In the last decade, there has been a great up-surge of interest in generation and use of ultrashort laser pulses [1], which has promoted the development of various diagnostic tools for spatio-temporal pulse characterization. In particular, there is an increasing trend in using ultrashort laser pulses having a desired amount of temporal asymmetry, in laser matter interaction experiments [2–3], laser based accelerators [4–5], nonlinear optics [6] etc. For instance, in self-modulated laser wake-field acceleration scheme, driving laser pulse of duration corresponding to several plasma wavelengths, but with a much sharper leading edge is found to result in a faster growth of the plasma wake-field leading to higher energy electrons after acceleration [4]. Similarly, variation in temporal pulse shape has been observed to be effective in controlling of laser heating in clusters [2]. Such asymmetric laser pulses may be generated by appropriate choice of dispersion and chirp in the laser system. For example, steepening of the leading edge of high contrast femtosecond pulses over a high dynamic range has been achieved through precise control of the third-order dispersion [3]. Further, presence of any pre-pulse in an ultrashort laser pulse may significantly alter the laser-matter interaction dynamics through generation of a pre-formed plasma. Simple methods capable of real time detection and measurement of temporal asymmetry and presence of any pre-pulses in ultrashort pulses are therefore desirable.

While simple diagnostic systems such as single-shot second order autocorrelators are widely used for measurement of pulse duration, angular chirp and pulse-front tilt [7–8] of ultrashort laser beams, they cannot provide information on the pulse shape and any temporal asymmetry in the laser pulse. A variety of techniques have been developed for detailed characterization of ultrashort laser pulses such as “frequency resolved optical gating” [9] (FROG), “time resolved optical gating” [10] (TROG), “spectral phase interferometry for direct electrical field reconstruction” [11] (SPIDER), and other variants [12–13], “phase and intensity from correlation and spectrum only” [14] (PICASO) etc. While these techniques are capable of providing a detailed information on the laser pulse shape, they generally require a rather sophisticated experimental set-up and involve complex phase retrieval algorithms. Further, it may not be convenient to use them for visual detection of temporal asymmetry, especially when the latter is of a very small magnitude of a few percent.

In principle, the temporal asymmetry in ultrashort laser pulses can be determined using third order [15–16] intensity correlation. However, recording of a higher order correlation has two disadvantages of requirement of a large amount of laser pulse energy and a somewhat more complex experimental set-up. In many situations, it is of main interest to detect presence of either any temporal asymmetry in the laser pulse or the occurrence of double pulses, rather than to carry out a detailed temporal characterization. It may be pertinent to recall that one of the early techniques for pulse shape and chirp measurement was based on second order interferometric cross-correlation [17]. It will be interesting if one could detect and measure the pulse asymmetry in a simpler way using the lowest (i.e., second) order autocorrelation. For instance, interferometric autocorrelation (IAC) can be obtained in a compact [18] set-up using a very small amount of pulse energy [19]. However the IAC signals have a rather poor sensitivity to pulse temporal asymmetry and chirp. Recently, the sensitivity of the IAC signals to chirp was significantly increased by using a modified spectrum auto-interferometric correlation [20–21] (MOSAIC). The MOSAIC signals are generated by modifying the spectral contents of the IAC signals. However, these MOSAIC signals do not provide any information about pulse asymmetry.

In this paper, we present a simple and sensitive method for visual detection of pulse asymmetry in ultrashort laser pulses, based on an unbalanced modified spectrum auto-interferometric correlation (UMOSAIC). This basically involves unbalancing the two pulse intensities in second order interferometric correlation to detect temporal asymmetry, followed by modification of the spectrum to enhance the sensitivity. The usefulness of this method in visual detection of pulse asymmetry over third order intensity correlation is illustrated by considering various examples of pulse asymmetries.

Let the electric field amplitude E(t) be expressed as E(t)=[I(t)]^1/2 exp[i{ω_o t+ϕ(t)}], where ω_o is the central frequency, I(t) is the pulse intensity envelope function, and ϕ (t) is the phase function. The second order interferometric autocorrelation function, S_2IAC(τ), for unequal intensities of the two laser pulses, assuming ∫I²(t) dt=1, may be written as

where k is the intensity ratio of the two pulses, and the three functions are

The above stated functions G₂(τ), F₂₁(τ) and F₂₂(τ) comprising the IAC signal are oscillating at different frequencies viz dc, ω_o and 2ω_o respectively. Hence these envelope functions can be easily derived from the observed IAC signal either through fast Fourier transforms or using appropriate electronic hardware. Further, while the G₂(τ) term, representing the second order intensity autocorrelation, contains information about the laser pulse duration, the interference terms F₂₁(τ) and F₂₂(τ) are governed by the laser pulse shape as well as the chirp present in the pulse. It may be noted from eq.2 and eq.4 that the functions G₂(τ) and F₂₂(τ) are symmetric for any value of k. Hence they cannot be used for detection of any pulse asymmetry. On the other hand, the function F₂₁(τ), comprising of two terms in the integrand of eq.3 viz k^1/2 I^3/2(t) I^1/2(t-τ) and k^3/2 I1/2(t) I^3/2(t-τ), becomes asymmetric for asymmetric laser pulses for k≠1, i.e., when the intensities of the two interfering beams are unequal. Therefore, in principle, pulse asymmetry can be detected by looking at either the asymmetry of the second order IAC signal or the asymmetry of the unbalanced function F₂₁(τ) alone. However, as stated earlier, the IAC signals have a poor sensitivity to pulse temporal asymmetry, which renders its detection to be very difficult.

The sensitivity of IAC signals towards detection of pulse asymmetry can be greatly enhanced by using spectrally modified signals, which may enable one to visually detect even a small amount of pulse asymmetry of a few percent. We first normalize the three functions G₂(τ), F₂₁(τ) and F₂₂(τ) w.r. to their respective peak values, and denote them by g₂ (τ), f₂₁ (τ) and f₂₂ (τ) respectively. We then generate spectrally modified signal S(τ) using the above normalized functions such that the signal S(τ) exhibits an asymmetry for asymmetric laser pulse, and it becomes zero at the center (i.e, .τ=0). This can be accomplished by subtracting one of the two normalized symmetric functions g₂ (τ) and f₂₂ (τ) from the only asymmetric function f₂₁ (τ) to make the signal zero at τ=0. Two such signals are

and

The above unbalanced modified spectrum auto-interferometric correlation (UMOSAIC) signals are asymmetric for unequal intensities of the two asymmetric laser pulses. They show different peaks occurring on leading and trailing edges of the correlation signal, and therefore they can be used for visual detection of pulse asymmetry. The UMOSAIC signal S₁(τ) may be preferred over S₂(τ) as the latter involves subtraction of dc background to obtain G₂ (τ) from the IAC signal. It may be mentioned here that one may generate another function S₃(τ) using the two symmetric functions,

While this cannot be used for detection of any pulse asymmetry, it is useful in estimation of various order chirp [21] present in the laser pulse.

We now illustrate the effect of pulse asymmetry on UMOSAIC signals and compare them with that for other unbalanced second order signals. We consider a laser pulse of the form $I(t)\propto \frac{1}{{\left[\exp \left(c_{1}t\right)+\exp \left(-c_{2}t\right)\right]}^2}$, where the parameters c₁ and c₂ govern the half width at half maximum (HWHM) durations Δt₁ and Δt₂ on the two sides of the peak of the pulse. One may define the pulse asymmetry parameter (t_asm) as t_asm=(Δt₁/Δt₂)-1. Fig.1(a) depicts an asymmetric pulse with t_asm=0.1 (i.e. 10% pulse asymmetry) for no chirp condition (i.e., ϕ(t)=constant). The time variation in this figure is expressed as a normalized parameter t/t_p, where t_p defines an ideal symmetric sech² (t/t_p) pulse of a full width at half maximum (Δt) equal to that of the asymmetric pulse i.e. Δt=Δt₁+Δt₂. Various unbalanced second order correlation signals viz the intensity correlation signal, the IAC signal, and the UMOSAIC signal S₁(τ) for an intensity ratio k of 0.36 of the two overlapping beams are shown in Fig. 1(b), Fig.1(c), and Fig. 1(d) respectively. As expected, the intensity correlation (Fig. 1(b)) is symmetric, hence it cannot be used for detection of any pulse asymmetry. Next, the IAC signal (Fig. 1(c)) does not show any visually detectable asymmetry. In comparison to these, the UMOSAIC signal S₁(τ) (Fig. 1(d)) clearly depicts the effect of pulse temporal asymmetry. Moreover, this UMOSAIC signal can also distinguish the direction of temporal asymmetry. For instance, Fig. 1(e) shows an asymmetric laser pulse which has the same magnitude of the temporal asymmetry as that shown in Fig. 1(a) but in the opposite direction of time. The corresponding S₁(τ) signal shown in Fig. 1(f) also gets temporally reversed w.r. to that shown in Fig. 1(d).

Next, we examine the sensitivity of UMOSAIC signals on pulse asymmetry and intensity unbalance of the two laser pulses. The ratio of the peak amplitudes of the UMOSAIC signals P₁/P₂ (as indicated in Fig. 1(d)) may be taken as an index of the sensitivity of this signal towards pulse asymmetry. Fig. 2(a) shows dependence of the peak amplitudes ratio on the intensity ratio k of the two laser pulses for t_asm of 0.05. It is clear that the sensitivity increases as k decreases. However, the IAC signal contrast ratio decreases as k decreases and the same is shown in Fig. 2(a). Hence for detection of a very small amount of pulse asymmetry, the value of k should be chosen to be appropriately small. Next, Fig. 2(b) shows dependence of the peak amplitude ratio P₁/P₂ on the pulse asymmetry parameter t_asm for a fixed vale of k=0.04. It is seen that the ratio P₁/P₂ increases with increasing value of pulse asymmetry parameter. It may be noted that the ratio P₁/P₂ is large enough to permit visual detection of the pulse asymmetry even when the latter is of a few percent magnitude.

It may be useful to see the effect of using a very small value of intensity ratio k on various auto-correlation signals for asymmetric laser pulses. Figure 3 shows IAC signal, intensity correlation signal, UMOSAIC signal F₂₁(τ), UMOSAIC signal F₂₂(τ), and UMOSAIC signal S₁(τ) for t_arm=0.05 and k=0.04. As also noted earlier from Fig. 2(a), the contrast ratio of IAC signal (Fig. 3(a)) is now reduced to ~1.2. The peak values of various UMOSAIC signals are G₂(τ) ~6×10^-3, F₂₁(τ) ~0.16, and F₂₂(τ) ~3×10^-3. These small values may require a phase-sensitive detection system. However, this can be avoided by choosing a higher value of k as illustrated later in this paper.

 

Fig. 1. Effect of temporal asymmetry on various unbalanced second order correlation signals for k=0.36 : (a) A laser pulse with asymmetry parameter of 0.1; (b) intensity correlation signal; (c) IAC signal; (d) UMOSAIC signal S₁(τ); (e) An asymmetric laser pulse with opposite direction of time ; and (f) corresponding UMOSAIC signal. The symmetric sech² (t/t_p) pulse of the same FWHM duration as that of the asymmetric pulses (a and e) is also shown by dash-dot lines for visual illustration.

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Next, we compare the pulse asymmetry sensitivity of the UMOSAIC signals with that of the unbalanced third order intensity correlation [16]. To illustrate this, we consider an ultrashort laser pulse with t_asm of 0.05. The normalized third order intensity correlation and the UMOSAIC signal S₁(τ) for k=0.36 are shown in Fig. 4(a) and Fig. 4(b) respectively. While the third order intensity correlation (Fig. 4(a)) does not visually show any asymmetry, Fig. 4(b) clearly shows asymmetry. It may be pointed out that a much larger intensity unbalance would be required for visual detection of pulse asymmetry from the third order intensity correlation. For instance, even for an intensity ratio of k=0.04, the detection of pulse asymmetry by third order intensity correlation would be limited to a value of asymmetry parameter t_asm ≥ 0.1. Moreover, in the latter case, a reduction in the signal contrast ratio for small values of k may necessitate use of the phase lock detection to experimentally record the unbalanced correlation signal.

 

Fig. 2. Variation of the ratio of UMOSAIC signal peaks (i.e. P₁/P₂) : (a) with intensity ratio k; (b) with pulse asymmetry parameter t_asm. The variation of IAC signal contrast ratio with intensity ratio k is also shown in Fig.2.(a)

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Fig. 3. Various autocorrelation and UMOSAIC signals for k=0.04 and t_asm=0.05 : (a) IAC signal; (b) Intensity correlation ; (c) UMOSAIC signal F₂₁(τ); (d) UMOSAIC signal F₂₂(τ); (e) UMOSAIC signal S₁(τ)

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Fig. 4. A comparison of the UMOSAIC and third order intensity correlation signals for detection of pulse asymmetry (t_asm=0.05) for k=0.36: (a) normalized unbalanced third order intensity correlation; (b) normalized UMOSAIC signal S₁(τ).

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Finally, we examine the suitability of the UMOSAIC signals to detect the presence of double pulses in ultrashort pulse laser beams. We consider such double pulses of the form I(t)∝sech²(t/t_p)+A sech²((t±t_d)/t_p), where A is the relative amplitude of the pre (or post) pulse and td is the delay between the peak of the pre (or post) pulse w. r. to the peak of the main pulse. Figure 5(a) shows a double pulse with A=10^-3 and t_d=5 t_p. The corresponding unbalanced third order intensity correlation, UMOSAIC signal F₂₁(τ) and the UMOSAIC signal S₁(τ) calculated for k=0.04 are shown in Fig. 5(b), Fig. 5(c) and Fig. 5(d) respectively. It may be noted from these figures that while the hump in UMOSAIC signal S₁(τ) has an amplitude of ~0.25, the corresponding value for the third order intensity correlation signal, and the UMOSAIC signal F₂₁(τ) are ~10^-3 and 7×10^-2 respectively. Hence, the UMOSAIC signals are much more sensitive in detection of double pulses as compared to the third order intensity correlation.

The UMOSAIC signals can be easily derived from experimentally observed unbalanced IAC signals. Though in all our calculations, we have used analytical formulas, the various UMOSAIC signals can be retrieved using suitable digital signal processing filters as described in our earlier work [21]. This is because of the fact that the envelope functions G₂(τ), F₂₁(τ) and F₂₂(τ) are located at different frequencies, and the same can be obtained from experimentally observed IAC signals by either using a simple computer program based on fast Fourier transform or using suitable electronic hardware. In addition to visual detection of the asymmetry from UMOSAIC signals, one can estimate the pulse asymmetry parameter t_asm (see Fig.2(b)) for a known value of intensity ratio k from the observed ratio of the UMOSAIC signal peaks. Moreover, the presence of double pulses can also be visually detected and the quantitative extent of the second pulse can be estimated from the magnitude of the hump observed.

The present method is illustrated using chirp-free pulses. It may be pointed out that while the presence of chirp would slightly change the ratio P₁/P₂, shape of the UMOSAIC signal S₁ (τ) remains mostly unaltered. If the magnitude of the chirp and power spectrum are independently determined, then by taking the chirp into account, one can get a correct estimate of the pulse asymmetry. This will be addressed separately in detail in future.

It may be pertinent to address to the effect of noise on various UMOSAIC signals. In our earlier work [21], it has been shown that a random noise of ±5% does not affect the MOSAIC signals. Moreover, the noise can also be suppressed during digital processing of the signals. However, detailed investigation of the effect of different types of noise may be required.

The above method can be applied to a broad range of pulse duration of ultrashort laser pulses. The lower limit is governed by the condition that the spectral bandwidth should be smaller than the central frequency. In the visible-IR spectral region, this would put the lower limit to be ~1–2 fs. Hence this method should be practically applicable for pulse duration exceeding ~5 fs.

 

Fig. 5. A comparison of the second order UMOSAIC and third order intensity correlation signals for detection of double pulse for k=0.04: (a) A double pulse with A=10^-3 and t_d=5t_p; (b) normalized unbalanced third order intensity correlation; (c) normalized UMOSAIC signal f₂₁(τ). (d) normalized UMOSAIC signal S₁(τ). (Note that logarithmic scale is used on vertical axis to show the small amplitude of the second pulse)

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In conclusion, we have presented a new, simple and sensitive method for visual detection of pulse asymmetry in ultrashort laser pulses. This is based on second order unbalanced modified spectrum auto interferometric correlation (UMOSAIC). The suitability of the method is illustrated by considering examples of chirp-free asymmetric ultrashort laser pulses. The presence of chirp can also be estimated [21] with high sensitivity using UMOSAIC signal S₃(τ). Since the second order UMOSAIC signals are much more sensitive to pulse asymmetry in comparison to that for unbalanced third order intensity correlation, this technique can be used for real time visual detection of the pulse asymmetry and presence of multiple pulses. It may also be helpful in optimizing femtosecond laser pulse oscillators or chirped pulse amplification based laser systems to generate ultrashort laser pulses with a desired level of pulse asymmetry which are increasingly required for various applications.