Lohit Malik
email: lohit.malik@gmail.com

Alexandre Escarguel

Mayank Kumar

Abhishek Tevatia

Rajpal Singh

Uncovering remarkable contribution of lasers peak intensity region in holography

Keywords: Laser beams, peak intensity region, frequency difference, intensity pattern, modulation depth, holography

For the scheme of two colour lasers in interference and holography, we uncover remarkable contribution of focusing region (spread) of the peak intensity of superposing laser beams based on the numerical studies and confirm it by mathematical calculations. The pulses having peak intensity for a wider region create better interference pattern, leading to higher modulation depth. Also, such beams having a stronger intensity gradient enhance the range of frequency detuning (difference), which would ease the experimentation in getting better holograms. A comparison of the results achieved with such beams is done with the most commonly used Gaussian beams for the better understanding of the concept for the advancement in holography.

Introduction

A monochromatic light wave described by 𝐸(𝑟 ⃗, 𝑡) = 𝐸 0 cos(𝜔𝑡 -𝑘 ⃗⃗ • 𝑟 ⃗) exists over the whole space for all times, but it does not exist in practice. In practice we have quasimonochromatic wave, which has limited extent in space, means the coherence length is limited. When we use two waves of slightly different frequencies and superpose them, the obtained interference pattern is not stationary but moving. Since we cannot record such a pattern on a twodimensional detector, the variation of the time signal is followed at a point, and this type of interferometry is called heterodyne interferometry.

For the better-quality holograms, generally the two superposing lightwaves are coherently obtained from a single light source (laser). When the light is in the form of a pulse of time duration  or the coherence length of c, a hologram can be recorded on a twodimensional detector so long the path difference between the two superposing pulses remains within the coherence length. It is however experimentally difficult to perform and in the case of femtosecond lasers the difficulty level rises up. Still, the femtosecond holography has validated the recording of static holograms [START_REF] Abramson | Motion picture of short pulses[END_REF][START_REF] Hsieh | Femtosecond holography in lithium niobate crystals[END_REF]. The femtosecond lasers have proved to be a useful tool in both the transparent and opaque materials with respect to the patterning diffractive elements and in obtaining efficient radiation [START_REF] Malik | Tunable and collimated terahertz radiation generation by femtosecond laser pulses[END_REF][START_REF] Malik | Investigations on terahertz radiation generated by two superposed femtosecond laser pulses[END_REF] and wake field in nonlinear media [START_REF] Dorranian | Radiation from highintensity ultrashort-laser-pulse and gas-jet magnetized plasma interaction[END_REF][START_REF] Dorranian | Generation of short pulse radiation from magnetized wake in gas-jet plasma and laser interaction[END_REF].

In a single beam femtosecond laser direct writing technique, a double-layer hologram inside the polymethyl methacrylate sample and a triple layer hologram inside LiNbO3 sample have been constructed [START_REF] Sohn | Three-dimensional hologram printing by single beam femtosecond laser direct writing[END_REF]. Based on the femtosecond lasers, nonvolatile spectral holography has been conducted in photorefractive Lithium Niobate crystals. With the use of stoichiometric 𝐿𝑖𝑁𝑏𝑂 3 : 𝑇𝑏, 𝐹𝑒, the two colour hologram multiplexing has been achieved successfully [START_REF] Lee | Two-color hologram multiplexing from the colored state in stoichiometric LiNbO3 : Tb[END_REF]. Apart from these enticing applications, a method based on computational adaptive optics was proposed to rectify the deviations in digital holography [START_REF] Matkivsky | Correction of aberrations in digital holography using the phase gradient autofocus technique Laser[END_REF]. Improvement in quality is always fruitful for imaging techniques. An algorithm was used to improve image resolution based on phase shifting holography [START_REF] Leihong | Study of key technology of ghost imaging via compressive sensing for a phase object based on phase-shifting digital holography[END_REF]. An amplitude-only Fermat-spiral sieve has also been used for holographic imaging that eliminated the limitation at shorter wavelengths [START_REF] Zhang | Single-shot phase-shifting digital holography with amplitude-only Fermat-spiral sieves[END_REF].

Two femtosecond laser beams with significant frequency difference have been employed for the holography and permanent holographic gratings have been prepared, but the quality could be achieved by keeping the two frequencies very close to each other [START_REF] Odoulov | Interference and holography with femtosecond laser pulses of different colours[END_REF]. As an attempt for the improvement by reducing the null regions, which were developed in such a scheme, we have proposed skewness in the Gaussian pulses [START_REF] Malik | Role of the temporal profile of femtosecond lasers of two different colours in holography[END_REF] for obtaining the betterquality holograms. Also, the femtosecond lasers with zero intensity at their central region have been proved to be better candidates for realizing holograms for large frequency differences [START_REF] Malik | Dark hollow lasers may be better candidates for holography[END_REF]. However, now for the first time we show that the spread or focusing region of the peak intensity of lasers is very important for enhancing not only the region of frequency difference but also the modulation depth and intensity pattern and hence, in achieving the better-quality holograms. We make precise numerical studies of the problem and confirm the results and the proposal with the mathematical calculations. Comparative study is done with the most commonly used Gaussian pulses of the lasers in order to prove how important is this proposal for the advancement of the holography technology.

Interference pattern by lasers with wider peak intensity region

A Gaussian pulse profile (temporal) of the laser is described by

𝐸(𝑡) = 𝐸 0 𝑒 -( 𝑡 𝜏 ) 2
, where 𝐸 0 is the amplitude of its electric field and  is the pulse duration. This profile is shown by the dotted line in figure 1, from which this is clear that the peak of its field (or intensity) stays for a smaller region. However, the expression 𝐸(𝑡) = 𝐸 0 𝑒 -( 𝑡 𝜏 ) 4 for another kind of profile is plotted in this figure, which shows that the peak electric field (intensity) stays for a longer region. This kind of pulse is called as super-Gaussian pulse, where the higher even powers of 𝑡 𝜏 , such as 6, 8, 10, etc. (index of the laser), further enlarge this region. A comparison of the two graphs reveals that the peak intensity region for the super-Gaussian beam is much more than that of the Gaussian beam which is relatively pointed. Third graph in this figure is for 𝐸(𝑡) = 𝐸 0 cos ℎ 2 (0.935 𝑡 𝜏 ) 𝑒 -𝑡 2 𝜏 2 , which is called cosh 2 profile. This kind of profile is such that its focused region has the same spread as that of the super-Gaussian beam (index 4) but the rise and fall rate of its intensity is appreciably slow. It means the gradient in intensity of the super-Gaussian profile is higher than that of the cosh 2 profile and the Gaussian profile. We will make use of these laser beams for obtaining the interference pattern and then the modulation depth from the energy density corresponding to the pattern. The modulation depth that represents the interference modulation efficiency is the main quantity for judging the quality of the holograms obtained. It is one of the most important quantities that manipulates the signal-to-noise ratio of an interferogram and clearly impacts the sensitivity of the spectrometer. This is in general the coefficient of the cosine term in the expression of interference pattern 𝐼 = (𝐼 1 + 𝐼 2 )[1 + 𝑀 𝑑 cos ] obtained when two coherent beams of intensities I1 and I2 and phase difference  are used to interfere. This expression is derived from the resultant intensity 𝐼 = 𝐼 1 + 𝐼 2 + 2(𝐼 1 𝐼 2 ) 1/2 𝑐𝑜𝑠 and is the wave vector of the pattern, representing the periodicity of the pattern. In order to plot this intensity pattern, we consider the same parameters as used by Odoulov et al. [START_REF] Odoulov | Interference and holography with femtosecond laser pulses of different colours[END_REF] in their experiment, i.e. Figures 2 and3 show the intensity patterns in space and time obtained when two Gaussian beams (of either 100 fs or 150 fs pulse duration) superpose, whereas figure 4 shows the pattern for the case of two super-Gaussian beams. Focusing along any one inclined intensity region, one observation in figures 2 and 3 is that the intensity is initially maximum (yellow colour) which gradually falls down (blue colour) and this blue region is laid out till long until the intensity turns zero. Similar observation is made with the super-Gaussian beams, but the rate of fall in the intensity is relatively abruptly to almost zero. It means that the intensity of the pattern in this case is very confined, means the fringes with higher degree of bright dark contrast are obtained with the lasers having wider peak intensity region. Focusing on the maximum intensity regions (bright yellow) in figures 2 -4, similar observation is made that these regions are found to climb upwards leading to a larger area covered by them as the pulse duration is raised from 100 fs to 150 fs. Since the fringe spacing is determined by

𝑀 𝑑 = (𝐼 𝑚𝑎𝑥 -𝐼 𝑚𝑖𝑛 )/ (𝐼 𝑚𝑎𝑥 + 𝐼 𝑚𝑖𝑛 ) = 2(𝐼 1 𝐼 2 ) 1 2 / (𝐼 1 +𝐼 2 ) .
2  𝑘
, it remains the same in all the cases irrespective of the profile of the laser beams and their pulse duration.

Numerical evaluation of modulation depth

To specifically study the fascinating role of the focusing region, we need to precisely calculate and plot the modulation depth. The formulated intensity pattern is used to calculate the energy density using 𝐸 𝑛 (𝑟 ⃗) = ∫ 𝐼(𝑟 ⃗, 𝑡)𝑑𝑡 ∞ -∞ , which finally leads to the modulation depth (say 𝑀 𝑑 ). Using MATLAB, we could build pertinent programs and accurately solved this problem numerically. The said numerical method integrates the function within the specified limits using the global adaptive quadrature rule and error tolerances. An approximate value 𝐼 𝑎𝑝𝑝𝑟𝑜𝑥 of the integral is estimated. This is subtracted from the actual value 𝐼 to obtain the error estimate as 𝜀 = |𝐼 𝑎𝑝𝑝𝑟𝑜𝑥 -𝐼|. If 𝜀 is greater than the desired error tolerance, the interval is divided, and the process is repeated on both the newly created sub-intervals. Finally, the sum of the calculations on each sub-interval is returned.

The numerical calculations reveal that 𝑀 𝑑 is a function of both 𝜏 and  𝜔 . For observation we fix 𝜏 = 75 𝑓𝑠 and the graph is plotted between 𝑀 𝑑 and  𝜔 in figure 5. The horizontal line denotes the modulation depth of 0.45, above which holograms obtained are considered to be of appreciable quality. Interestingly, the vertical line dropped from the point of intersection of 𝑀 𝑑 = 0.45 and the super-Gaussian beam-curve to the X-axis is ahead as compared to the case of the Gaussian beam-curve. Hence, the flexibility for choosing two lasers of significant frequency difference is higher in the case of super-Gaussian beams. This also translates to a higher area under the super-Gaussian beam-curve above 𝑀 𝑑 of 0.45 relative to the area under the Gaussian beam-curve. Hence, the two-colour holography can be easily applied in view of these results, which was necessary for grating recording to improve the holographic sensitivity of the medium [START_REF] Buse | Non-volatile holographic storage in doubly doped lithium niobate crystals[END_REF]. An additional advantage of using super-Gaussian beams is clear from its secondary peak in figure 5 (and figure 9, later), which rises with the lasers having higher index. This means that it would be possible to generate holograms for a higher frequency difference also, which is impossible in the case of Gaussian beams. Now to meticulously uncover the possible reason for these, we delved deeper and found that all of this is related to the focusing region of the beam. As the region of peak intensity of the Gaussian beams is smaller, it leads to a relatively unconfined intensity pattern (figures 2 and 3) and finally a smaller flexibility or a smaller modulation depth. The modulation depth also vanishes at particular values of  𝜔 in the case of super-Gaussian beams. Actually the regions very close to these points correspond to the null regions, as discussed in [START_REF] Malik | Role of the temporal profile of femtosecond lasers of two different colours in holography[END_REF][START_REF] Malik | Dark hollow lasers may be better candidates for holography[END_REF].

Comparison and confirmation by mathematical calculations

Now the intriguing question arises: is the above behaviour due to the widening of peak intensity or due to the fast variation in the rate of rise and fall of lasers intensity and our numerical results are correct? To verify this, we obtain analytical solution to the problem by using cosh 2 profile beams that have the same spread of peak intensity (field) as the super-Gaussian beams but smaller intensity gradient. With the said expression of cosh 2 Putting the limits, we get 𝐼 01 == √𝜋 𝜏 2 (𝑒 0.87 + 1).

Similarly, solving the second integral without limits (say 𝐼 2 ), we get Considering 𝛿 𝑘 ⃗⃗⃗⃗⃗  𝑟 ⃗ = 𝑝, we solve these integrals.

Then putting the limits we obtain From the above expression, we find the modulation depth, as The interference pattern obtained by 100 fs cosh 2 beams is shown in figure 6. A comparison of this figure with the earlier ones infers the better confinement of the intensity (yellow colour) in the case of cosh 2 beams than the Gaussian beams but not the super-Gaussian beams. In view of the profiles of cosh 2 and super-Gaussian beams, this can be concluded that the reason for better interference patters / fringes is the rise and fall in the lasers intensity / field and not the region of their peak intensities.

𝑀 𝑑 = [1+𝑒 0.87 cos(0.935𝛿 𝜔 𝜏)] (𝑒 0.87 +1) 𝑒 -𝛿 𝜔 2 𝜏 2 4 (1)
Lasers with stronger intensity gradients produce precise fringes. This observation further strengthens when we compare figures 7 and 8 for the variation of the modulation depth with the pulse duration and frequency difference for cosh 2 and super-Gaussian beams, respectively. Super-Gaussian beams stay the best as they can produce accurate holograms based on the higher modulation depth for broader frequency difference. Such an observation is more evident from figure 9, where the secondary peak in the modulation depth curve is also found to be higher in the case of super-Gaussian beams. Another observation is that the pulses with smaller durations lead to the wider frequency difference for better holograms, consistent to the work discussed in [START_REF] Malik | Role of the temporal profile of femtosecond lasers of two different colours in holography[END_REF] where the role of temporal profile was talked about in detail. The temporal envelopes of the short pulses have also been processed with the help of spectral holography [START_REF] Oba | Nonvolatile photorefractive spectral holography[END_REF]. Recently ultrashort pulses (attosecond signals) have also been considered for realizing double-blind holography, where the reconstruction of missing spectral phases and the characterization of unknown signals could be possible for a finite pulse duration [START_REF] Pedatzur | Doubleblind holography of attosecond pulses[END_REF]. Since the pulse duration in the present work manipulates the modulation depth, interference pattern and frequency difference (detuning), our proposal might play a vital role in the reconstruction of missing phases and signal characterization in double-blind holography. Above discussion reveals that the better results are obtained for the lasers whose intensity carries stronger gradient. Such lasers have been found to create electron vortices [START_REF] Punia | Generation and regulation of electron vortices in an underdense plasma by Laguerre-Gaussian laser pulses Results[END_REF] and create plasma channels [START_REF] Dwivedi | Dynamics of channel formation and self-focusing of q-Gaussian laser pulse[END_REF] in addition to their self-focusing and frequency shifting [START_REF] Malik | Relativistic self-focusing and frequency shift of super-Gaussian laser beam in plasma Results[END_REF]. The point of greater flexibility in frequency difference in the present proposal can be exploited for the application in radiation generation [START_REF] Malik | Generalized treatment of skew-cosh-Gaussian lasers for bifocal terahertz radiation[END_REF]. . When we plotted this for observing its variation with the frequency difference and pulse duration, it exactly matched with the blue dashed line in figure 5 for 75 fs pulse duration. This confirms that our numerical calculations are accurately done.

The occurrence of points corresponding to particular combinations of 𝜏 and  𝜔 that lead to zero modulation depth in figures 5 and 9 can be explained based on the term [1 + 𝑒 0.87 cos(0.935𝛿 𝜔 𝜏)] in Eq. ( 1). Here cosine term evolves due to the cosh 2 term appearing in the expression of cosh 2 beam profile. Since cosh 2 term is responsible for the spread of lasers peak intensity region, this is evident that the periodic/oscillating nature of 𝑀 𝑑 curves (about points 𝑀 𝑑 = 0) is attributed to this intensity spread only. This is further justified when we look at 𝑀 𝑑 curves for the super-Gaussian beams and the spreading of peak intensity region of these lasers (figure 1). We can also establish a correlation between the interference pattern and the modulation depth by making a comparative study for Gaussian, super-Gaussian and cosh 2 beams. This reveals that the intensity of the pattern in the case of super-Gaussian beams (wider intensity region and higher intensity gradient) is very confined and the fringes with higher degree of bright dark contrast are obtained compared to the cases of Gaussian and cosh 2 beams. Since the highest modulation depth is also obtained in the case of super-Gaussian beams, this is inferred that higher modulation depth is directly related to the confined and better fringe pattern and in this case better holograms can be prepared. On the other hand, this is worth mentioning that the said phenomenon can also be realized with the lasers having spatial profiles in place of the said temporal profiles. In the present case of temporal profile laser beams, the product of "frequency difference  𝜔 and pulse length 𝜏 in time" is important. However, then the product of "wavelength difference   and pulse length L in space" will play the same role.

Conclusions

The interference / intensity pattern and the modulation depth were engineered for the lasers having focused intensity (the Gaussian), lasers with wider peak intensity region and stronger intensity gradient (super-Gaussian), and the lasers with wider peak intensity region but weaker intensity gradient (cosh 2 ) for revealing the remarkable contribution of spreading / widening of the peak intensity in obtaining precise holograms and for advancement of this technique when the two lasers with significant frequency differences are employed. Our numerical and mathematical calculations show that the confined intensity and high-quality fringes of the interference pattern are obtained for the lasers having wider peak intensity region.

The situation becomes superior when the intensity gradient of these lasers is raised and their pulse duration is decreased. The modulation depth also assumes higher values for wider frequency differences, when such laser pulses are used. The enhancement in the lasers intensity gradient also leads to the evolution of secondary region of frequency difference for which appreciable value of the modulation depth can be achieved, which would open new regions for exploring the holography technique and further improve the present holographic measurement methods.

Figure 1 .

 1 Figure 1. Field profiles for cosh 2 beam, super-Gaussian beam of index 4 and Gaussian beam.

  At first we consider two Gaussian beams (n = 2) or super-Gaussian beams (n = 4) having electric fields 𝐸 𝑗 (𝑟 ⃗, 𝑡) = 𝐸 0𝑗 𝑒 -( 𝑡 𝜏 ) 𝑛 cos(𝜔 𝑗 𝑡 -𝑘 𝑗 ⃗⃗⃗⃗  𝑟 ⃗), which on superposition yield the following intensity pattern 𝐼(𝑟 ⃗, 𝑡) = 𝑒 -02 cos(  𝑡 -𝛿 𝑘 ⃗⃗⃗⃗⃗  𝑟 ⃗)]. Here  𝜔 = 𝜔 2 -𝜔 1 is the frequency difference (detuning) and 𝛿 𝑘 ⃗⃗⃗⃗⃗ = 𝑘 2 ⃗⃗⃗⃗⃗ -𝑘 1 ⃗⃗⃗⃗⃗

  𝐸 01 = 𝐸 02 = 4.38 × 10 9 𝑉/𝑚,  𝜔 = 12 × 10 13 𝑟𝑎𝑑/𝑠,  𝑘 = 4 × 10 5 /𝑚 for the lasers having pulse duration of 𝜏 = 100 𝑓𝑠 and 150 𝑓𝑠.

Figure 2 .

 2 Figure 2. Intensity pattern obtained by Gaussian beams at 𝜏 = 100 𝑓𝑠.

Figure 3 .

 3 Figure 3. Intensity pattern obtained by Gaussian beams at 𝜏 = 150 𝑓𝑠.

Figure 4 .

 4 Figure 4. Intensity pattern obtained by super-Gaussian beams at 𝜏 = 100 𝑓𝑠.

Figure 5 .

 5 Figure 5. 𝑀 𝑑 vs  𝜔 for Gaussian beams and super-Gaussian beams at 𝜏 = 75 𝑓𝑠.

  𝑡) 𝑑𝑡 + 𝐶. 𝐼.

2 [ 1 +

 21 𝑒 0.87 cos(0.935𝛿 𝜔 𝜏)] cos(𝛿 𝑘 ⃗⃗⃗⃗⃗  𝑟 ⃗) 𝑒 0.87 cos(0.935𝛿 𝜔 𝜏)] cos(𝛿 𝑘 ⃗⃗⃗⃗⃗  𝑟 ⃗)

Figure 6 .

 6 Figure 6. Intensity pattern obtained by cosh 2 beams at 𝜏 = 100 𝑓𝑠.

Figure 7 .

 7 Figure 7. 𝑀 𝑑 vs   (rad/s) and 𝜏 (s) for cosh 2 beams.

Figure 8 .

 8 Figure 8. 𝑀 𝑑 vs   (rad/s) and 𝜏 (s) for super-Gaussian beams.

Figure 9 .

 9 Figure 9. 𝑀 𝑑 vs   for cosh 2 beams (dotted lines) and super-Gaussian beams (solid lines). IG stands for intensity gradient, IG1 (n = 4) < IG2 (n = 6).

Finally, we𝜏 2

 2 verify the accuracy of our numerical calculations done for the Gaussian and super-Gaussian beams. This is clear that the expression 𝐸(𝑡) = 𝐸 0 cos ℎ 2 (0for cosh 2 beams reduces to that of the Gaussian beams if the cosh 2 term reduces to 1. Using this concept in Eq. (1), modulation depth for the Gaussian beams is obtained as 𝑀 𝑑 = 𝑒 -𝛿 𝜔 2 𝜏 2 4

Acknowledgments

The author LM acknowledges Aix Marseille University, Marseille, France and Indian Institute of Technology Delhi, India for providing opportunity to work.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.