The diffraction of light is a basic optical phenomenon subject to a long history of study and research. As is well-known, when light impinges upon a particle, an obstacle, a grating, and so on, complex diffraction occurs to take light in a significant departure from its original transport path. Moreover, for a grating involving a periodic microstructure, another distinct physical effect, the interference of light, takes place simultaneously. In the regime of linear optics, these linear optical diffraction (LOD) phenomena, rules, and laws have been well established^[1–3]. Recent studies have demonstrated that when a particle, obstacle, or grating have both linear and nonlinear optical responses to the incident laser beam of considerable intensity and power, various nonlinear optical diffraction (NOD) phenomena and processes take place, with nonlinear Raman–Nath diffraction^[4–6] and nonlinear Cherenkov radiation (NCR)^[7–9] of second-harmonic generation (SHG) being some prominent examples.

White lights involve a continuous band of colored light, say, red, orange, yellow, malachite, green, bluish, and violet light for visible light. They can be generated in several ways, including thermal radiation, arc discharge, electroluminescence, supercontinuum laser light^[10–15], and femtosecond white lasers^[16–20]. When sunlight, the most ordinary white light originating from thermal radiation, impinges upon a regular reflective metal diffraction grating, this grating will be subject to the LOD effect and serve as a powerful optical dispersion device to separate different colored light and redistribute them in space, forming a rainbow-like pattern. It is interesting to examine what happens when different white lights shine on a diffraction grating. Also, it is interesting to see what happens if the grating has a nonlinear optical response in addition to a linear optical response, which is called a dual linear–nonlinear grating. In this work, we consider four types of white light sources: ordinary halogen lamp (WL1), NKT supercontinuum laser (WL2), femtosecond white laser obtained by sending a Ti:sapphire femtosecond pulse laser beam through a fused silica plate (WL3), and femtosecond white laser obtained by sending a Ti:sapphire femtosecond pulse laser beam through a cascaded fused silica plate and chirped periodically poled lithium niobate (CPPLN) module (WL4). We shine them upon a PPLN dual linear–nonlinear thin-plate grating with weak surface corrugation and perform a systematic comparison examination on their LOD and NOD properties.

The experimental setup for the LOD and NOD experiment of the PPLN nonlinear grating is schematically illustrated in Fig. 1(a). After focusing, the white light beam is incident upon the transparent PPLN nonlinear grating and a reference reflective Au linear grating, and then the transmission and reflection diffraction patterns are projected upon an observation screen, observed by naked eyes, and recorded by a digital camera. In addition, the spectrum of these four white light beams is analyzed by an ordinary visible band spectrometer (Ocean Optics, USB4000) with very low detection efficiency above 900 nm.

Figure 1(b) displays the photograph of the naked-eye white light spot from four different pump light sources, WL1–WL4, from left to right. The ordinary halogen lamp light beam (WL1) shows a gentle and warm white color and a nearly circular shape in naked-eye view; the NKT photonic crystal glass-fiber supercontinuum source (WL2) exhibits a shining yellow-red color and irregular shape. In comparison, the two femtosecond white laser sources display shining and cold white colors and nearly perfect circular shape surrounded by a magenta-blue (WL3) and blue-violet (WL4) diffraction halo. Overall, WL1 and WL2 are warm white light sources, while WL3 and WL4 are cold white light sources.

The reflective gold grating has 1200 lines/mm and a period of 833 nm and serves as a good visible band dispersion device. The PPLN nonlinear grating [Fig. 1(c)] is a 1-mm-thick LN plate with 3.48 µm positive (up arrows) and 3.48 µm negative (down arrows) nonlinear polarization domains consecutively and periodically arrayed along a horizontal direction (say, x axis) and perpendicular to the pump laser direction (z axis). Besides, there exists very shallow (67 nm in depth) surface corrugation on one surface, meaning that this PPLN plate is a dual linear-nonlinear diffraction grating with a period of 6.96 µm, which might involve fruitful LOD and NOD phenomena.

Figures 2(a)–2(c) illustrate the zeroth and first-order LOD patterns for the WL1 beam (with a power of 48.7 mW and a spot diameter of around 8 mm at the point of incidence) transmitting through the PPLN grating and reflected by the gold grating, and the measured spectral profile. Figures 2(d)–2(f) illustrate the LOD pattern for the WL2 beam (with a power of 52.3 mW and a spot diameter of around 2.5 mm at the point of incidence) transmitting through the PPLN grating and reflected by the gold grating, and the measured spectral profile. The WL1 has a Gaussian-like spectral profile centered at 650 nm, and a 10 dB bandwidth ranging from 470 to 917 nm, while the WL2 has a double-peak spectral profile centered at 540 and 720 nm, respectively, and a 10 dB bandwidth from 500 to 929 nm. Thus, WL1 is a more balanced white source than WL2. The measured spectral profiles agree well with the dispersion pattern analyzed by the gold diffraction grating displayed in Figs. 2(b) (with apparent balanced red, yellow, green, and blue colors) and 2(e) (only with limited red, yellow, and green colors). The external angle of the diffraction beams in the grating satisfies the well-known Bragg’s law in the form of dsinθ=mλ, m=0,±1,±2…, where d is the period of the grating and m is the diffraction order. This indicates that the primary peaks of different wavelengths at the same order emerge in different directions, with the diffraction angle being larger for the long wave and smaller for the short wave. This analysis is fully confirmed by the prominent dispersive features shown in Figs. 2(b) and 2(e).

The diffraction patterns for WL1 and WL2 passing through the large-period (about 10 times the average incident wavelength) shallow grating are distinctly different, in that WL1 only has a series of Bragg scattering and diffraction spots expanding to connect each other, having gentle brightness and weak color dispersion, while WL2 shows a series of concentrated and bright Bragg scattering and diffraction spots, with each spot having a smooth color dispersion close to that displayed in Fig. 2(e), gradually changing from red to green. The contrast can be ascribed to the bad coherence of WL1 (thermal radiation source) and the good coherence of WL2 (supercontinuum laser source). WL2 is subject to double-fold coherent Bragg scattering and dispersive diffraction, while WL1 is only subject to bad-collimation-induced imperfect Bragg scattering. This contrast also discloses the important role of light coherence in shaping the ordinary LOD properties. Another notable feature of the LOD patterns from Figs. 2(d) and 2(e) is that, within the receiving field of view, the larger-period PPLN nonlinear grating allows for the observation of more diffraction order patterns accompanied by larger automatic dispersion in space, further fully verifying the diffraction characteristics of the grating in terms of the period d, diffraction order m, and wavelength λ.

Now we proceed to see what happens when femtosecond pulse white lasers shine upon the PPLN dual linear–nonlinear grating, which should have a much higher peak power and intensity than the NKT nanosecond laser source. Figures 3 and 4 display the observed diffraction pattern for the WL3 femtosecond pulse white laser of two different power levels passing through the transparent PPLN grating. WL3 is generated by sending a homemade Ti:sapphire femtosecond pulse laser (Fogbow-HP, Guangdong Jingqi) beam with 800 nm center wavelength, 2.78 mJ pulse energy, 1 kHz repetition rate, and 50 fs pulse duration, through an 8-mm-thick fused silica plate^[19]. Due to the significant third-order nonlinear (3rd-NL) self-phase modulation (SPM) effect^[17–20], a supercontinuum laser with significantly broadened spectral bandwidth is output, as witnessed by Figs. 3(d) and 3(e). The SPM effect is a nonlinear optical effect that arises during the transmission of an optical signal through a nonlinear medium. As a light pulse travels through a medium, the high-intensity portion of the light field causes a slight variation in the medium’s refractive index over time or space. This refractive index alteration results in phase distortion of the light, subsequently impacting the spectral properties of the light. As expected, Fig. 3(e) shows a 10 dB bandwidth of 434–864 nm, which is about 10 times the pumped 800 nm femtosecond laser, while Fig. 3(d) shows the color band pattern analyzed by the reflective gold grating, which involves continuously visible color bands varying from purple to red but with obviously uneven intensity. The spectral profile in Fig. 3(e) further indicates that the spectral peak is still located at the 800 nm pump laser center wavelength, although a larger fraction of energy is converted and transferred to shorter and longer wavelength parts.

In our experiment, the WL3 femtosecond white laser has a maximum pulse energy of 1.72 mJ. We first focus the WL3 laser beam with full power, shine it upon the PPLN grating (with an incident spot diameter of about 8 mm), and examine both the LOD and NOD patterns. Figure 3(a) illustrates the full-scenery photograph of the optical diffraction pattern, which consists of a number of bright and sharp colored Bragg scattering and dispersive diffraction spots in the center region of the observation screen, and a weak but definitely apparent blue-violet diffraction pattern across a broad spatial range in the far side of the observation screen. The enlarged pictures are displayed in Figs. 3(b) and 3(c), respectively, where more details can be seen and allow for deeper analysis. The former is categorized as the LOD of WL3 beam against the large-period shallow surface-corrugation PPLN linear grating [showing similar LOD features as Fig. 2(d)], while the latter is recognized as the NCR, originating from the SHG interaction of the pump femtosecond white laser with the PPLN nonlinear grating^[9,21,22]. The emitted angle of NCR radiation is measured to be approximately 49° to 79°, which aligns well with the purple band of NCR SHG, ranging from 455 to 375 nm.

We then place a diaphragm to allow the WL3 beam with a smaller pulse energy of 0.8 mJ to pass and shine on the PPLN nonlinear grating. The observed LOD and NOD patterns are displayed in Fig. 4. The LOD patterns illustrated in Figs. 4(a), 4(b), and 4(d) are similar to Figs. 3(a), 3(b) and 3(d), only with an almost linear reduction in brightness. In contrast, the NCR pattern, shown in Fig. 4(c), changes very much in the profile; besides, the intensity is no longer subject to a linear reduction but rather to a square-law reduction compared with Fig. 3(c). This observation is in agreement with the fact that NCR is intrinsically a second-order nonlinear optical process (i.e., SHG) and its diffraction into various longitudinal phase matching orientations that satisfy the condition of 2k1z=k2z, where k1z and k2z are the wave vectors of the pump laser and SHG beam projected along the pump light direction (z axis)^[9]. The NCR radiation angle decreases to around 49°–75°, due to the weakening of the pump laser intensity. Further, we gradually decrease the white laser total energy from 0.5 to 0.12 W to observe and determine the peak power threshold of nonlinear diffraction of the PPLN nonlinear grating, as shown in Figs. 4(e)–4(h). The peak intensity of the pump white laser corresponds to 21–88GW/cm2, respectively. It can be observed that the NCR intensity gradually decreases as the peak power density diminishes, eventually fading away, as seen in the case of 0.12 W input, corresponding to the peak power threshold of 21GW/cm2. Overall, the PPLN nonlinear grating enables dual-mode diffraction phenomena through its hybrid architecture: the LOD arises from the 67 nm shallow surface corrugation (period ∼6.96μm), producing sub-40° angular dispersion with linear power dependence, while the NCR originates from the 3.48 µm periodic nonlinear domains, generating 49°–79° angular dispersion in visible light (455–375 nm) via ultrafast pulse-driven noncollinear SHG in domain wall regions. This distinction highlights two fundamental differences between linear and nonlinear diffractions: 1) structural origin—Bragg-transmission-dominated linear diffraction relies on a shallow surface grating of a depth only 67 nm to induce directional coherent scattering, whereas NCR emerges from internal nonlinear polarization domains of the PPLN nonlinear grating with a 6.97 µm period; 2) functional behavior—linear diffraction maintains fixed angular dispersion and scalable efficiency, whereas nonlinear diffraction exhibits power-dependent saturation (complete extinction below 0.12 W total pump energy) and enables broadband spectral engineering.

Finally, we perform the LOD and NOD experiment of the WL4 beam against the PPLN grating. WL4 is generated by sending the above-mentioned homemade Ti:sapphire femtosecond pulse laser beam (pulse energy 2.78 mJ) through a cascaded silica-CPPLN module^[19]. Due to the double-fold nonlinear optical interaction of significant 3rd-NLSPM by the silica plate and ultrabroadband high-efficiency SHG effect by the CPPLN crystal^[17–20], femtosecond white laser WL4 with a high pulse energy (0.8 mJ) and high flatness spectral profile encompassing ultraviolet, visible, and near-infrared bands can be achieved, as witnessed by Figs. 5(c) and 5(d). The spectral profile in Fig. 5(d) indicates that the spectral peak is still located at the pump laser center wavelength of 800 nm, but a larger fraction of energy is converted and transferred to blue and violet bands compared with WL3. The overall performance of spectral bandwidth and spectral flatness is much higher than those of the WL1 thermal radiation source and the WL2 and WL3 laser sources. Accordingly, the dispersive diffraction pattern shown in Fig. 5(c) exhibits very bright blue and violet colors, balanced short and long wavelength components, and an overall white color much colder than the other three sources. The LOD patterns illustrated in Figs. 5(a) and 5(b) further strengthen the idea that WL4 has a flat spectral profile and balanced energy distribution across the entire visible band because a series of Bragg scattering spots appear and exhibit a dispersive colorful (from red on the left to violet on the right) diffraction pattern connecting with each other one by one.

A remarkable thing for the WL4 experiment is that we do not observe a naked-eye visible NCR pattern, as apparently seen in the WL3 experiments of Figs. 3(a), 3(c), 4(a), and 4(c). We believe that this is due to the major difference in the pump laser energy for the wavelength components around 800 nm and that it takes longer to generate the diffraction pattern of the SHG signal that is visible to naked eyes (light components around 400 nm and longer). Comparing Figs. 3(e) and 5(d), one finds that both WL3 and WL4 exhibit a spectral peak at 800 nm, rightly being the wavelength of the Ti:sapphire femtosecond laser. Yet, more energy is concentrated at this spectral peak in WL3 than in WL4 because WL4 has more energy transferred from the peak to the short wavelengths around 400–500 nm than WL3. At the same time, the pulse width of WL4 would be further expanded upon with WL3 since WL4 is just WL3 further passing through the CPPLN crystal, and would be further increased to ∼500fs^[19]. It can be inferred that the 800 nm peak energy for WL4, with a pulse energy of 0.8 mJ (Fig. 5), is approximately 10GW/cm2, which is below the peak power threshold of NCR, about 20GW/cm2 (Fig. 4). Moreover, WL3 in Fig. 3 has about twice the pulse energy of that in Fig. 4, so the NCR intensity is much stronger (about 4 times) in Fig. 3 than in Fig. 4. All these analyses agree well with what we observe in Figs. 3–5 about nonlinear diffraction. This theory can also help to interpret the complete absence of nonlinear diffraction in WL1 and WL2 experiments discussed in Fig. 2 because their 800 nm components energy and peak power density is too weak to ignite any SHG signal and its nonlinear diffraction.

In summary, we have made a comparison study of linear and nonlinear diffraction by a PPLN dual linear–nonlinear grating with weak surface corrugation against four types of white light sources, the ordinary halogen lamp thermal radiation source (WL1), silica photonics crystal fiber supercontinuum nanosecond laser source (WL2), 3^rd-NL femtosecond white laser (WL3), and 2^nd-NL/3^rd-NL synergy femtosecond white laser (WL4). We have found that the coherence, peak power, spectral bandwidth, profile, and flatness of the pump white light all contribute to shaping the characteristics of LOD and NOD patterns. In particular, for WL1, WL2, and WL4, because the peak power is not sufficiently large, only linear diffraction occurs, and nonlinear diffraction is absent. For incoherent WL1, only linear diffraction occurs, while the interference is absent. For WL2 and WL4, both linear diffraction and interference take place simultaneously. Remarkably, when the femtosecond white laser with sufficiently large peak power (WL3) shines upon the PPLN dual linear–nonlinear grating, bright and sharp linear Bragg scattering spots, dispersive colored linear diffraction patterns, and NCR nonlinear diffraction patterns are all observed simultaneously by naked eyes. As this study has disclosed many interesting properties of light–matter interaction with linear and nonlinear diffraction gratings, it would deepen our physical and optical understanding of LOD and NOD upon a dual linear–nonlinear grating. From a more fundamental physics point of view, the current experimental observation would enrich our knowledge about the fundamental diffraction and interference phenomena of various white light sources against microstructure obstacles.