Generation of Higher-Order Hermite–Gaussian Modes via Cascaded Phase-Only Spatial Light Modulators

Abstract

The spatial distribution of higher-order Hermite–Gaussian (HG) modes is more complicated than the fundamental mode, and the characteristics of different modes and their orthogonal characteristics have essential applications in the fields of measurement, imaging, and large-capacity communications. The main issue in future applications is how to efficiently generate higher-order HG modes. The spatial light modulator has high spatial resolution and excellent maneuverability, so it is a simple and effective way to generate higher-order HG modes. Here, the efficient generation of higher-order HG modes via cascaded spatial light modulators was experimentally investigated. For the HG90 mode, the conversion efficiency is 61%. The method with high conversion efficiency has promising application potentials in biophotonics, laser physics, and quantum information.

1. Introduction

Lasers are widely used in various fields of optics. The field distribution perpendicular to the propagation direction of the laser is called the transverse mode, and different transverse modes correspond to different spatial distributions. The Hermite–Gaussian (HG) modes comprise a complete set of spatial modes with orthogonal bases, and any spatially distributed pattern can be expanded under the HG mode basis [1]. The higher-order HG modes have opened up the path for new applications in laser physics [2,3], precise measurement [4], and other fields due to their complex spatial distribution. In addition, the combination of higher-order HG modes and quantum technology can produce many novel quantum states. These quantum states have brought new applications for high-dimensional quantum communication [5] and quantum computing [6].

The spatial squeezed state is a critical quantum state, and the HG mode squeezed state makes up a core part. Taylor et al. applied the spatial squeezed light to biological measurement, achieving a biological measurement beyond the quantum limit [7]. Coupled with the higher-order HG mode, squeezed light can further improve the measurement accuracy. An optical parametric oscillator (OPO) is a widely used way of generating a higher-order HG mode squeezed state, which requires the higher-order HG mode to be used as the optimal pump beam [8]. For example, to generate HG40 mode squeezed light, the HG80 mode is required as the optimal pump beam. Therefore, to generate higher-order HG mode squeezed light, a high-quality and high-power higher-order HG mode needs to be used as the pump beam. In gravitational wave interferometers, higher-order HG modes can effectively reduce the loss caused by mode mismatch [9] and can reduce the mirror thermal noise [10,11]. Therefore, how to efficiently generate a higher-order HG mode has received widespread attention.

Four methods are commonly used to generate higher-order HG modes: specially designed lasers [12,13,14,15], mode mismatch [8,16], phase plates [17,18], and phase diffraction elements [19,20,21,22,23,24,25]. Chu et al. studied the excitation of higher-order HG modes in end-pumped lasers with an opaque wire inserted into a laser cavity [13]. By controlling the transverse position of the pump beam and the opaque wire, a different-order HG mode output was achieved, due to the additional optical element inside the resonant cavity, resulting in low output power of the higher-order HG mode. Mode mismatch requires locking the cavity length of the mode conversion cavity. The mode conversion cavity needs to be locked to ensure stable operation under the influence of various disturbances, and the higher the mode order, the smaller the signal-to-noise ratio of the error signal used for locking the cavity. Therefore, the mode conversion cavity is suitable for generating low-order HG modes. Higher-order HG modes can be efficiently generated using a specially designed phase plate, but the generated mode is not a standard mode. In addition, the phase distribution of the higher-order HG mode is more complex, which requires higher processing technology of the phase plate, and each specially designed phase plate can only produce specific higher-order HG modes.

With the advancement of science and technology, optical diffraction elements such as spatial light modulator (SLM) and deformable mirror have become useful tools in generating higher-order HG modes, and the desired light field can be generated by controlling the wavefront of the incident light. Trep et al. reflected the incident light field on a deformable mirror multiple times to generate the HG10 mode, the HG20 mode, and the HG30 mode [19]. Limited by the number of pixels, deformable mirror is not suitable for generating a complex distribution light field. Compared with the deformable mirror, the SLM has more pixels. Stefan et al. generated higher-order HG modes based on complex amplitude modulation with a single phase-only SLM, and the mode conversion efficiency was a few percent [10].

The squeezed states of higher-order HG modes are useful for applications in quantum information processing. A major challenge for future applications is efficient generation. Here, we used cascade SLMs to efficiently generate higher-order HG modes, the method theoretically allows for an efficiency of almost 100%. We experimentally demonstrate that cascaded phase-only SLMs can efficiently generate higher-order HG modes. We verified the intensity distribution and phase distribution of the generated HG modes by comparing the measured intensity pattern and interferograms with the corresponding theoretical results. For the HG90 mode, the mode conversion efficiency is 61%.

2. Theory

The principle of the cascaded SLM system is shown in Figure 1. Simultaneously modulating the amplitude and phase distribution of the incident light field using a cascaded phase-only SLM system, this scheme is theoretically lossless and the mode conversion efficiency can reach almost 100%. Two SLMs and two Fourier transforming lenses form a 4f system, and the two SLMs are in the conjugate Fourier plane of lens1. To create the complex amplitude field $a\left(x,y\right)$ in the target plane, a definite complex amplitude field $A\left(u,v\right)$ is required in the plane of SLM2, which can be written as [26]

$$A\left(u,v\right)=\frac{1}{\lambda f}{{\displaystyle \int }}_x{{\displaystyle \int }}_{y}a\left(x,y\right)exp[-i\frac{2\pi }{\lambda f}\left(ux+yv\right)]dxdy$$

(1)

where $\lambda$ and $f$ denote the laser wavelength and the focal length of the Fourier transforming lens, respectively. The SLM2 can shape the phase distribution of the field $A\left(u,v\right)$, so the amplitude distribution of $A\left(u,v\right)$ must be created by SLM1 using the Gerchberg–Saxton (GS) algorithm [27]. After k iterations of the GS algorithm, the optical field $A_k\left(u,v\right)$ in the plane of SLM2 is produced, which is given by [28]

$$A_k\left(u,v\right)=\frac{1}{\lambda f}{{\displaystyle \int }}_x{{\displaystyle \int }}_{y}U(\epsilon ,\mu )exp[i{\varphi }_k\left(\epsilon ,\mu \right)]exp[i\frac{2\pi }{\lambda f}\left(u\epsilon +y\mu \right)]d\epsilon d\mu$$

(2)

where $U(\epsilon ,\mu )$ is the input fundamental beam, and ${\varphi }_k\left(\epsilon ,\mu \right)$ is the phase distribution on SLM1 created from the k iterations of the GS algorithm. The amplitude distribution of the field $A_k\left(u,v\right)$ is almost approximated to the amplitude distribution of the field $A\left(u,v\right)$. The random phase ${\psi }_k\left(u,v\right)$ of the field $A_k\left(u,v\right)$ produced by this optimization procedure is uncontrollable but known. To obtain the phase $\phi \left(u,v\right)$ of the field $A\left(u,v\right)$ outgoing from SLM2, we correct the phase distribution by loading a phase correction hologram $\Phi \left(u,v\right)$ on the SLM2, in the following form:

$$\Phi \left(u,v\right)=\phi \left(u,v\right)-{\psi }_k\left(u,v\right)$$

(3)

Ultimately, the desired field $a\left(x,y\right)$ is generated in the target plane. From the theoretical analysis, it can be seen that $a\left(x,y\right)$ can be an arbitrary complex amplitude field; in other words, we can generate an arbitrary complex amplitude field through the cascaded SLM system.

3. Experimental Setup

Figure 2 shows the experimental setup for generating higher-order HG modes using a cascaded SLM system. The fundamental mode is expanded by a telescope system consisting of 5 cm and 20 cm lenses, and the distance between the lenses is the sum of the focal lengths. After expansion by the telescope system, the fundamental mode is expanded into near-parallel light with a waist of 5 mm and incident on the cascaded SLM system. The polarization of the fundamental mode incident to the cascaded SLM system is consistent with the SLM working polarization. The cascaded SLM system consists of two phase-only SLMs (Hamamatsu, Japan, X10648-03, consisting of 792 ∗ 600 pixels) and two Fourier transforming lenses (focal length of 75 cm).

SLM1 is illuminated with an expanded and collimated fundamental mode, and the diffracted light is subsequently reflected onto SLM2 such that the desired amplitude emerges in the plane of SLM2. There, SLM2 reshapes the random phase from the incoming mode. Then, through the Fourier transforming lens, the desired complex amplitude field appears on the charge-coupled device (CCD, Hamamatsu, Japan, C10633).

The critical part of the cascaded SLM system is whether the light field on the SLM2 plane is consistent with the theoretically calculated spot size and phase distribution. If it is inconsistent, it cannot be corrected, and ultimately, the expected complex amplitude light field cannot be generated at the target plane. We can generate a hologram with a special pattern using the GS algorithm, load it onto the SLM1, and use the intensity distribution captured by the CCD at the plane of SLM2 to measure the actual spot size and to compare it with the theoretical spot size. In addition, the light field at the plane of SLM2 needs to be able to correspond point to point with the hologram loaded onto the SLM2, and the accuracy needs to reach a few microns. This can be achieved by carefully moving the hologram loaded onto the SLM2 for alignment to find the best position so that the random phase can be corrected.

4. Experimental Results and Discussion

The performance of the cascaded SLM system is shown in Figure 3. To show the manipulation of the higher-order Laguerre-Gaussian (LG) modes, the LG33 mode is taken as an example. Figure 3a shows the distribution of the intensities for the HG30 mode, HG60 mode, HG90 mode, and LG33 mode generated. The left column shows the theoretical results used for comparison, and the right column shows the experimental results. It can be seen that the experimental results are in agreement with the theoretical results. As a comparison, shown in Figure 3b,c are the holograms used to generate the HG30 mode and the HG90 mode, respectively. The first line consists of the holograms loaded onto SLM1, obtained after 200 iterations of the GS algorithm. The second line consists of the holograms loaded onto SLM2. By comparing the two sets of holograms, it can be seen that the higher the order of the modes generated, the more complex the corresponding holograms.

It can be seen from Figure 3 that the intensity distributions of the experimentally generated HG modes are in agreement with that from the theory. Next, to verify the distribution of the phases for the generated HG modes, using a Gaussian beam as a reference, we measured the interference between the HG modes generated and the Gaussian beam. Here, we took the HG30 mode as an example to verify the phase distribution of the generated modes, and the interferograms are shown in Figure 4. For higher-order HG modes, the phase difference between adjacent spots is π, so that we can see the interference fringes between bright and dark. It can be seen from Figure 3 and Figure 4 that the intensity distribution and phase distribution of the HG modes generated by the cascaded SLM system agree with those of the theoretical results.

The mode conversion efficiency is the ratio of optical power of the output HG mode to that of the input mode, we measured the mode conversion efficiency of higher-order HG modes generated by the cascaded SLM system: HG30 mode, 72 ± 2%; HG60 mode, 64 ± 2%; and HG90 mode, 61 ± 2%. No hologram was loaded onto the SLM. The ratio of optical power of the output beam to that of the input beam was measured, and the SLM reflected 97% of the input beam. After that, a blazed grating was loaded onto the SLM, and the modulated light was transferred to the first-order diffraction. The ratio of optical power of the modulated light to that of the output beam was measured, and the diffraction efficiency of the SLM was 93%. Therefore, the main optical losses were due to the absorption and imperfect diffraction efficiency of the two SLMs, with an experimentally measured loss of about 20%. The optical components in the optical path caused 2% transmission loss. In addition, the limited aperture of the optical element led to diffraction loss of a part of the high spatial frequency light, and the more complex the generated HG mode, the greater the diffraction losses. Therefore, we can see that as the mode order increases, the conversion efficiency decreases.

It can be seen from Figure 3 that there are some differences between the intensity distribution of the HG modes generated and the theoretical results, especially for the modes with complex spatial distribution, such as the HG90 mode. Such phenomena might be further explained by three aspects. First, there is crosstalk between SLM pixels [29]. When the phase difference between adjacent pixels is significant, the loaded driving voltage has a corresponding difference. Then, adjacent pixels will influence one another, causing a deviation between the hologram loaded onto the SLM and the one we set to load onto the SLM; the greater the phase change, the greater the effect of pixel crosstalk. It can be seen from Figure 3b,c that the holograms loaded onto the SLM1 and SLM2 to generate the HG90 mode is more complex than that of the HG30 mode. Next, limited by the resolution of the SLMs, the correction of random phase with an accuracy that strongly depends on the resolution of the SLM2. It is obvious that this factor results in an inevitable deterioration in the quality of the mode generated, especially for higher-order (more complexly distributed) modes. Additionally, higher orders lead to higher spatial frequency and, thus, more sensitive to the accurate alignment of SLM2. Finally, the diffraction efficiency of the SLMs is not 100%, and a part of the unmodulated light enters the final measurement, which causes the quality of the HG modes generated to decrease [30].

Compared with other higher-order HG mode generation methods, the cascaded SLM system has the following two advantages: On the one hand, the cascade SLM system can achieve efficient amplitude and phase modulation of the input beam. In the specially designed laser method, due to the additional optical elements added in the resonator to increase the optical loss, the mode conversion efficiency for the specially designed laser method is below 30% [13,14]. In the mode mismatch method, different orders of HG modes are generated, and the resonator needs to be locked to output the desired mode. The mode conversion efficiency for the HG30 mode is 20%, and as the mode order increases, the mode conversion efficiency decreases [16]. In the phase plate method, the light field generated by the phase plate is not a standard higher-order HG mode, and each phase plate can only generate a specific HG mode. No reports on HG mode generation present a mode higher than the third order using a phase plate. In the phase diffraction element method, for the deformable mirror, due to the limited number of pixels, only the first three-order HG modes are generated [19]. For the SLM, higher-order HG modes can be generated by loading the phase factor and complex amplitude modulation with a single spatial light modulator. Loading the phase factor is similar to the phase plate method, although the generated beam has a perfectly phase structure and the mode conversion efficiency can reach more than 95%, the intensity distribution does not match that of a standard HG mode [16]. The complex amplitude modulation with a single spatial light modulator has a large optical loss, the mode conversion efficiency is below 10% [10]. The cascaded SLM system can independently control the phase distribution and amplitude distribution of the incident beam, theoretically, higher-order HG modes can be generated with high quality (almost the standard HG modes), and the mode conversion efficiency can reach almost 100%.

On the other hand, the cascaded SLM system can generate not only higher-order HG modes, but also LG modes, even arbitrary complex amplitude distribution fields [31]. Different spatial modes can be generated simply by loading different holograms onto the SLM without further modifications to the setup. In the future, with the improvements in optical elements and SLM performance, the efficiency of the system will be improved, and the quality of the generated modes will also be improved.

5. Conclusions

In this study, we demonstrated that higher-order HG modes can be efficiently generated by the cascaded SLM system, and that the conversion efficiency for the HG90 mode can reach 61%. Different-order HG modes can be generated just by loading different holograms onto the SLMs.

The efficient generation of higher-order HG modes has relevant applications in quantum information processing, such as super-resolution imaging [32]; reducing the mirror thermal noise and mode matching loss in gravitational wave detection; and showing promise in applications to multimode optical quantum information systems [5].