1. Introduction

Using Young’s interference to measure the spatial coherence of laser beams has a long history [1]. Recently, this is extended to measure the tilt of femtosecond laser pulses since the optical-path difference in the Young’s interference introduces a time delay [2]. The spectrum of an extreme-ultraviolet laser can be also measured by a pinhole pair, offering a convenient method of making absolute wavelength and relative spectral intensity [3]. In the conventional spectrum measurement with the Young’s interference, the interference pattern is analyzed in the spatial frequency domain by directly taking a Fourier transform of the interference pattern. The Fourier transform spectrum involves a spike portion and two spectrum portions. Therefore, the measurement accuracy is primarily determined by the mutual coherence portion as the spectrum is convolved by the Fourier transform of the mutual coherence term or contrast ratio of the interference. It is necessary to cancel the influence of the mutual coherence portion to obtain a pure spectrum. In this paper, we describe a curve-fitting method for the spectrum measurement of femtosecond laser pulses with a pinhole pair to reduce the influence of the mutual coherence portion.

2. Curve-fitting method

The measurement scheme is the same as the conventional Young’s double-slit experiment (Fig. 1). Laser pulses fall normally on a pinhole pair and the diffracted laser pulses are directly superposed on a line CCD sensor. If the pinhole diameter is small enough or the distance from the pinhole pair to the CCD is large enough, the diffracted laser beams can be approximately considered as two spherical waves. An interference pattern will be observed from the overlapped region of the two diffracted beams near the central axis if the two beams transmit through the same optical lengths approximately. In addition, the interference pattern is also modulated by the frequency bandwidth of the incident laser pulses.

 

Fig. 1. Experimental setup for the measurement of a pure spectral profile.

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The intensity signal I_CCD displayed on the CCD is given by

where E_out (t) is the total amplitude of the laser beam diffracted by the pinhole pair. According to the Parseval’s theorem, the signal received by the CCD can be expressed as

where ω is the frequency, E_in (ω) and E_out (ω) are the spectral distributions of the input and diffracted laser pulses, U(ω) is the transmission function of the pinhole pair and is expressed as

where x is the axis along the transverse direction of the CCD, d is the separation of the two pinholes, z is the distance between the pinhole pair and the CCD, U ₁(ω) and U ₂(ω) are the diffracted beam amplitudes, and c is the light velocity. In the spectrum measurement, the interference region of the two diffracted beams should be less than the Airy disk region to obtain the whole information of the laser spectrum. Under the Fraunhofer diffraction approximation at a given central wavelength λ ₀, the transverse region Δx of the diffracted laser beam displayed on the CCD is approximately given by Δx= 1.22λ ₀ z/a. The spatially overlapped region Axs corresponding to the laser bandwidth Δλ is approximately given by Δx _s=(4ln2/π)${z\lambda }_0^2$/(dΔλ), provided that the laser spectrum has a initial Gaussian profile. As Δx should be larger than Δx _s we have d/a=(4ln2/1.22π)(λ ₀/Δλ)n, where n is the ratio of the diffracted beam to the interference range. This means that a broader spectrum requires a small separation d for a given pinhole diameter a. In addition, as the two diffracted beams should be also less than the receiving range Δx_ccd of the CCD, we obtain z=Δx _ccd(a/λ ₀)/(1.22m), where m is the ratio of the diffracted beam to the receiving range of the CCD. Let λ ₀=800 nm, Δλ ₀=12.5 nm, a=50 μm, Δx_ccd =2.1 cm, n=5, and m=2, we have z=54.6 cm and d=11.6 mm, respectively. The distance z is primarily determined by the receiving range of the CCD, and the parameters d and a are determined by the incident laser parameters such as the central wavelength and bandwidth. Eq. (1) can be simply expressed as

The difference between two calculation results with Eqs. (1) and (3) is less than 0.1% for spectral bandwidths ranging from 1 nm to 90 nm. Therefore, Eq. (3) can be used as a standard analytical function to process the experimental data under the Fraunhofer approximation.

The laser spectrum can be extracted by taking a Fourier transform of Eq. (3) as a conventional Fourier-transform spectrometer, in which the spectrum is determined by the mutual coherence term. To increase the measure accuracy, the variation of the mutual coherence term U ₁ U ₂ with the transverse position x of the CCD should be minimum, and the optimization for the parameters z and d is necessary. This can be achieved by taking a differential calculation of U ₁ U ₂ and the z and d with respect to the most minimum value of dU ₁ U ₂/dx are the optimized parameters. For z=65 cm, λ ₀=800 nm, and a=50 μm, the optimized d is 12.25 mm. After the determination for the d, the final spectral accuracy is proportional to the distance z and inversely proportional to the pinhole diameter a.

If we can previously obtain the information on the mutual coherence term U ₁ U ₂, the pure spectrum can be expected. For this purpose, changing eq. (3) gives,

A Fourier transform of Eq. (4) gives

where F[ ] denotes the Fourier transform with respect to the transverse position x of the CCD, f_x is the spatial frequency. Therefore, we can extract the power spectrum without the influence of the mutual coherence term. The diffracted patterns of |U ₁|² and |U ₂|² can be measured through the experiment by cutting one of the two diffracted laser beams, respectively. Such measurement will be affected by the temporal variation in the laser power. In this paper, we propose a data-processing method to obtain |U ₁|² and |U ₂|² by directly taking a curve fitting of the experimental data involving the mutual coherence portion. In the curve-fitting process, we use the following forms as the fitting functions of |U ₁|² and |U ₂|², respectively,

where J ₁ is the Bessel function of the first order, I _j0 (j=1, and 2) is the coefficient relating with the peak intensity of the diffracted beam, α describes the noise level of measurement setup, α_j is the coefficient with respect to the center of the single pinhole, and β is the coefficient used for further improving the accuracy of a/z. These coefficients are determined completely by the raw experimental data. As the division by the mutual coherence term U ₁ U ₂ is used in the data processing, it is necessary to ensure that the Airy disk due to the pinhole diffraction should be large enough so that U ₁ U ₂ is certainly over zero.

3. Experimental results and discussion

In our experiments, laser pulses with good spatial and temporal coherences are generated from a Ti:sapphire laser system, which operates at a repetition rate of 80 MHz, a pulse duration of 80 fs, and a central wavelength of 802 nm. The spectral bandwidth (FWHM) is 12.5 nm which was measured by a commercial spectrometer. The interference signal of the diffracted laser beam is recorded by a 3000-pixel line CCD sensor (7 μm × 200 μm pixel size). A pinhole pair with a diameter of 50 μm and a separation distance of 12.5 mm is used. The distance between the pinhole pair and the CCD is 69.53 cm which is finely calibrated by using a standard laser beam.

 

Fig. 2. Measured experimental data.

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Fig. 3. Curve-fitting results for (i) |U ₁|²+|U ₂|², (ii) |U ₁|², and (iii) |U ₂|².

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Figure 2 shows the experimental result of the interference signal. The corresponding spectral distribution convolved by the mutual coherence portion can be obtained by directly taking a Fourier transform of the raw data like the data processing with the conventional method [3], resulting in a spectral bandwidth (FWHM) of 12.3 nm. This result agrees well with that measured with the commercial spectrometer. Figure 3 is the curve-fitting result with the form of Eq. (6). The standard deviation of the curve-fitting result for the sum |U ₁|²+|U ₂|² is 1.24 %, which is estimated by a standard deviation between the experimental and fitted data except for the mutual coherence portion. The pure spectrum shown in Fig. 4 is obtained by firstly processing the experimental data as shown in Eq. (4) and then taking a Fourier transform of the processed raw experimental data. The extracted spectrum [Fig. 4(b)] is near the Gaussian with a spectral range (FWHM) of 11.67 nm, and the obvious noise is accompanied by a factor of 1/100 in comparison with the spectral level.

 

Fig. 4. (a) Fourier transform of the experimental result by canceling the mutual coherence portion. (b) Extracted spectral profile of the incident laser beam shown in Fig. 4(a).

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In the spectrum calculation with Eq. (4), the value of I_s is sensitive to the mutual coherence term U ₁ U ₂ as I_s is divided by U ₁ U ₂ if U ₁ U ₂ has a value near zero over the receiving range of the line CCD sensor. We estimated the variation of the mutual coherence term U ₁ U ₂ with the pinhole separation d and its rate of change around the minimum value and in particular the effect of noise on the assessment of the minimum. If the minimum value of U ₁ U ₂ is near zero and located within the receiving range of the CCD, noise in the raw data I _CCD is enhanced around the minimum position of U ₁ U ₂ during the data processing and will affect the fitted power spectrum. To reduce the influence of the zero point of the mutual coherence term on the data processing, it is generally necessary to ensure that the zero point of the mutual coherence term is outside the receiving range of the CCD. At least the zero point position of U ₁ U ₂ should be outside the spatially overlapped region corresponding to the laser spectrum so that we can cancel the zero-point effect by covering the raw data with a spatial window filter during the data processing. A small pinhole diameter a, a small pinhole separation d and a large distance z are generally available in the design of the pinhole pair under the condition that the spectral region is completely displayed on the CCD.

The measurement accuracy is primarily influenced by the parameters of the pinhole pair and CCD such as a, z, and d at the given raw experimental data [3]. In addition, the laser intensity should be strong to increase the signal-noise ratio. And the pinhole pair can be replaced by a slit pair to increase the intensity of the diffracted laser beam on the CCD, in which the Bessel function should be exchanged by a Sinc function with respect to the slit diffraction.

4. Conclusions

We described a design method for the spectrum measurement of femtosecond laser pulses with a pinhole pair and presented a curve-fitting method for the spectrum measurement without the influence of the mutual coherence portion.