure technique can obtain higher accuracy and enlarge the sensitivity range. More importantly, this method can be employed in the pulse-width detection for any wavelength, provided that Fresnel diffraction theory works. Typically, this method can be employed in infrared and x-ray cases, for which nonlinear methods fail to work for lack of a proper nonlinear medium.

Acknowledgments

The authors acknowledge the support of the National Science Foundation of China under Outstanding Youth Program (60125512, 60177016) and the Shanghai Science and Technology Committee (011661032, 012261011).

References and links

1.	A. Brun, P. Georges, G. L. Saux, and F. Salin, “Single-shot characterization of ultrashort light pulses,” J. Phys. D 24, 1225–1233 (1991). [CrossRef]
2.	D. J. Kane and R. Trebino, “Single-shot measurement of intensity and phase of an arbitrary ultrashort pulse by using frequency-resolved optical gating,” Opt. Lett. 18, 823–825 (1993). [CrossRef] [PubMed]
3.	P. O’Shea, M. Kimmel, X. Gu, and R. Trebino, “Highly simplified device for ultrashort-pulse measurement,” Opt. Lett. 26, 932–934 (2001). [CrossRef]
4.	D. N. Fittinghoff, J. L. Bowie, J. N. Sweetser, R. T. Jennings, M. A. Krumbügel, K. W. Delong, R. Trebino, and I. A. Walmsley, “Measurement of the intensity and phase of ultraweak, ultrashort laser pulses,” Opt. Lett. 21, 884–886 (1996). [CrossRef] [PubMed]
5.	M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64, 270–272 (1994). [CrossRef]
6.	S. Prein, S. Diddams, and J.-C. Diels, “Complete characterization of femtosecond pulses using an all-electronic detector,” Opt. Commun. 123, 567–573 (1996). [CrossRef]
7.	Z. Jiang, R. Jacquemin, and W. Eberhardt, “Time dependence of Fresnel diffraction of ultrashort laser pulses by a circular aperture,” Appl. Opt. 36, 4358–4361 (1997). [CrossRef] [PubMed]
8.	M. Gu and X. S. Gan, “Fresnel diffraction by circular and serrated apertures illuminated with an ultrashort pulsed-laser beam,” J. Opt. Soc. Am. A 13, 771–778 (1996). [CrossRef]
9.	H. Wang, C. Zhou, S. Zhao, P. Xi, and L. Liu, “The temporal Fresnel diffractive field of a grating illuminated by an ultrashort pulsed-laser beam,” J. Opt. A: Pure Appl. Opt. 3, 159–163 (2001). [CrossRef]
10.	H. Wang, C. Zhou, J. Li, and L. Liu, “Talbot effect of a grating under ultrashort pulsed-laser illumination,” Micro. Opt. Tech. Lett. 25, 184–187 (2000). [CrossRef]
11.	V. Wong and I. A. Walmsley, “Linear filter analysis of methods for ultrashort-pulse-shape measurements,” J. Soc. Am. B 12, 1491–1499 (1995). [CrossRef]
12.	A. W. Lohmann and J. A. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt. 29, 4337–4340 (1990). [CrossRef] [PubMed]

OCIS Codes
(070.6760) Fourier optics and signal processing : Talbot and self-imaging effects
(320.7100) Ultrafast optics : Ultrafast measurements

ToC Category:
Research Papers

History
Original Manuscript: August 30, 2002
Revised Manuscript: September 23, 2002
Published: October 7, 2002

Citation
Peng Xi, Changhe Zhou, Enwen Dai, and Liren Liu, "Novel method for ultrashort laser pulse-width measurement based on self-diffraction effect," Opt. Express 10, 1099-1104 (2002)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-20-1099

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References

A. Brun, P. Georges, G. L. Saux, and F. Salin, "Single-shot characterization of ultrashort light pulses," J. Phys. D 24, 1225-1233 (1991). [CrossRef]
D. J. Kane and R. Trebino, "Single-shot measurement of intensity and phase of an arbitrary ultrashort pulse by using frequency-resolved optical gating," Opt. Lett. 18, 823-825 (1993). [CrossRef] [PubMed]
P. O'Shea, M. Kimmel, X. Gu, and R. Trebino, "Highly simplified device for ultrashort-pulse measurement," Opt. Lett. 26, 932-934 (2001). [CrossRef]
D. N. Fittinghoff, J. L. Bowie, J. N. Sweetser, R. T. Jennings, M. A. Krumbugel, K. W. Delong, R. Trebino, and I. A. Walmsley, "Measurement of the intensity and phase of ultraweak, ultrashort laser pulses," Opt. Lett. 21, 884-886 (1996). [CrossRef] [PubMed]
M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, "Time-to-frequency converter for measuring picosecond optical pulses," Appl. Phys. Lett. 64, 270-272 (1994). [CrossRef]
S. Prein, S. Diddams, and J. -C. Diels, "Complete characterization of femtosecond pulses using an allelectronic detector," Opt. Commun. 123, 567-573 (1996). [CrossRef]
Z. Jiang, R. Jacquemin, and W. Eberhardt, "Time dependence of Fresnel diffraction of ultrashort laser pulses by a circular aperture," Appl. Opt. 36, 4358-4361 (1997). [CrossRef] [PubMed]
M. Gu and X. S. Gan, "Fresnel diffraction by circular and serrated apertures illuminated with an ultrashort pulsed-laser beam," J. Opt. Soc. Am. A 13, 771-778 (1996). [CrossRef]
H. Wang, C. Zhou, S. Zhao, P. Xi, and L. Liu, "The temporal Fresnel diffractive field of a grating illuminated by an ultrashort pulsed-laser beam," J. Opt. A: Pure Appl. Opt. 3, 159-163 (2001). [CrossRef]
H. Wang, C. Zhou, J. Li, and L. Liu, "Talbot effect of a grating under ultrashort pulsed-laser illumination," Micro. Opt. Tech. Lett. 25, 184-187 (2000). [CrossRef]
V. Wong and I. A. Walmsley, "Linear filter analysis of methods for ultrashort-pulse-shape measurements," J. Soc. Am. B 12, 1491-1499 (1995). [CrossRef]
A. W. Lohmann and J. A. Thomas, "Making an array illuminator based on the Talbot effect," Appl. Opt. 29, 4337-4340 (1990). [CrossRef] [PubMed]

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