Generation of novel partially coherent truncated Airy beams via Fourier phase processing

Xin Liu; Xin Liu; Dening Xia; Dening Xia; Yashar E. Monfared; Chunhao Liang; Chunhao Liang; Chunhao Liang; Fei Wang; Yangjian Cai; Yangjian Cai; Yangjian Cai; Yangjian Cai; Pujuan Ma; Pujuan Ma; Pujuan Ma

doi:10.1364/OE.390477

1. Introduction

Wave packets normally spread out during their propagation in free space as a result of medium dispersion in the Schrödinger equation [1]. Airy wave packets are particular solutions of the liner Schrödinger equation which can exhibit remarkable non-diffracting, self-accelerating, and self-healing features [2–4]. Based on these prime characteristics, Airy beams have found applications in laser filamentation, super-resolution imaging, materials processing, particle trapping and light bullet generation [3–9]. However, it is impossible to realize an ideal Airy beam in practice as an ideal Airy beam carries infinite energy. Truncated Airy beams were proposed as a practical realization of such beams in the pioneer work of Siviloglou in 2007 [10]. Following by this study, Ngcobo and Porat designed and characterized the truncated Airy beam lasers [11,12]. Such practical Airy beams were achieved via the Fourier transform of a cubic phase modulated Gaussian beam. While this topic attracted a great amount of interest, the beam profile of truncated Airy beams always tends to evolve into a Gaussian-like profile during propagation in the free space [2]. Furthermore, the non-diffracting and self-healing abilities of these so-called fully coherent truncated Airy beams (FCTABs) reduce significantly upon propagation which reduce their attractiveness for a range of applications [3,4,13].

Recently, partially coherent Schell-model beams have found various applications, such as diffractive imaging and particle trapping [14]. Partially coherent truncated Airy beams (PCTABs) were proposed as well [15–17]. The initial studies on PCTABs show that they can exhibit the same prime characteristics of FCTABs like self-accelerating. Additionally, PCTABs can reduce the turbulence-induced negative effects such as beam wander and scintillation on propagation in the turbulent medium [17]. The optical frame can be produced in the far field by partially coherent dual and quad Airy beams [18]. However, it has been shown that the evolution process of the beam intensity profile from Airy distribution to Gaussian-like distribution accelerates as the spatial coherence decreases. Therefore, the performance of PCTABs for realizing non-diffracting and self-healing will be worse than FCTABs which is a limit factor for PCTABs applications [16].

One of the experimental setups which has a great potential to realize NPCTABs in laboratory is 4f optical system. A 4f optical system is used for different applications including optical imaging, phase retrieval and optical measurement [19,20]. In such systems, a pupil at the spatial frequency plane usually plays the role of a low-pass filter by removing the stray light spots [21,22]. Nixon et al. utilized with such modified 4f optical system in a laser cavity to generate a degenerate laser beam which can directly emit the partially coherent beams with controllable coherence [23]. They reported a tunable coherence (from completely incoherent to fully coherent) where the coherence of the output laser beam only depends on the transmission area of the pupil. Moreover, they showed that by adopting different transmittance functions via different masks in the spatial frequency plane, one can effectively modulate the degree of coherence distribution of the output laser beam [24]. In addition, some groups previously reported the possibility of modulation of the incident scalar and vector partially coherent beams through a 4f optical system with an amplitude mask, including the intensity, degree of coherence, polarization, and optical spectrum [25–28]. Surprisingly, only amplitude type masks are considered for these studies to date.

To the best of our knowledge, in all the previous studies PCTABs are considered to have a Gaussian degree of coherence distribution upon propagation in free space. In this manuscript, we propose, for the first time, the generation of NPCTABs by a 4f optical system with a phase screen in its spatial frequency plane. Such beams have Airy-like distributions for both intensity and degree of coherence. Through studies on NPCTABs, we find that their beam properties can be adjusted by varying the coherence width of the source beam and the phase screen parameter. In addition, in comparison to the FCTABs, such novel beams are more robust under propagation in the free space and can maintain their Airy-like profile. Hence, we expect these new beams can find applications in broad range of optical experiments including particle trapping and optical imaging.

2. Theoretical analysis

Figure 1 shows the modified 4f optical system which consists of two lenses (L1 and L2) with the same focal length f. In the spatial frequency plane, a phase screen is placed to modulate and wrap the phase of the incident partially coherent beams. In this setup, the front focal plane of the first thin lens (L1) and the back focal plane of the second thin lens (L2) are treated as the source plane and output plane, respectively. First, we start with the incident partially coherent source in the source plane. In the space-frequency domain, the second-order statistical properties of the partially coherent beams (PCBs) are characterized by the cross-spectral density (CSD) function [29]. For the Schell-model beams, CSD function is defined as

(1)$$\begin{array}{ll} {W_i}({{{\textbf r}_1},{{\textbf r}_2}} )&= {\tau ^\ast }({{{\textbf r}_1}} )\tau ({{{\textbf r}_2}} ){\mu _i}({{{\textbf r}_2} - {{\textbf r}_1}} )\\ \textrm{ } &= {\tau ^\ast }({{{\textbf r}_1}} )\tau ({{{\textbf r}_2}} )\tilde{g}({{{\textbf r}_2} - {{\textbf r}_1}} ), \end{array}$$

here ${{\textbf r}_j} = ({{x_j},{y_j}} ),\textrm{ }j = 1,2$ denotes arbitrary position vectors in the source plane; $\tau ({\textbf r} )$ is a complex amplitude function; The tilde denotes the Fourier transform and ${\mu _i}$ stands for the degree of coherence (DOC) function which can be represented as the Fourier transform of the non-negative function g [29,30]. According to Ref. [31], Eq. (1) can be rewritten as

(2)$${W_i}({{{\textbf r}_1},{{\textbf r}_2}} )= \left\langle {{\Gamma ^\ast }({{{\textbf r}_{\textbf 1}}} )\times \Gamma ({{{\textbf r}_{\textbf 2}}} )} \right\rangle ,$$

with

(3)$$\Gamma ({\textbf r} )= \tau ({\textbf r} )\times {F_T}\left[ {\sqrt g \times {C_n}} \right],$$

where ${F_T}$ indicates Fourier transform. Here ${C_n}$ is circular complex Gaussian random variable whose probability density functions of the intensity and phase obey the negative exponent and uniform distribution in the interval of (0, 2${\pi }$), respectively [32]. Note that $\Gamma ({\textbf r} )$ is the random electric field function of the PCBs. The CSD function of the PCBs in the output and source planes are related through the following integral [33]

(4)$${W_o}({{{\boldsymbol{\mathrm{\rho}}}_1},{{\boldsymbol{\mathrm{\rho}} }_2}} )= \int {W({{{\textbf r}_1},{{\textbf r}_2}} ){h^\ast }({{{\textbf r}_1},{{\boldsymbol{\mathrm{\rho}}}_1}} )h({{{\textbf r}_2},{{\boldsymbol{\mathrm{\rho}}}_2}} ){d^2}{{\textbf r}_1}{d^2}{{\textbf r}_2}} ,$$

where ${{\boldsymbol{\mathrm{\rho}}}_1}$ and ${{\boldsymbol{\mathrm{\rho}}}_2}$ are two arbitrary position vectors in the output plane. $h({{\textbf r},{\boldsymbol{\mathrm{\rho}}}} )$ is an impulse function of the optical system described in Fig. 1, which can be defined as [33]

(5)$$\begin{array}{ll} h({{\textbf r},{\boldsymbol{\mathrm{\rho}}}} )&={-} \frac{1}{{{\lambda ^2}{f^2}}}\int {P({\boldsymbol{\mathrm{\xi}}} )\exp \left[ { - \frac{{ik}}{f}({{\textbf r} - {\boldsymbol{\mathrm{\rho}}}} ){\boldsymbol{\mathrm{\xi}}}} \right]{d^2}{\boldsymbol{\mathrm{\xi}}}} \\ &={-} \frac{1}{{{\lambda ^2}{f^2}}}\tilde{P}[{{{({{\textbf r} - {\boldsymbol{\mathrm{\rho}}}} )} \mathord{\left/ {\vphantom {{({{\textbf r} - {\boldsymbol{\mathrm{\rho}}}} )} {\lambda f}}} \right.} {\lambda f}}} ], \end{array}$$

here ${\boldsymbol{\mathrm{\xi}}} = ({{\xi_x},{\xi_y}} )$ is the transverse coordinate in the spatial frequency plane. $P({\boldsymbol{\mathrm{\xi}}} )$ is used to characterize the phase distribution of the phase screen and $k = {{2\pi } \mathord{\left/ {\vphantom {{2\pi } \lambda }} \right.} \lambda }$ is the wave number with the carrier wavelength $\lambda$.

Fig. 1. Modified 4f optical system which consists of the source plane, output plane and two lenses L1 and L2 with the same focal length f. A phase screen is located at the center of the spatial frequency plane.

Download Full Size | PDF

By combining Eqs. (2), (3) and (5) and replacing them into Eq. (4), we can obtain the CSD function of the PCBs in the output plane as

(6)$${W_o}({{{\boldsymbol{\mathrm{\rho}}}_{\textbf 1}},{{\boldsymbol{\mathrm{\rho}}}_{\textbf 2}}} )= \left\langle {{\eta^\ast }({{{\boldsymbol{\mathrm{\rho}}}_{\textbf 1}}} )\times \eta ({{{\boldsymbol{\mathrm{\rho}}}_{\textbf 2}}} )} \right\rangle ,$$

where

(7)$$\eta ({\boldsymbol{\mathrm{\rho}}} )= \Gamma ^{\prime}({\boldsymbol{\mathrm{\rho}}} )\otimes h({\boldsymbol{\mathrm{\rho}}} ),$$

Here we assume $\Gamma ^{\prime}({\textbf r} )= \Gamma ({ - {\textbf r}} )$, and “⊗” denotes convolution. After characterizing the generated beams, we now study the beam properties during propagation in free space using the following Huygens-Fresnel integral,

(8)$$\begin{array}{ll} T({{\textbf v},z} )&= ({ - {i \mathord{\left/ {\vphantom {i {\lambda z}}} \right.} {\lambda z}}} )\exp ({{{i\pi {{\textbf v}^2}} \mathord{\left/ {\vphantom {{i\pi {{\textbf v}^2}} {\lambda z}}} \right.} {\lambda z}}} )\int {R({\boldsymbol{\mathrm{\rho}}} )\exp ({ - {{i2\pi {\boldsymbol{\mathrm{\rho}}} \cdot {\textbf v}} \mathord{\left/ {\vphantom {{i2\pi {\boldsymbol{\mathrm{\rho}}} \cdot {\textbf v}} {\lambda z}}} \right.} {\lambda z}}} )} {d^2}{\boldsymbol{\mathrm{\rho}}}\\ \textrm{ } &= ({ - {i \mathord{\left/ {\vphantom {i {\lambda z}}} \right.} {\lambda z}}} )\exp ({{{i\pi {{\textbf v}^2}} \mathord{\left/ {\vphantom {{i\pi {{\textbf v}^2}} {\lambda z}}} \right.} {\lambda z}}} )\tilde{R}({{{\textbf v} \mathord{\left/ {\vphantom {{\textbf v} {\lambda z}}} \right.} {\lambda z}}} ), \end{array}$$

where $R({\boldsymbol{\mathrm{\rho}}} )= \eta ({\boldsymbol{\mathrm{\rho}}} )\exp ({{{i\pi {{\boldsymbol{\mathrm{\rho}}}^2}} \mathord{\left/ {\vphantom {{i\pi {{\boldsymbol{\mathrm{\rho}}}^2}} {\lambda z}}} \right.} {\lambda z}}} )$ and v = (v_x,v_y) is an arbitrary position vector in the receiver plane at propagation distance z. According to above equation, we can obtain the CSD function of such beams in the receiver plane with a distance separation from output plane by ${W_z}({{{\textbf v}_1},{{\textbf v}_2}} )= \left\langle {{T^\ast }({{{\textbf v}_1},z} )\times T({{{\textbf v}_2},z} )} \right\rangle$. Usually, we can simply obtain the intensity and DOC functions based on the CSD function. However, one should use caution in this case due to the circular complex Gaussian random variables. Here we utilized a mass of realizations of speckle patterns to calculate the intensity and the DOC distributions for the statistically stationary beams. For example in the output plane, for the intensity ${I_o}({\boldsymbol{\mathrm{\rho}}} )\approx \frac{1}{M}\sum\limits_{i = 0}^M {{S_{oi}}({\boldsymbol{\mathrm{\rho}}} )}$ and ${S_{oi}}({\boldsymbol{\mathrm{\rho}}} )= {|{{\eta_i}({\boldsymbol{\mathrm{\rho}}} )} |^2}$ for the DOC, ${|{{\mu_o}({{{\boldsymbol{\mathrm{\rho}}}_1},{{\boldsymbol{\mathrm{\rho}}}_2}} )} |^2} \approx \left\{ {{{\sum\limits_{i = 1}^M {[{{S_{oi}}({{{\boldsymbol{\mathrm{\rho}}}_1}} ){S_{oi}}({{{\boldsymbol{\mathrm{\rho}}}_2}} )} ]}} \mathord{\left/ {\vphantom {{\sum\limits_{i = 1}^M {[{{S_{oi}}({{{\boldsymbol{\mathrm{\rho}}}_1}} ){S_{oi}}({{{\boldsymbol{\mathrm{\rho}}}_2}} )} ]}} {M{I_o}({{{\boldsymbol{\mathrm{\rho}}}_1}} ){I_o}({{{\boldsymbol{\mathrm{\rho}}}_2}} )}}} \right.} {M{I_o}({{{\boldsymbol{\mathrm{\rho}}}_1}} ){I_o}({{{\boldsymbol{\mathrm{\rho}}}_2}} )}}} \right\} - 1$[31]. The equations provided here also work for the intensity and DOC in the receiver plane. Note the number of realizations M should be large enough to make the approximation valid.

Based on the derived formulas, we will explore the effect of the phase modulation in the spatial frequency plane on the incident PCBs. In particular, the generation of the NPCTABs through Fourier phase processing will be examined. Furthermore, the propagation properties of such beams will be compared to FCTABs and the role of the initial coherence width and the phase parameter will be studied in detail.

3. Generation of novel partially coherent truncated Airy beams (NPCTABs)

In this paper, we used a Gaussian Schell-Model (GSM) beam as the incident beam. Therefore, the corresponding functions g and $\tau$ in Eq. (1) for the beams are given by

(9)$$g({\boldsymbol{\mathrm{\kappa}}} )= 2\pi {\delta ^2}\exp ({ - 2{\pi^2}{\delta^2}{{\boldsymbol{\kappa}}^2}} ),$$

and

(10)$$\tau ({\textbf r} )= \exp ({ - {{{{\textbf r}^2}} \mathord{\left/ {\vphantom {{{{\textbf r}^2}} {\omega_0^2}}} \right.} {\omega_0^2}}} ),$$

where $\delta$ and ${\omega _0}$ are the source coherence width and beam width, respectively. The phase function of the phase screen in the spatial frequency plane is assumed to be

(11)$$P({\boldsymbol{\mathrm{\xi}}} )= \exp \{{ - ik[{{{({a{\xi_x}} )}^3} + {{({a{\xi_y}} )}^3}} ]- i\Phi } \}.$$

where a is a constant with a unit of m^−2/3 and $\Phi $ is an insignificant phase constant caused by the thickness of the phase screen. We ignore the effect of the phase constant $\Phi $ in our simulations for simplicity. The values of the focal length, excitation wavelength and beam width in the simulations are considered to be $f = 150\textrm{mm}$, $\lambda \textrm{ = 532nm}$ and ${\omega _0} = 1\textrm{mm}$, respectively.

In Fig. 2, we show the density plots of the normalized intensity distributions and the square of the modulus of the degree of coherence distributions for different values of the initial coherence width $\delta$. The value of the parameter a of the phase screen is assumed to be a = 350m^−2/3. As seen in Figs. 2(a1)–2(d1), intensity distributions show the Airy-like profiles for different initial coherence width $\delta$ ranging from infinity (FCTABs) to 0.25mm (NPCTABs). One finds that for the FCTABs in Fig. 2(a1), the cross line of intensity distribution (white curve) displays strong oscillations with the intensity dropping to zero between successive oscillations. As the initial coherence width $\delta$ of the incident GSM beam decreases, the oscillations progressively weaken and produce long tails along vertical and horizontal axes as displayed in Figs. 2(b1)-(d1). Moreover, we notice that the Airy-like distributions of the square of the modulus of the DOC appear as the initial coherence width decreases. As seen in Figs. 2(b2)–2(d2), lower coherence width produces a more refined structure for DOC distributions.

Fig. 2. Density plots of the intensity distribution (a1-d1) and the square of the modulus of the DOC distribution (a2-d2) of the NPCTABs in the output plane with a = 350m^−2/3 for different values of the initial coherence width $\delta$ of the incident GSM beam, where (a1) and (a2) $\delta = \textrm{infinity}$; (b1) and (b2) $\delta = 1\textrm{mm}$; (c1) and (c2) $\delta = 0.5\textrm{mm}$; (d1) and (d2) $\delta = 0.25\textrm{mm}$. Note that the intensity distributions are normalized to the peak intensity. The corresponding cross lines are shown as white solid curves.

Download Full Size | PDF

In Fig. 3, we demonstrate the effect of the parameter a of the phase screen on the generated NPCTABs properties. For these simulations, the initial coherence width $\delta$ is set to be $\delta = 1\textrm{mm}$. It can be clearly inferred from the Fig. 3 that the number of oscillations of the intensity increases by an increase in the value of a. Therefore, one can conclude that the parameter a works as the truncation factor for FCTABs [13]. Finally, the DOC distributions become more uniform by an increase in parameter a.

Fig. 3. Density plots of the intensity distribution (a1-d1) and the square of the modulus of the DOC distribution (a2-d2) of the NPCTABs in the output plane with the initial coherence width $\delta = 1\textrm{mm}$ for different values of the phase screen parameter a, where (a1) a = 250m^−2/3, (b1) a = 300m^−2/3, (c1) a = 350m^−2/3, (d1) a = 400m^−2/3. The corresponding cross lines are shown as white solid curves.

Download Full Size | PDF

Overall, these results indicate that NPCTABs show Airy-like distributions for the intensity and the square of the modulus of DOC in the output plane through Fourier phase processing. The numerical results also imply that the intensity distribution and the square of the modulus of DOC distribution of the NPCTABs can be easily controlled by adjusting the initial coherence width $\delta$ and the parameter a of the phase screen.

4. Propagation properties of NPCTABs

FCTABs exhibit parabolic self-deflecting properties during propagation and their transverse profile gradually evolves into a Gaussian-like profile [2]. The self-deflecting can be characterized by the deflection of the largest peak. An interesting question here is that what will happen to our proposed NPCTABs during propagation in the free space. To provide a response to this fundamental question, we plot of the deflection of such novel beams as a function of the propagation distance z and with the different values of a and $\delta$ in Fig. 4.

Fig. 4. Deflection of the largest peak of the NPCTABs as a function of the propagation distance z during propagation in free space for different values of the parameter a of phase screen and the source coherence width $\delta$.

Download Full Size | PDF

We start to analyze the results by considering the case of FCTABs. In the inset of Fig. 4, we display almost perfect agreement between the numerical simulations for deflection curves of the FCTABs during propagation (shown in a square black curve) and the corresponding theoretical fitting (shown in a solid green curve) achieved by Eq. (21) in Ref. [34]. One can observe a few fluctuations starting from around z = 60m away from the output plane due to the emergence of competition between secondary and main peaks of the beam intensity, because the beam intensity will gradually evolve into Gaussian-like profile during propagation which is also demonstrated in Fig. 5(b). The results for the case of FCTABs are shown in the main plot (square black).

Fig. 5. Normalized intensity of the NPCTABs during propagation in free space with a = 250m^−2/3 and different values of the initial coherence width $\delta$ in (a) the output plane and (b) the receiver plane at z = 60 m. Note that the beam intensity is normalized by its peak intensity.

Download Full Size | PDF

In the case of NPCTABs (red and blue curves), one can clearly see that the fluctuations starting point in the deflection curve require longer propagation distances. These fluctuations appear even at longer propagation distances for the lower initial coherence width $\delta $. These results imply that Airy-like distribution of the NPCTABs with a lower coherence width $\delta$ can maintain for a longer distance. Further, the position of the maximum value will deviate from its original trajectory to the optical axis as the beam intensity distribution changes from Airy profile to Gaussian-like distribution on propagation. Hence, we find the acceleration of a NPCTAB becomes larger for a lower source coherence $\delta$. In addition, we study the effect of the parameter a on the deflection of NPCTABs during propagation. We plot the deflection curves of the NPCTABs, with the initial coherence width $\delta = 0.5\textrm{mm}$ for different values of the parameter a where $a = 250{\textrm{m}^{\textrm{ - 2/3}}}$ (shown in dotted blue curves), $a = \textrm{30}0{\textrm{m}^{\textrm{ - 2/3}}}$ (shown in dash-dot blue curves), and $a = \textrm{3}50{\textrm{m}^{\textrm{ - 2/3}}}$ (shown in solid blue curves), as a function of the propagation distance z. The results reveal that the deflection of NPCTABs on propagation also depends on the parameter a. By increasing a, the deflection becomes smaller in magnitude as seen in Fig. 4. The evolution of the normalized intensity profile of NPCTABs during propagation in free space compared to FCTABs are illustrated in Fig. 5. The intensity distributions of NPCTABs with different coherence width at z = 0 (in the output plane) and z = 60 m (at a distance from output plane) are demonstrated in Fig. 5(a) and Fig. 5(b), respectively. Note the solid curves are achieved by the numerical simulation, and the dotted green curves are obtained by the analytical expression for the FCTABs (Eqs. (7) and (9) in Ref. [34]).

As seen in Fig. 5, the intensity distribution of the FCTABs gradually evolves from Airy profile [Fig. 5(a)] into a Gaussian-like profile [Fig. 5(b)]. Interestingly, NPCTABs with lower coherence width demonstrate weaker oscillations, longer tails and as a result, higher stability during propagation in the output plane. When such beams propagate to z = 60 m, we can infer from Fig. 5(b) that the Airy-like intensity distributions preserve its shape for the case of low coherence widths in comparison with the fully coherent case. This phenomenon can be explained by reciprocity between their intensity and coherence properties where the intensity is affected by the source DOC on propagation [35]. We want to stress that this fact is in sharp contrast to the expected results for PCTABs with a Gaussian profile DOC distribution [16]. Therefore, we can conclude that NPCTABs are more stable than the FCTABs during propagation in the free space, and their non-diffracting and self-healing abilities better than that of FCTABs can be anticipated.

5. Conclusion

We have explored, for the first time to our best knowledge, the generation of the novel partially coherent truncated Airy beams (NPCTABs) through Fourier phase processing. Such beams display Airy-like profiles for both intensity and degree of coherence distributions. We investigate the effect of the initial coherence width and the phase screen on the beam properties of NPCTABs. We reveal the crucial role of source coherence in a way that a reduction in source coherence width can result in increasing the number and level of side lobes for the intensity distribution and weaker corresponding intensity fluctuations. The lower coherence produces a more refined structure for DOC distributions. Moreover, the parameter a which is related to phase screen can be used as the truncation factor for NPCTABs. Therefore, the phase screen parameter determines the number of the side lobes for both intensity and degree of coherence distributions. Finally, we study propagation properties of such novel beams in free space in comparison with the fully coherent truncated Airy beams. The derived results illustrate that NPCTABs are more stable than the fully coherent ones in free space and can preserve their Airy-like profiles for longer propagation distance. As a result, we believe the self-healing property of such NPCTABs is better than one of fully coherent truncated Airy beams or conventional partially coherent truncated Airy beams as well. The proposed novel beams with tunable properties may find different applications including particle trapping and optical imaging.

Funding

National Key Research and Development Program of China (2019YFA0705000); National Natural Science Foundation of China (11525418, 11874046, 11947239, 11974218, 91750201); Innovation group of Jinan (2018GXRC010); China Postdoctoral Science Foundation (2019M662424).

Disclosures

The authors declare no conflicts of interest.

References

1. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979). [CrossRef]

2. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007). [CrossRef]

3. N. K. Efremidis, Z. Chen, M. Segev, and D. N. Christodoulides, “Airy beams and accelerating waves: an overview of recent advances,” Optica 6(5), 686–701 (2019). [CrossRef]

4. Y. Zhang, H. Zhong, M. R. Belić, and Y. Zhang, “Guided self-accelerating Airy beams—a mini-review,” Appl. Sci. 7(4), 341 (2017). [CrossRef]

5. P. Polynkin, M. Kolesik, and J. Moloney, “Filamentation of femtosecond laser Airy beams in water,” Phys. Rev. Lett. 103(12), 123902 (2009). [CrossRef]

6. S. Jia, J. C. Vaughan, and X. Zhuang, “Isotropic three dimensional super-resolution imaging with a self-bending point spread function,” Nat. Photonics 8(4), 302–306 (2014). [CrossRef]

7. A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. Lacourt, and J. Dudley, “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101(7), 071110 (2012). [CrossRef]

8. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wave packets,” Nat. Photonics 2(11), 675–678 (2008). [CrossRef]

9. A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010). [CrossRef]

10. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007). [CrossRef]

11. S. Ngcobo, I. Litvin, L. Burger, and A. Forbes, “A digital laser for on-demand laser modes,” Nat. Commun. 4(1), 2289 (2013). [CrossRef]

12. G. Porat, I. Dolev, O. Barlev, and A. Arie, “Airy beam laser,” Opt. Lett. 36(20), 4119–4121 (2011). [CrossRef]

13. X. Chu, G. Zhou, and R. Chen, “Analytical study of the self-healing property of Airy beams,” Phys. Rev. A 85(1), 013815 (2012). [CrossRef]

14. X. Lu, Y. Shao, C. Zhao, S. Konijnenberg, X. Zhu, Y. Tang, Y. Cai, and H. P. Urbach, “Noniterative spatially partially coherent diffractive imaging using pinhole array mask,” Adv. Photonics 1(1), 016005 (2019). [CrossRef]

15. Y. Lumer, Y. Liang, R. Schley, I. Kaminer, E. Greenfield, D. Song, X. Zhang, J. Xu, Z. Chen, and M. Segev, “Incoherent self-accelerating beams,” Optica 2(10), 886–892 (2015). [CrossRef]

16. S. Cui, Z. Chen, K. Hu, and J. Pu, “Investigation on partially coherent Airy beams and their propagation,” Acta. Phys. Sin. 62(9), 094205 (2013). [CrossRef]

17. W. Wen and X. Chu, “Beam wander of partially coherent Airy beams,” J. Mod. Opt. 61(5), 379–384 (2014). [CrossRef]

18. Z. Pang and D. Zhao, “Partially coherent dual and quad Airy beams,” Opt. Lett. 44(19), 4889–4892 (2019). [CrossRef]

19. G. Boudebs and S. Cherukulappurath, “Nonlinear optical measurements using a 4f coherent imaging system with phase objects,” Phys. Rev. A 69(5), 053813 (2004). [CrossRef]

20. W. Han, W. Cheng, and Q. Zhan, “Design and alignment strategies of 4f systems used in the vectorial optical field generator,” Appl. Opt. 54(9), 2275–2278 (2015). [CrossRef]

21. C. Ping, C. Liang, F. Wang, and Y. Cai, “Radially polarized multi-Gaussian Schell-model beam and its tight focusing properties,” Opt. Express 25(26), 32475–32490 (2017). [CrossRef]

22. C. Liang, G. Wu, F. Wang, W. Li, Y. Cai, and S. A. Ponomarenko, “Overcoming the classical Rayleigh diffraction limit by controlling two-point correlations of partially coherent light sources,” Opt. Express 25(23), 28352–28362 (2017). [CrossRef]

23. M. Nixon, B. Redding, A. A. Friesem, H. Cao, and N. Davidson, “Efficient method for controlling the spatial coherence of a laser,” Opt. Lett. 38(19), 3858–3861 (2013). [CrossRef]

24. R. Chriki, M. Nixon, V. Pal, C. Tradonsky, G. Barach, A. A. Friesem, and N. Davidson, “Manipulating the spatial coherence of a laser source,” Opt. Express 23(10), 12989–12997 (2015). [CrossRef]

25. T. D. Visser, G. P. Agrawal, and P. W. Milonni, “Fourier processing with partially coherent fields,” Opt. Lett. 42(22), 4600–4602 (2017). [CrossRef]

26. T. Wu, C. Liang, F. Wang, and Y. Cai, “Shaping the intensity and degree of coherence of a partially coherent beam by a 4f optical system with an amplitude filter,” J. Opt. 19(12), 124010 (2017). [CrossRef]

27. X. Zhao, T. D. Visser, and G. P. Agrawal, “Controlling the degree of polarization of partially coherent electromagnetic beams with lenses,” Opt. Lett. 43(10), 2344–2347 (2018). [CrossRef]

28. C. Mi, C. Liang, F. Wang, L. Liu, Y. Gao, and Y. Cai, “Modulating the statistical properties of a vector partially coherent beam by a 4f optical system,” J. Quant. Spectrosc. Radiat. Transfer 222-223, 145–153 (2019). [CrossRef]

29. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007). [CrossRef]

30. C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine- Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014). [CrossRef]

31. P. Ma, B. Kacerovská, R. Khosravi, C. Liang, J. Zeng, X. Peng, C. Mi, Y. E. Monfared, Y. Zhang, F. Wang, and Y. Cai, “Numerical approach for studying the evolution of the degrees of coherence of partially coherent beams Propagation through an ABCD Optical System,” Appl. Sci. 9(10), 2084 (2019). [CrossRef]

32. J. C. Dainty, ed., Laser speckle and related phenomena (Springer-Verlag, New York, 1984).

33. D. Xu, X. Song, H. Li, D. Zhang, H. Wang, J. Xiong, and K. Wang, “Experimental observation of sub-Rayleigh quantum imaging with a two-photon entangled source,” Appl. Phys. Lett. 106(17), 171104 (2015). [CrossRef]

34. J. E. Morris, M. Mazilu, J. Baumgartl, T. Čižmár, and K. Dholakia, “Propagation characteristics of Airy beams: dependence upon spatial coherence and wavelength,” Opt. Express 17(15), 13236–13245 (2009). [CrossRef]

35. L. Ma and S. A. Ponomarenko, “Free-space propagation of optical coherence lattices and periodicity reciprocity,” Opt. Express 23(2), 1848–1856 (2015). [CrossRef]

Generation of novel partially coherent truncated Airy beams via Fourier phase processing

Abstract

1. Introduction

2. Theoretical analysis

3. Generation of novel partially coherent truncated Airy beams (NPCTABs)

4. Propagation properties of NPCTABs

5. Conclusion

Funding

Disclosures

References

Cited By

Figures (5)

Equations (11)

Optics Express