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Abstract
Holographic optical element (HOE) has the advantages of light weight, small volume and multiple functions, but the fixed focal length limits its application. The current methods of adjusting focal length suffer from the disadvantages of small adjustable range and the introduction of aberration. This paper proposes a design method for HOE, and it can compensate the aberrations of the reconstructed image for each distance in a large adjustable range. The focal length of HOE is adjusted by modulating the incident light through the wavefront modulator, and the aberration of the reconstructed image is corrected by optimizing the phase distribution of HOE and superimposing different compensation phases for different reconstruction distances. The reconstructed image experimentally moves without aberration in a large range, which demonstrates the feasibility of the proposed method. It is expected to be widely used in various optical fields where the focal length of HOE needs to be dynamically tuned.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Diffractive optical element (DOE) is a kind of element that regulate the light field through microstructure. It is widely used in beam splittin [1–3], beam shaping [4,5] and other aspects [6]. Focal length is an important parameter of DOE, which affects the distance from the reconstructed image to DOE. Adjusting the focal length of DOE is of great significance: it can expand the function of DOE and improve the performance of optical systems. As a kind of easily integrated focal length adjustable element, DOE is widely required in eye-tracking [7–9], micro-machining, particle manipulation [10,11] and many other tasks, especially in the systems for holographic projection [12,13], near eye display [14–17], imaging [18], microscopy [19–21], optical fiber sensing [22], integrated optoelectronic [23] and so on.
The traditional methods of adjusting the focal length can be classified into two categories: manipulating the element or modulating the wavefront. Rotating two moiré elements by a certain angle can adjust the focal length around the designed focus [24–28], but this method needs mechanical movement and the adjustable range of focal length is limited. Stretching the stretchable element is another way to manipulating the element [29], besides the above problem, it further suffers from the low stability. Modulating the wavefront can also realize the adjustable focal length [30–34]. The usual implementation method is to add a wavefront modulator in the optical path, such as liquid crystal lens or liquid lens, to change the wavefront curvature. However, the wavefront modulator will introduce aberration, and the larger the adjustable range, the greater the aberration.
Holographic optical element (HOE) is a kind of DOE based on holography, which not only has the characteristics of traditional DOE, but also has the advantages of low manufacturing cost and the potential of multiplexing. Correct the aberration while adjusting the focal length in a large range is the hardest part in the design and fabrication of HOE. Researchers have studied the aberrations of HOE in head mounted display and waveguide. The aberrations are mainly analyzed by formula derivation [35,36], ray tracing [37–39] and wavefront sensor detection [40], and the aberrations are corrected mainly by introducing cylindrical mirror [36] in the recording optical path or making pre-compensation [37–39] in the image source. However, global optimization for multiple aberrations and depths of HOE is still a problem to be solved.
This paper proposes a design method for HOE, which can adjust the focal length of HOE in a wide range and correct the aberrations at the same time. In this paper, formula derivation, numerical simulation and optical experiments are carried out.
This paper is arranged as follows: the second chapter describes the basic ideas and principles, deduces the relationship between the focal length of HOE and the distance of reconstructed image, designs a general algorithm to optimize the phase distribution of HOE, and analyzes the principles and compensation methods of aberration caused by oblique incidence; the third chapter shows and analyzes the experimental results of correcting the aberration at the design focal length and in the process of adjusting the focal length.
2. Principle of adjusting the focal length and correcting aberrations
The function of HOE is to reconstruct the target wavefront at the designed distance. As shown in Fig. 1(a), the HOE is an off-axis reflective DOE made by holographic principle and the fabrication process can be found in [41]. According to the basic principle of holography [42], if the illumination wave is the same as the reference wave, the object will be reconstructed at the designed distance. But when the wavefront of the illumination wave is modulated, the reconstructed wave will also be changed. The movement of the reconstructed image means the change of the focal length. To adjust the focal length of HOE in a wide range, the wavefront of the illumination wave needs to be modulated. As shown in Fig. 1(b), the spatial light modulator (SLM) is selected as the wavefront modulator due to its excellent modulation ability. When the modulation factor is loaded on the SLM, an additional phase distribution will be superimposed on the reconstructed wave. The parameters of modulation factor determine the curvature radius of illumination wave, which is related to the focal length of HOE. When the parameters of the modulation factor change, the position of the reconstructed image will be adjusted to different distances. It is worth mentioning that, for simplicity and clarity, only the positions of three reconstruction planes are drawn in Fig. 1(b). However, due to the high refresh rate of SLM and the continuity of compensation parameters, the number of reconstruction plane is unlimited, which can be continuously adjusted within a range, and the aberration at each position can be compensated.
Fig. 1. (a) shows the reconstruction process of HOE and (b) illustrates the process of focal length adjustment and aberration correction by SLM. $IW$: the illumination wave, $RI$: the reconstructed image, ${l_d}$: the designed focal length, $MF$: the phase distribution of modulation factor, $SLM$: spatial light modulator, ${I_1}{I_3}$: reconstructed images at different distances.
When the focal length of HOE is adjusted, the reconstructed image will have aberrations such as distortion, field curvature and astigmatism. Part of it comes from the manufacturing process and can be compensated by optimizing the phase distribution of HOE. The other part comes from the parameters of the modulation factor during reconstruction and the oblique incidence of the illumination wave, which can be corrected by superimposing the compensation phase on the SLM. The following sub-sections will describe the principles of adjusting the focal length of HOE and correcting aberration respectively.
2.1 Adjust the focal length of HOE
In order to describe the relationship between the parameters of illumination wave and the focal length of HOE accurately, the expression is derived based on Fresnel diffraction integral formula. Suppose the coordinates of the target light field are $({{x_1},{y_1}} )$, the coordinates of HOE are $({x,y} )$, and the coordinates of the reconstructed light field are $({x^{\prime},y^{\prime}} )$. Given that the designed focal length is ${l_d}$, the wavefront distribution of the target object is $o({{x_1},{y_1}} )$, the complex amplitude of the reference wave and the illumination wave are $R({x,y} )= |R |\cdot exp({i{\varphi_r}} )$ and $C({x,y} )= |C |\cdot exp({i{\varphi_c}} )$ respectively, where |R| and |C| represent the amplitude while ${\varphi _r}$ and ${\varphi _c}$ represent the phase of wavefront. The complex amplitude of the reconstruction image can be written as
Equation (2) shows the relationship between the distance of the reconstructed image and the focal length of HOE, where the object distance is infinite, and $1/({1/{l_c} + 1/{l_d}} )$ is equivalent to the focal length of HOE. The designed focal length ${l_d}$ is determined when designing the phase distribution, so the distance ${l_i}$ of the reconstructed image can be adjusted by changing the parameter ${l_c}$ of the illumination wave.
2.2 Correct the aberrations at the designed focal length
The aberration correction method in this paper mainly has two steps. One is to correct the aberrations at the designed focal length by optimizing the DOE phase distribution before manufacturing, and the other is to compensate the aberrations at each focal length by wavefront modulation element when adjusting the focal length. This section will describe the causes and correction methods of aberrations at the designed focal length.
SLM is used to load the phase distribution, so there exists zero-order light in the reconstructed image due to the dead zone of SLM. It will block the reconstructed image and reduce its diffraction efficiency. However, if the zero-order is directly filtered by high pass filter in the spatial spectrum plane of $4f$ system, the middle region of the reconstructed image will be missing. In this paper, a quadratic phase factor is superimposed on the original phase distribution, resulting in the separation of signal light and zero-order light in depth. Therefore, the zero-order light can be filtered out and has little effect on the signal light.
In addition, distortion is another aberration that affects the quality of reconstructed image. It is mainly introduced by the optical elements and can be corrected by pre-distorting the target image in the process of calculating the phase distribution of DOE [43]. The mathematical models of radial and tangential distortion can be written as
Astigmatism and field curvature are also aberrations that can’t be ignored. Astigmatism is introduced by the eccentricity and inclination of optical elements, which will make the horizontal and vertical lines in the reconstructed image not clear at the same time. Field curvature is introduced by the wavefront distortion caused by optical elements, and the existence of field curvature makes the center and edge of the reconstructed image not clear at the same time. Although they are formed for different reasons, astigmatism and field curvature can be corrected by superimposing compensation phase on the phase distribution of DOE. The compensated phase can be expressed as
The two exponential factors in Eq. (5) are used to correct astigmatism and field curvature respectively. Where ${l_{ax}}$ and ${l_{ay}}$ are the correction parameters in the x-z plane and y-z plane respectively, ${l_v}$ is the correction parameter of field curvature.
The evaluation function of astigmatism and field curvature can be defined as Eq. (6) and Eq. (7) respectively.
Since the correction of distortion, astigmatism and field curvature involves the optimization of compensation parameters, the general process of determining compensation parameters is introduced.
Figure 2 shows the process of optimizing parameters by dichotomy. The search range as input can be obtained by two methods, one is to carry out several pre-experiments and take the parameters over and under corrected aberration, the other is to introduce the camera and image processing program into the optimization process. Firstly, an initial value k and a search step s are selected to calculate value of the evaluation function when the parameters are k, $k + s$ and $k - s$ respectively, and then the direction in which the evaluation function decreases is selected as the search direction. When the evaluation function increases, $k \pm ns$ and k are taken as the two endpoints of the search range, where n is the number of optimization cycle. In the optimization process, the parameters for correcting aberrations can be found.
2.3 Correct the aberrations during focal length adjustment
Because the phase factors with different curvature radii are superimposed on the x-z and y-z planes during recording, if the parameters required to adjust the focal length are still calculated by Eq. (2), astigmatism will appear, and the astigmatism corresponding to different distances is different. To solve this problem, Eq. (2) is deformed to
The focal length f of HOE can be calculated by Eq. (9), where m can be x or y, representing the parameters in the x-z or y-z plane respectively. The parameter ${l_c}$ of the modulated illumination wave can be calculated according to the Eq. (8). The parameters in the x-z and y-z planes vary with the distance, and the corresponding parameters in the two planes are different at the same distance.
High diffraction efficiency can be obtained by using reflective HOE, but the illumination wave needs to be incident obliquely. If the illumination wave is an inclined spherical wave, the wavefront curvature on the x-z and y-z planes are different when incident on HOE, which will cause astigmatism. The greater the tilt angle and the larger the size of HOE, the more obvious the astigmatism will be.
As shown in Fig. 3, it is assumed that the transverse dimension of HOE is ${d_{hy}}$, the curvature radius of the illumination wave corresponding to the center is ${l_c}$, the included angle between the illumination wave and the z-axis is $\theta $.
Fig. 3. The oblique incidence of illumination wave into HOE introduces astigmatism. (b) is the top view of (a).
As described in 2.1, to adjust the focal length of HOE, a quadratic factor needs to be superimposed on the original phase distribution. The quadratic factor is equivalent to the phase distribution of a spherical wave perpendicular to the surface of HOE. However, the actual illumination wave is the inclined spherical wave ${\varphi _{ts}}$. when it subtracts the inclined phase factor ${\varphi _{tp}}$ of the reference wave, it is not equal to the spherical wave ${\varphi _s}$ which perpendicular to the surface of HOE, and the phase difference between them is ${\varphi _d}$.
As shown in Fig. 4, the numerical simulation results show that the actual and ideal illumination wavefront differ by a cylindrical phase factor.
To pre-compensate the phase difference caused by oblique incidence, the phase of illumination wave corresponding to the center of HOE is taken out, converted into height value, and then the illumination light curvature in this direction is calculated through geometric relationship. According to the above steps, the curve of the curvature radius of HOE in the y-z plane with the curvature radius of illumination light in the y-z plane can be drawn by numerical simulation.
First, a series of discrete data are calculated, as shown by the red dot in Fig. 5. Then, these discrete data are fitted to obtain the mathematical expression between $l{}_{c - HOE}$ and $l{}_{c - SLM}$. With this mathematical expression, phase parameter $l{}_{c - SLM}$ that needs to be loaded on SLM for any known $l{}_{c - HOE}$ can be directly calculated, instead of calling the numerical simulation program again when the focal length needs to be adjusted, which can improve the speed of adjustment. Since the distribution of discrete data is approximately linear, we choose the first-order equation for fitting and the coefficient can be obtained.
Fig. 5. The actual radius of curvature on HOE varies with the radius of curvature of the illumination light with the influence of oblique incidence angle.
As shown in Fig. 5, the slope is related to the inclination angle of the illumination wave. The greater the inclination angle is, the greater the slope is. The slope of y-z plane is expected to be close to that of x-z plane, which means that the astigmatism introduced by oblique incidence is small and the compensation effect is better. However, the inclination angle of the illumination wave cannot be too small, which will overlap with the area where the reconstructed image is located. Considering the above two points and the convenience of setting the recording light path, 45° is choosed as the inclination angle of the illumination wave.
3. Experimental results
In the second section, we divide the aberration correction of HOE into the correction at the design focal length and the correction in the adjustment process. This is because the recording and reconstruction process of HOE is independent. The recording process is to “freeze” the wavefront of the object wave in the holographic material through interference, and then “activate” the wavefront of the object wave by the illumination wave in the reconstruction process, to make it continue to propagate. The aberration correction at the design focal length is the correction in the recording process. After the correction, the aberration will not be introduced into the subsequent reconstruction process. However, the reconstruction process itself will also introduce new aberrations due to oblique incidence, and this aberration will change with the adjustment of focal length. Therefore, in the reconstruction process, it is necessary to compensate for each reconstruction distance separately. Therefore, the following sub-sections will correct the aberration in the order of first at the design focal length and then in the adjustment process according to the above description.
3.1 Correct the aberrations at the designed focal length
Figure 6 shows the process and results of correcting the aberrations at the designed focal length. Figure 6(b) is a reconstructed image before correction. Through the method described in section 2, zero-order light, distortion, astigmatism and field curvature are eliminated in Fig. 6(c)-(g) respectively. This can be seen more clearly through the partially enlarged picture. The difference between the width of horizontal and vertical line in Fig. 6(j) is obviously less than Fig. 6(h), indicating that astigmatism has been corrected. The comparison between Fig. 6(i) and Fig. 6(j) shows that the difference between the width of horizontal and vertical line at the edge is greater than that at the center, that is, there is still field curve in the reconstructed image, Fig. 6(k) and Fig. 6(l) show that the width of horizontal and vertical line at the edge and center is basically the same, indicating that the field curve is corrected.
Fig. 6. Aberration correction at the designed focal length. (a) Original image. (b) the reconstructed image before correction. (c) the reconstructed image after zero-order light elimination. (d) the original image with pre-distortion. (e) - (g) represents the reconstructed image after correcting distortion, astigmatism and field curvature respectively. (h) - (l) is a partial enlarged view of (e) - (g).
The reconstructed image before and after correction is evaluated by image processing program. The value of distortion evaluation function decreases from 4337.9 to 2865.6, the value of astigmatism evaluation function increases from 0.72 to 0.93, and the field curve evaluation function increases from 0.89 to 0.97. It shows that the distortion, astigmatism and field curve of the reconstructed image have been well corrected.
3.2 Correct the aberrations during focal length adjustment
Figure 1 shows the optical path of the experiment. The illumination wave is inclined and its wavefront is modulated by SLM. The reconstructed wave is emitted perpendicular to the surface of HOE, and a clear reconstructed image is formed at the position satisfying Eq. (8).
In the experiment, the designed reconstruction distance ${l_d}$ is 40cm, the transverse dimension ${d_{hy}}$ of HOE is 2.15mm, and the included angle $\theta $ between illumination light and z-axis is 45°. The expression fitted from the above data is
The curves of illumination wave parameters of HOE and SLM with distance are drawn respectively, as shown in Fig. 7.
As shown in Fig. 8, the reconstructed image is adjusted to 30 cm, 40 cm and 50 cm respectively, and the astigmatism introduced by focal length adjustment and inclined illumination wave is well corrected. To present the compensation results more clearly, the part at the same position in the reconstructed image is enlarged as a comparison.
Fig. 8. Reconstructed images at different distances after aberration compensation. The reconstructed images before and after compensation are shown in (a) - (c) and (d) - (f) respectively, where (a) (d), (b) (e) and (c) (f) correspond to the reconstructed images adjusted to 30 cm, 40 cm and 50 cm respectively.
In the experiment, the reconstructed image is moved in the range of 30∼50 cm by changing the focal lengths of x-z and y-z planes at 54.55∼200.00 cm and 23.35∼37.71 cm respectively, but due to the excellent wavefront modulation ability of SLM, theoretically, the adjustment range of the focal length of HOE is unlimited. However, some practical factors need to be considered in the experiment. For example, in the process of gradually increasing the reconstruction distance, the energy of the reconstructed pattern will gradually decrease due to the loss of power in the process of propagation and the expansion of the image size. In addition, the reduction of reconstruction distance will also lead to the reduction of reconstruction image size. Therefore, the adjustable range of focal length needs to be comprehensively considered according to the needs of the optical system. The actual optical system usually requires an adjustable range of no more than a few meters, and the proposed method can meet the needs of most actual optical systems. After the adjustable range of focal length is determined, the designed focal length of HOE can be selected in the middle of the adjustable range for subsequent adjustment and compensation.
Since there is no lens in the reconstruction process, distortion and field curvature are not introduced. Among the aberrations introduced in the reconstruction process, the astigmatism at different reconstruction distances caused by the oblique incidence of illumination light is the aberration that has the greatest impact on the quality of the reconstructed image, which is the main aberration concerned in this paper. The first row and the second row in Fig. 8 are the reconstructed images before and after correction, respectively. To fully illustrate the effect of astigmatism correction, the change of its evaluation function is calculated through the image processing program. The value of astigmatism evaluation function changes from 0.85 to 0.90 at 30cm, from 0.83 to 0.93 at 40cm and from 0.81 to 0.89 at 50cm. When the value of astigmatism evaluation function is greater than 0.9, it can be considered that astigmatism is not obvious. The experiment results show that the method in this paper can well compensate the aberrations at each focal length.
4. Conclusion
This paper proposes a method to correct the aberrations of HOE in the process of large-scale focal length adjustment. The relationship between the focal length of HOE and the parameters of modulation factor is derived. Combined with SLM which has excellent wavefront modulation ability, the theoretically designed adjustable range of focal length can be selected according to the requirements of optical system. The aberration introduced in the fabrication process is corrected by optimizing the phase distribution. The values of the evaluation functions of distortion, astigmatism and field curvature change experimentally by 33.94%, 29.17% and 8.99% respectively in the direction of aberration reduction. The effects of the parameters of modulation factor on aberration in the reconstruction process are analyzed. The value of astigmatism evaluation function of corrected reconstructed image increases by 6.00%, 11.94% and 10.44% at 30 cm, 40 cm and 50 cm in optical experiment, respectively. The increased values are close to or more than 0.9, which shows that the proposed method can realize the HOE with large range adjustment of focal length and aberration correction. In future, the design method for HOE is expected to match various optical fields with multifunctional diffractive optical elements, such as the holographic projection and the multi-depth structured light generation.
Funding
National Natural Science Foundation of China (61975014, 62035003); Beijing Municipal Science and Technology Commission, Administrative Commission of Zhongguancun Science Park (Z211100004821012).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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