Holographic Multi-Notch Filters Recorded with Simultaneous Double-Exposure Contact Mirror-Based Method

Abstract

This study presents a novel simultaneous double-exposure contact mirror-based method for fabricating holographic multi-notch filters with dual operational central wavelengths. The proposed method leverages coupled wave theory, the geometric relationships of K-vectors, and a reflection-type recording setup, incorporating additional reflecting mirrors to guide the recording beams. To validate the approach, a holographic notch filter was fabricated using photopolymer recording materials, resulting in operational wavelengths of 531.13 nm and 633.01 nm. The measured diffraction efficiencies at these wavelengths were η_s = 52.35% and η_p = 52.45% for 531.13 nm, and η_s = 67.30% and η_p = 67.40% for 633.01 nm. The component’s performance was analyzed using s- and p-polarized spectral transmission intensities at various reconstruction angles, revealing polarization-independent characteristics under normal incidence and polarization-dependent behavior under oblique incidence. The study also explored the relationships between recording parameters, such as incident angle, wavelength, emulsion expansion, and dispersion. The findings demonstrate that the first operational central wavelength is primarily influenced by the recording wavelength, while the second is primarily determined by the incident angle, covering a range from visible light to near-infrared. This method offers significant potential for cost-effective, mass-produced filters in optoelectronic applications.

1. Introduction

Filters can selectively transmit, reflect, or block specific wavelengths of electromagnetic radiation, playing a crucial role in various communication and imaging applications across a broad range of the electromagnetic spectrum [1,2,3,4]. These applications include optical communications [5,6,7], imaging and display [8,9,10,11], scientific research [3,12,13], and medical applications [9,14]. Based on their operating principles, filters can be broadly classified into three main types: absorption filters, scattering filters, and dichroic filters. Each type can be designed and fabricated according to different principles.

Traditional absorption filters include colored glass filters, plastic filters, and Wratten filters [15]. These filters control spectral properties by introducing elements, compounds, dyes, or other colorants into the substrate. Waveguide-type absorption filters, on the other hand, utilize the principle of guided mode resonances in waveguide structures to achieve spectral filtering [2,16]. Scattering filters embed small particles into the substrate and use the scattering properties of particles of different sizes to control the filter’s spectral characteristics [3,17].

Typical dichroic filters are made using optical coating techniques to create multilayer dielectric structures with alternating high and low refractive indices. They achieve selective transmission or reflection of specific wavelengths through the principle of thin-film interference [18]. When light is incident at a certain angle, different wavelengths are reflected or transmitted due to the interference effect. Since the number of coating layers can often exceed a hundred, the complexity and precision of the manufacturing process become a challenge, thus impacting the production cost. In contrast, volume holograms are fabricated using the double-beam interference principle, offering the advantages of ease of production and suitability for low-cost mass production [19,20,21]. Therefore, volume holograms provide a viable alternative to dichroic filters.

Research on holographic reflection filters can be traced back to the 1970s and 1980s [22,23]. To achieve high diffraction efficiency by ensuring better interference fringe contrast, the production of holographic reflection filters often adopts a symmetric double-beam interference recording setup [24,25]. To address the drawbacks of this setup’s large size and operational complexity, the single-beam Lippmann configuration has been employed. In this improved setup, to overcome the limitation of the incident beam exceeding the critical angle at the air–emulsion interface, a triangular prism is introduced to guide the beam [26]. A cylindrical prism is used to guide the recording beam, resulting in a holographic photopolymer linear variable filter with enhanced blue performance and significantly improved filter efficiency [27]. However, existing holographic reflection filters are typically designed to operate at a single central wavelength. Developing filters with dual-band capability would enhance their flexibility and application potential.

Accordingly, this paper proposes a recording method for fabricating holographic notch filters with two distinct operational central wavelengths. The holographic multi-notch filters produced using this method are flexible, easy to manufacture, cost-effective, and suitable for mass production. The designed operational wavelengths range from 0.45 μm in the visible spectrum to 1.80 μm in the near-infrared, demonstrating significant potential for modern optoelectronic applications.

2. Principles

Figure 1 and Figure 2 illustrate the schematic setups for the construction and reconstruction of holographic reflection filters based on the contact mirror method. The contact mirror-based method is a reflection-type holographic recording method using a single incident light beam. In the recording setups, the incident expanded beam (IEB) can be normally (IEB_n) or obliquely (IEB_θ) incident from air onto the holographic recording material (H) lying on the x-y plane, as shown in Figure 1a and Figure 2a. Before reaching the holographic recording material, the incident linearly polarized laser beam with a wavelength of λ₁ passes through a half-wave plate (HWP), an electronic shutter (ES), a spatial filter (SF), and a collimating lens (CL). The application of the half-wave plate (HWP) ensures that the polarization of the incident laser beam is s-polarized (along the y-direction). Figure 1d and Figure 2d illustrate the reconstruction of the fabricated filters under the two aforementioned conditions.

2.1. Normally Incident Condition

In Figure 1a, the IEB_n is normally incident onto H with a reflection mirror (M₁) attached at the back. For ease of understanding, the protective coversheet (PC) (polycarbonate) of the recording material and the refractive index matching liquid are omitted. The details of geometric relations and material-related parameters are shown in Figure 1b. The thickness of the photosensitive emulsion (PE) in the recording material is d₁, with a refractive index of n_f1 (at λ₁), and the refractive index matching liquid (RM) has a refractive index of n_rm (at λ₁). The IEB_n serves as the reference beam R_1,n, and the normally reflected light by the mirror serves as the object beam O_1,n. R_1,n and O_1,n are two parallel beams in opposite directions inside the recording material. Accordingly, a set of interference fringes parallel to the emulsion plane can be easily recorded. The corresponding K-vector diagram and geometric relations are shown in Figure 1c, where k_R1,n and k_O1,n (with |k_R1,n| = |k_O1,n| = 2πn_f1/λ₁) are the propagation vectors for the reference and object beams, respectively, and K_1,n (with |K_1,n| = 2π/Λ_1,n = 2|k_R1,n|) is the grating vector with a grating period of Λ_1,n, which is perpendicular to the emulsion plane.

Figure 1. (a) Schematic setup for the construction of holographic reflection filters under the normally incident condition (IEBn). (b) Geometric relations and material-related parameters for the normally incident condition. (c) K-vector diagram and geometric relations for the normally incident condition. (d) Reconstruction setup for the fabricated filter under the normally incident condition. (e) K-vector diagram for the reconstruction of the fabricated filter under the normally incident condition.

Figure 1. (a) Schematic setup for the construction of holographic reflection filters under the normally incident condition (IEBn). (b) Geometric relations and material-related parameters for the normally incident condition. (c) K-vector diagram and geometric relations for the normally incident condition. (d) Reconstruction setup for the fabricated filter under the normally incident condition. (e) K-vector diagram for the reconstruction of the fabricated filter under the normally incident condition.

In Figure 1d, the fabricated filter is reconstructed using a normally incident, unpolarized reference beam R_2,n with a wavelength of λ_2,n. The object beam O_2,n diffracts in the opposite direction at an angle of θ_d = 180° inside the hologram. The non-diffracted light passes through the element directly. Consider the changes in thickness and average refractive index of the photosensitive emulsion caused by exposure. The thickness of the photosensitive emulsion is d₂, with an average refractive index of n_f2,n (at λ_2,n). The corresponding K-vector diagram is shown in Figure 1e, where k_R2,n and k_O2,n are the propagation vectors for the reference and object beams (|k_R2,n| = |k_O2,n| = 2πn_f2,n/λ_2,n, with n_f2,n as the average refractive index of the emulsion at λ_2,n), and K_2,n (|K_2,n| = 2π/Λ_2,n = 2|k_R2,n|) is the grating vector with a period of Λ_2,n perpendicular to the emulsion plane. Accordingly, the relationship fulfilling the Bragg condition between the reconstruction wavelength λ_2,n and the recording wavelength λ₁ can be expressed as:

$${\lambda }_{2,n}={\displaystyle \frac{n_{f2,n}}{n_{f1}}}{\displaystyle \frac{{\Lambda }_{2,n}}{{\Lambda }_{1,n}}}{\lambda }_1.$$

(1)

2.2. Obliquely Incident Condition

In Figure 2a, the IEB_θ is obliquely incident onto H at an angle θ. The details of geometric relations and material-related parameters are shown in Figure 2b. The thickness of the photosensitive emulsion in the recording material is d₁, with a refractive index of n_f1 (at λ₁). The IEB_θ serves as the reference beam R_1,θ, and the light reflected by the mirror serves as the object beam O_1,θ. The incident and reflection angles of R_1,θ and O_1,θ are θ in the air with respect to the surface normal of the recording material; after refraction, the corresponding incident and refraction angles in the emulsion of the recording material are θ′. R_1,θ and O_1,θ are two obliquely incident and reflected beams inside the recording material. Accordingly, a set of interference fringes parallel to the emulsion plane can be easily recorded. The corresponding K-vector diagram and geometric relations are shown in Figure 2c, where k_R1,θ and k_O1,θ (with |k_R1,θ| = |k_O1,θ| = 2πn_f1/λ₁) are the propagation vectors for the reference and object beams, respectively, and K_1,θ (with |K_1,θ| = 2π/Λ_1,θ = 2|k_R1,θ|cosθ′) is the grating vector with a grating period of Λ_1,θ, which is perpendicular to the emulsion plane.

Figure 2. (a) Schematic setup for the construction of holographic reflection filters under the obliquely incident condition (IEB_θ). (b) Geometric relations and material-related parameters for the obliquely incident condition. (c) K-vector diagram and geometric relations for the obliquely incident condition. (d) Reconstruction setup for the fabricated filter under the obliquely incident condition. (e) K-vector diagram for the reconstruction of the fabricated filter under the obliquely incident condition.

Figure 2. (a) Schematic setup for the construction of holographic reflection filters under the obliquely incident condition (IEB_θ). (b) Geometric relations and material-related parameters for the obliquely incident condition. (c) K-vector diagram and geometric relations for the obliquely incident condition. (d) Reconstruction setup for the fabricated filter under the obliquely incident condition. (e) K-vector diagram for the reconstruction of the fabricated filter under the obliquely incident condition.

In Figure 2d, the fabricated filter is reconstructed using a normally incident, unpolarized reference beam R_2,θ with a wavelength of λ_2,θ. The object beam O_2,θ diffracts in the opposite direction at an angle of θ_d = 180° inside the hologram. The non-diffracted light passes through the element directly. Consider the changes in thickness and average refractive index of the photosensitive emulsion caused by exposure. The thickness of the photosensitive emulsion is d₂, with an average refractive index of n_f2,θ (at λ_2,θ). The corresponding K-vector diagram is shown in Figure 2e, where k_R2,θ and k_O2,θ are the propagation vectors for the reference and object beams (|k_R2,θ| = |k_O2,θ| = 2πn_f2,θ/λ_2,θ, with n_f2,θ as the average refractive index of the emulsion at λ_2,θ), and K_2,θ (|K_2,θ| = 2π/Λ_2,θ = 2|k_R2,θ|) is the grating vector with a period of Λ_2,θ perpendicular to the emulsion plane. Accordingly, the relationship fulfilling the Bragg condition between the reconstruction wavelength λ_2,θ and the recording wavelength λ₁ can be expressed as:

$${\lambda }_{2,\theta }={\displaystyle \frac{n_{f2,\theta }}{n_{f1}}}{\displaystyle \frac{{\Lambda }_{2,\theta }}{{\Lambda }_{1,\theta }}}{\displaystyle \frac{{\lambda }_1}{\cos {\theta }^{\prime }}},$$

(2)

and when the operating wavelength of the filter is designed to be λ_2,θ, the incident angle θ of the expanded beam IEB_θ can be expressed as:

$$\theta ={\sin }^{-1}\left\{n_{f1}{\left[1-{\left({\displaystyle \frac{n_{f2,\theta }}{n_{f1}}}{\displaystyle \frac{{\Lambda }_{2,\theta }}{{\Lambda }_{1,\theta }}}{\displaystyle \frac{{\lambda }_1}{{\lambda }_{2,\theta }}}\right)}^2\right]}^{1/2}\right\}.$$

(3)

2.3. Simultaneous Double-Exposure Contact Mirror-Based Method

Based on the principles introduced in Section 2.1 and Section 2.2, combining the normally incident condition and the obliquely incident condition results in the method described in this section, as shown in Figure 3. This method guides the light beam by introducing two additional mirrors, M₂ and M₃. M₂ is perpendicular to M₁, and the angle between M₃ and M₂ is (90° − θ/2). Consequently, the expanded beam IEB_θ is obliquely incident onto H at an angle θ and then reflected by mirror M₁, completing the recording of the first set of interference fringes. The reflected beam is incident on mirror M₂ at an angle of (90° − θ) and then reflected. This reflected beam is incident on M₃ at an angle of θ/2, reflected again to become IEB_n, and finally perpendicularly incident onto H, and reflected by M₁, completing the recording of the second set of interference fringes. Therefore, a holographic multi-notch filter with two operational central wavelengths, λ_2,n and λ_2,θ, can be fabricated, utilizing multiple holograms.

According to the coupled wave theory [19], when the holographic filter is reconstructed with reference beam R₂ incident perpendicularly and object beam O₂ diffracted in the opposite direction with a diffraction angle θ_d = 180°, as shown in Figure 1d and Figure 2d, the s- and p-polarized components of R₂ have the same diffraction efficiency. Accordingly, for the two operational central wavelengths λ_2,n and λ_2,θ, the diffraction efficiencies of the s- and p-polarized components can be expressed as:

$${\eta }_{s,p,l}={\tanh }^2\left[{\displaystyle \frac{\pi \mathsf{\Delta }{n}_l{d}_2}{{\lambda }_{2,l}}}\right]={\tanh }^2\left[\pi \mathsf{\Delta }{N}_l\right],$$

(4)

where l represents n or θ, Δn_n and Δn_θ are the modulation refractive index amplitudes, d₂ is the emulsion thickness, λ_2,n and λ_2,θ are the wavelengths of the reconstruction light, and ΔN_n and ΔN_θ are the effective refractive modulation.

Figure 3. Schematic setup for the simultaneous double-exposure contact mirror-based method, combining both normally and obliquely incident conditions.

Figure 3. Schematic setup for the simultaneous double-exposure contact mirror-based method, combining both normally and obliquely incident conditions.

3. Experimental Results and Discussion

To verify the feasibility of the method proposed in this study, a holographic multi-notch filter with dual operational central wavelengths was fabricated. The optical recording setup is shown in Figure 3. In this setup, the incident expanded beam (IEB_θ) was incident onto the holographic recording material (H) and M₁ at an oblique angle of θ = 55°. M₂ was perpendicular to M₁, and the angle between M₃ and M₂ was 62.5°. A photopolymer recording material (C-RT20, Litiholo, Hampton, VA, USA) with a thickness of 16 μm, which does not require chemical post-processing, was attached to the first mirror (M₁). Water (n_rm = 1.33 at 532 nm) was used as the refractive index matching liquid. A 50 mW diode-pumped solid-state laser (GLM-SOPT-50, LEADLIGHT TECHNOLOGY, INC., Taoyuan City, Taiwan) with a linearly polarized output at a wavelength of 532 nm was used for recording. The recommended exposure energy for C-RT20 at this wavelength is approximately 30 mJ/cm². The irradiance of the incident expanded beam (IEB_θ) was 0.639 mW/cm². The actual exposure time was 300 s, with an exposure energy of 192 mJ/cm². Since the Litiholo film develops as it is exposed, any additional laser exposure will have only a minor effect after the film has used up all of its exposure capacity. After exposure and drying, as shown in Figure 4, the fabricated holographic multi-notch filter (with dimensions approximately 8 mm × 7 mm) exhibited a brilliant rainbow appearance under white light illumination that varied with the observation angle, confirming the successful recording of interference fringes.

3.1. Measurement of Transmittances and Diffraction Efficiencies

The fabricated holographic multi-notch filter, mounted on a rotation stage, was tested at different reconstruction incident angles (θ_i) using a linearly polarization-controlled tungsten halogen white-light source (HL-2000, Ocean Optics, Dunedin, FL, USA) combined with a spectrometer (HR4000CG-UV-NIR, Ocean Optics), as shown in Figure 5. During the operation of the measurement system, only the holographic filter was rotated, while all other components remained fixed. The transmitted light was collected by the fiber optic, which had a collection microlens in front of it, and then directed to the spectrometer. To reveal the characteristics and details of the component and this recording method, the reconstruction incident angles (θ_i) were set to 0°, 10°, 20°, 30°, 40°, and 50°. The s- and p-polarized spectral intensities of the white-light source, I_in,s,p(λ), and the transmitted light through the holographic filter, I_out,s,p(λ), were measured. Using the measured data, the transmittance T_s,p,θi(λ) can be expressed as:

$$T_{s,p,\theta i}\left(\lambda \right)=t_{1,s,p,\theta i}\left(\lambda \right)t_{2,s,p,\theta i}\left(\lambda \right)=\left[{\displaystyle \frac{I_{\mathrm{o}\mathrm{u}\mathrm{t},s,p}\left(\lambda \right)}{I_{\mathrm{i}\mathrm{n},s,p}\left(\lambda \right)}}\right]\times 100\%,$$

(5)

where the subscripts θ_i, s, and p represent the values of the incident angle (θ_i) and the s- and p-polarized components, respectively. $t_{1,s,p,\theta i}\left(\lambda \right)$ represents the transmittance caused by the material, and $t_{2,s,p,\theta i}\left(\lambda \right)$ represents the transmittance caused by grating diffraction. $t_{2,s,p,\theta i}\left(\lambda \right)$= 1 − η_s,p,θi(λ), where η_s,p,θi(λ) is the spectral diffraction efficiency within the emulsion of the fabricated holographic filter. Accordingly, the spectral diffraction efficiencies η_s,p,θi(λ) can be expressed as:

$${\eta }_{s,p,\theta i}\left(\lambda \right)=\left\{1-\left[{\displaystyle \frac{I_{\mathrm{o}\mathrm{u}\mathrm{t},s,p}\left(\lambda \right)}{I_{\mathrm{i}\mathrm{n},s,p}\left(\lambda \right)t_{1,s,p,\theta i}\left(\lambda \right)}}\right]\right\}\times 100\%.$$

(6)

Using Equations (5) and (6), Figure 6 and Figure 7 show the transmittances for s- and p-polarizations and the spectral diffraction efficiencies within the emulsion for the holographic filter at normal incidence (θ_i = 0°) and at various reconstruction angles of incidence (θ_i = 10°, 20°, 30°, 40°, and 50°), respectively. The incident angles and corresponding dual operational central wavelengths, s- and p-polarization transmittances and diffraction efficiencies, and full widths at half maximum (FWHMs) are summarized in

Table 1. Summary of the corresponding wavelengths, transmittances, diffraction efficiencies, and full widths at half maximum (FWHMs) of the holographic filter at various incident angles.

Incident Angle (°)	Wavelength (nm)	Ts (%)	Tp (%)	ηs (%)	ηp (%)	FWHM(s/p) (nm)
0	531.13	33.36	33.28	52.35	52.45	10.3/10.3
	633.01	22.89	22.82	67.30	67.40	13.2/13.2
10	527.52	33.01	33.71	51.52	51.71	11.6/11.9
	630.21	26.12	26.54	61.65	61.98	14.8/14.8
20	517.01	30.87	35.06	54.13	50.62	10.6/11.1
	617.26	22.18	26.90	67.05	62.11	13.2/13.2
30	501.28	29.07	39.67	57.38	48.42	10.3/10.6
	598.16	19.39	30.66	71.57	60.13	13.5/12.5
40	480.35	27.56	46.35	60.12	42.05	9.67/14.8
	572.60	18.64	42.12	73.02	47.47	14.4/13.6
50	457.73	25.93	55.37	60.94	30.57	12.0/17.7
	545.11	13.35	48.20	80.21	39.51	14.9/12.6

.

Figure 5. Experimental setup for testing the fabricated holographic multi-notch filter at different reconstruction incident angles (θ_i).

Figure 5. Experimental setup for testing the fabricated holographic multi-notch filter at different reconstruction incident angles (θ_i).

3.1.1. Analysis of Transmittances and Diffraction Efficiencies Under the Normally Incident Reconstruction Condition

In Figure 6a, the holographic filter demonstrates polarization-independent dual operational central wavelengths at ${\lambda }_{2,n}$ = 531.13 nm and ${\lambda }_{2,\theta }$ = 633.01 nm, with transmittances T_s,p,0 of 33% and 23%, respectively. Due to the lack of anti-reflection coating on the component surface, the material transmittance of the non-filtered wavelength range is between $t_{1,s,p,0}\left(425\right)$ = 66% and $t_{1,s,p,0}\left(675\right)$ = 71%, which can be improved with an anti-reflection coating. Figure 6b shows the diffraction efficiencies η_s,p,0 within the emulsion for these two operating wavelengths, which are η_s,0 = 52.35% and η_p,0 = 52.45% for 531.13 nm, and η_s,0 = 67.30% and η_p,0 = 67.40% for 633.01 nm, with full width at half maximum (FWHM) values of 10.3 nm and 13.2 nm, respectively. These results confirm the dual operational central wavelength filtering function of the fabricated holographic filter. In the presentation of Figure 6b, to avoid negative values for the diffraction efficiency in the longer wavelength range, the transmittance $t_{1,s,p,\theta i}\left(\lambda \right)$ was set to 70% when calculating the diffraction efficiency curve using Equation (6).

Due to the limitations imposed by the emulsion thickness, the diffraction efficiencies for these two operational central wavelengths are relatively low. Using Equation (4), considering λ_2,n = 531.13 nm, λ_2,θ = 633.01 nm, η_s,p,n = 52.4%, η_s,p,θ = 67.35%, and d₂ = 16 μm, solving the equations yields Δn_n = 0.00968 and Δn_θ = 0.01459, and ΔN_n = 0.91610 and ΔN_θ = 1.15855. Due to λ_2,n being allocated a smaller effective refractive modulation compared to λ_2,θ, it has a lower diffraction efficiency. Using these parameters, the relationship between η_s,p and d₂ was plotted. As shown in Figure 8, when d₂ > 32 μm, the diffraction efficiencies for these two operational central wavelengths can exceed 90%. Therefore, higher diffraction efficiencies can be achieved by using the same commercial recording material with a greater thickness, or by increasing the refractive index modulation strength of the recording material.

In practice, in the recording setup of Figure 3, in addition to the two sets of gratings recorded by the interference of IEB_θ and IEB_n and their beam reflected by M₁, there should be four additional sets, two transmission gratings and two reflection gratings, resulting from the mutual interference between IEB_θ, IEB_n, and their reflected beams. These four sets of gratings do not diffract light at 180° during reconstruction like the original two sets; however, this does not affect the functionality of the filter. Furthermore, the theoretical reconstruction wavelengths for all four sets of gratings are close to 532 nm. However, as seen in Figure 6a, the transmission dip at 531.13 nm is not as pronounced as the one at 633.01 nm. A reasonable explanation is that after penetrating multiple interfaces, IEB_θ becomes IEB_n, and the ratio of the irradiances of IEB_θ and IEB_n is so disparate that it cannot maintain optimum modulation to form clear grating structures.

3.1.2. Analysis of Transmittances and Diffraction Efficiencies Under the Obliquely Incident Reconstruction Conditions

As shown in Figure 7a,b, it is evident that as the reconstruction incident angle increases from 0° to 50°, the dual operational central wavelengths of the multi-notch filter shift from 531.13 nm and 633.01 nm to shorter wavelengths of 457.73 nm and 545.11 nm, as summarized in Table 1. Due to the oblique incidence, the reconstruction beam R₂ and the diffracted beam O₂, which satisfy the Bragg condition, must have larger propagation vectors (|k′_R2,n| = |k′_O2,n| = 2πn′_f2/λ′_2,n and |k′_R2,θ| = |k′_O2,θ| = 2πn′_f2/λ′_2,θ) to form a closed isosceles triangle vector geometry with the inherent grating vector K_2,n and K_2,θ, as shown in Figure 9a,b. Therefore, a larger reconstruction incident angle corresponds to a larger propagation vector, i.e., shorter reconstruction wavelengths (${\lambda }_{2,n}^{\prime }\mathrm{a}\mathrm{n}\mathrm{d}{\lambda }_{2,\theta }^{\prime }$). In Figure 9, the angle ${\theta }_i^{\prime }$ is the refraction angle of the test light in the emulsion after it enters the component from the air.

In Figure 7a and Table 1, the transmittances for s- and p-polarizations of the filter show opposite trends as the reconstruction incident angle increases from 0° to 50°. This opposite trend is consistent with Fresnel transmittances (air to emulsion), where T_s decreases with increasing incident angle, and T_p increases with increasing incident angle between 0° and Brewster’s angle. The transmittance for s-polarization (T_s) of the dual operational central wavelengths generally decreased from 33.36% and 22.89% at 0° to 25.93% and 13.35% at 50°. The transmittance for p-polarization (T_p) of the dual operating wavelengths showed an increasing trend from 33.28% and 22.82% at 0° to 55.37% and 48.20% at 50°. In addition, at a larger reconstruction incident angle of 50°, the dispersion characteristics are significant, with T_p increasing with wavelength.

In Figure 7b and Table 1, the s- and p-polarization diffraction efficiencies within the emulsion for the dual operating wavelengths also show opposite trends as the reconstruction incident angle increases from 0° to 50°. The diffraction efficiency for s-polarization (η_s) of the dual operating wavelengths generally increased from 52.35% for 531.13 nm and 67.30% for 633.01 nm at 0° to 60.94% for 457.73 nm and 80.21% for 545.11 nm at 50°. The diffraction efficiency for p-polarization (η_p) of the dual operating wavelengths showed a decreasing trend from 52.45% for 531.13 nm and 67.40% for 633.01 nm at 0° to 30.57% for 457.73 nm and 39.51% for 545.11 nm at 50°. The full widths at half maximum (FWHMs) for s- and p-polarizations of these dual operating wavelengths slightly fluctuated between 9.67 nm and 14.9 nm as the reconstruction incident angle increased from 0° to 50°. In the presentation of Figure 7b, to avoid negative values for the diffraction efficiency in the longer wavelength range, the transmittances $t_{1,s,p,\theta i}\left(\lambda \right)$ were set to values between 67% to 80% when calculating the diffraction efficiency curve using Equation (6).

According to the coupled wave theory [19], as shown in Figure 5 and Figure 9, when test light is incident at an oblique angle ${\theta }_i^{\prime }$ and is diffracted at an angle θ_d = 180° − ${\theta }_i^{\prime }$ within the emulsion, the diffraction efficiencies for the s- and p-polarizations of the holographic filter can be expressed as:

$${\eta }_{s,p}={\displaystyle \frac{1}{1+\frac{1}{{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}}^2\upsilon }}},$$

(7)

where

$$\upsilon ={\upsilon }_s={\displaystyle \frac{i\pi \mathsf{\Delta }n{d}_2}{{\lambda }_r{\left(-{\mathrm{c}\mathrm{o}\mathrm{s}}^2{\theta }_i^{\prime }\right)}^{1/2}}},$$

(8)

and

$$\upsilon ={\upsilon }_p={-\upsilon }_s\mathrm{c}\mathrm{o}\mathrm{s}\left(2{\theta }_i^{\prime }\right),$$

(9)

where υ_s and υ_p are the coupling coefficients for the s- and p-polarizations, Δn is the modulation refractive index amplitude, d₂ is the emulsion thickness, and λ_r is the wavelength of the reconstruction light. When ${\theta }_i^{\prime }$ = 0°, the Equations (7)–(9) can be simplified into the form of Equation (4). Considering λ_r = 531.13 nm, Δn = 0.00968, and d₂ = 16 μm, the relationship between ${\theta }_i^{\prime }$ and the diffraction efficiency η_s,p can be obtained using Equations (7)–(9), as shown in Figure 10. From Figure 10, it is evident that the holographic filter is polarization-independent with the same diffraction efficiencies for s- and p-polarizations when the reconstruction light is incident perpendicularly (${\theta }_i^{\prime }$ = 0°). When the reconstruction light is incident at a large angle (${\theta }_i^{\prime }$ approaching 80°), the component remains polarization-independent with nearly 100% diffraction efficiency. For any other incident angles (0° < ${\theta }_i^{\prime }$ < 80°), the component exhibits polarization dependence, with the diffraction efficiencies for s- and p-polarizations being functions of the incident angle (${\theta }_i^{\prime }$). This explains the varying trends in s- and p-polarization diffraction efficiencies observed in Figure 7b and Table 1. Additionally, Figure 10 shows that the diffraction efficiency for p-polarization is zero when the incident light is at ${\theta }_i^{\prime }$ = 45°.

3.2. Relationships Between Recording Incident Angle, Recording Wavelength, and Dual Operational Central Wavelengths

From Equations (1)–(3), it is evident that the two operational central wavelengths of the fabricated holographic multi-notch filter, λ_2,n and λ_2,θ, are related to the recording wavelength λ₁, the dispersion characteristics of the photosensitive emulsion used in the holographic recording material, the emulsion’s expansion and contraction, and the recording incident angle θ of the exposure beam IEB_θ.

To understand the feasible wavelength range for the holographic multi-notch filter fabricated by the method proposed in this study, we consider simplified general conditions to explore the relationships between the recording incident angle θ (referred to as θ′ in the emulsion), the recording wavelength λ₁, and the corresponding normal reconstruction dual wavelengths λ_2,n and λ_2,θ.

Using the dispersion data of poly(methyl methacrylate) (PMMA), the main component of pan-photopolymer recording materials, we simulate the relationships between the recording incident angle, the recording wavelength, and the dual operational central wavelengths. The dispersion formula for PMMA can be expressed as [28]:

$$n^2-1={\displaystyle \frac{1.1819{\lambda }^2}{{\lambda }^2-0.011313}}$$

(10)

By substituting the experimental parameters λ₁ = 532 nm and λ_2,n = 531.13 nm into Equation (1) and ignoring the emulsion’s expansion and contraction, we find that the refractive index of the photopolymer after exposure is less than that before exposure, with a ratio of n_f2,n/n_f1 = 0.998. Therefore, the refractive index change approximates 0.0024.

According to Equations (1), (2) and (10), and as shown in

Table 2. Summary of wavelengths (λ₁), initial refractive indices (n_f1), calculated first operational wavelengths (λ_2,n), and refractive indices after exposure (n_f2,n) for common commercial holographic laser sources using pan-photopolymer recording materials. Refractive index change value equals 0.0024. Grating period ratio: Λ_2,n/Λ_1,n = 1.

λ1 (μm)	nf1	λ2,n (μm)	nf2,n
0.450	1.5006	0.449	1.4982
0.532	1.4937	0.531	1.4913
0.633	1.4887	0.632	1.4863

, we consider common commercial holographic laser sources with wavelengths λ₁ = 0.450  μm, 0.532  μm, and 0.633 μm. Using the simulation parameters n_f1 = 1.5006 (at λ₁ = 0.450  μm), 1.4937 (at λ₁ = 0.532  μm), 1.4887 (at λ₁ = 0.633  μm), a refractive index change value of 0.0024, and a grating period ratio Λ_2,n/Λ_1,n = 1, the first operational central wavelengths λ_2,n of the dual operating wavelength holographic notch filter are calculated to be 0.449 μm, 0.531 μm, and 0.632 μm, respectively.

As shown in Figure 11, the second operational central wavelength λ_2,θ increases with the recording incident angle θ′ in the emulsion, covering a range from visible light to the near-infrared. Compared to the refractive index of the photopolymer recording material, the second operational central wavelength λ_2,θ is primarily determined by the recording incident angle θ′. Longer recording wavelengths λ₁ correspond to larger operational wavelengths λ_2,θ. In cases of larger incident angles θ′, an isosceles trapezoid prism can be used to guide the recording beam for better component recording effects, as shown in Figure 12.

4. Conclusions

This study introduces a method for recording holographic multi-notch filters using simultaneous double-exposure with a contact mirror setup. The fabricated component’s characteristics were analyzed using polarized spectral transmittance at different reconstruction angles. The component exhibited polarization-independent transmission and diffraction efficiency under normal incidence but displayed polarization-dependent behavior under oblique incidence. When the emulsion thickness exceeded 32 µm, the diffraction efficiency could exceed 90%, significantly enhancing the notch filtering performance. The first operational wavelength is mainly determined by the recording wavelength, while the second is influenced by the incident angle, covering a range from visible light to near-infrared. This method offers significant potential for cost-effective, mass-produced filters in optoelectronic applications.