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Showing papers on "Diffraction efficiency published in 2004"


Patent
Koichi Maruyama1, Daisuke Koreeda1
16 Jun 2004
TL;DR: In this article, an objective lens of an optical pick-up that converges light beams of two different wavelengths onto recording layers of optical discs of different recording densities is described.
Abstract: Disclosed is an objective lens of an optical pick-up that converges light beams of two different wavelengths onto recording layers of optical discs of two different recording densities, respectively. The objective lens includes a refractive lens whose one surface is divided into a common area through which a light beam of a low NA, which is necessary and sufficient for an optical disc having low recording density, passes and a high NA exclusive area through which a light beam of a high NA, which is necessary only for an optical disc having high recording density, passes. A diffractive lens structure is formed in both the areas. The diffractive lens structure in the common area maximizes the diffraction efficiency of the first order and that in the high NA exclusive area maximizes the diffraction efficiency of the second or third order at the wavelength corresponding to the optical disc having high recording density.

198 citations


Journal ArticleDOI
TL;DR: By preventing the appearance of the nematic phase during the curing process, it is possible to avoid the formation of liquid-crystal droplets and obtain a sharp and uniform morphology, which reduces scattering losses and increases diffraction efficiency.
Abstract: We present a new kind of UV-cured holographic grating that consists of polymer slices alternated with pure nematic films. By preventing the appearance of the nematic phase during the curing process, it is possible to avoid the formation of liquid-crystal droplets and obtain a sharp and uniform morphology, which reduces scattering losses and increases diffraction efficiency.

181 citations


Journal ArticleDOI
TL;DR: In this paper, volume holographic recording in silica-nanoparticle-dispersed methacrylate photopolymers with reduced scattering loss as low as 2% was demonstrated.
Abstract: We demonstrate volume holographic recording in silica-nanoparticle-dispersed methacrylate photopolymers with reduced scattering loss as low as 2%. This is made possible by use of 13-nm silica nanoparticles. As a result a net diffraction efficiency near 100% is achieved for a transmission volume hologram of 45-μm thickness. Grating buildup dynamics are measured for various nanoparticle concentrations, and the effects of nanoparticle size on refractive-index modulation and polymerization shrinkage are also evaluated.

159 citations


Journal ArticleDOI
TL;DR: This work proposes two simple alternative ways to focus radially polarized beams: a parabolic mirror and a flat diffractive lens, because of their large apodization factor for a high NA.
Abstract: Recently, Dorn [Phys. Rev. Lett.91, 233901 (2003)] demonstrated the significance of radially polarized doughnut beams in obtaining very small focal spots (with an area of ∼0.26λ2) with high-numerical-aperture (NA) aplanatic microscope objectives. We propose two simple alternative ways to focus such radially polarized beams: a parabolic mirror and a flat diffractive lens. Because of their large apodization factor for a high NA, a significant further reduction in spot area (up to a factor of 1.76 at a NA of 1) compared with the aplanatic system can be achieved.

130 citations


Journal ArticleDOI
TL;DR: Multilevel phase-type diffractive lenses were fabricated by translating a filament of a femtosecond laser pulse into three-dimensional space inside synthesized silica inside synthesised silica by use of a multi-level approximation to a kinoform lens.
Abstract: Multilevel phase-type diffractive lenses were fabricated by translating a filament of a femtosecond laser pulse into three-dimensional space inside synthesized silica. The profile of the lenses was designed by use of a multilevel approximation to a kinoform lens. Two-level diffractive lenses with multiple layers along the optical axis provided a maximum efficiency of 37.6% at a wavelength of 632.8 nm. A four-level diffractive lens provided a maximum efficiency of 56.9%. The lenses fabricated with filamentation were free from birefringence.

110 citations


Journal ArticleDOI
TL;DR: A bandpass filter based on guided-mode resonance effects in a single-layer periodic waveguide is presented and it is shown that one resonance provides a broad low-transmission band, whereas the other supplies the transmission peak through its asymmetrical line shape.
Abstract: A bandpass filter based on guided-mode resonance effects in a single-layer periodic waveguide is presented. Strong refractive-index modulation is used to support the excitation of a resonance pair in which leaky waveguide mode TE0 is excited by the second diffraction order and TE2 by the first diffraction order. It is shown that one resonance provides a broad low-transmission band, whereas the other supplies the transmission peak through its asymmetrical line shape. A bandpass filter with central wavelength at 1.55 µm is presented to demonstrate this concept.

109 citations


Journal ArticleDOI
TL;DR: In this paper, the first-order diffraction efficiencies of volume phase holographic (VPH) transmission gratings are discussed and the authors show that the variation of the Kogelnik efficiency versus Bragg angle depends only on one parameter, given by Ptune = (Δnd)/(nΛ), where n is the average index, d is the thickness of the active layer, and Λ is the grating period.
Abstract: We discuss the polarization properties and first-order diffraction efficiencies of volume phase holographic (VPH) transmission gratings, which can be exploited to improve the throughput of modern spectrographs. The wavelength of peak efficiency can be tuned by adjustment of the incidence angle. We show that the variation of the Kogelnik efficiency versus Bragg angle depends only on one parameter, given by Ptune = (Δnd)/(nΛ), where Δn is semiamplitude of the refractive index modulation, n is the average index, d is the thickness of the active layer, and Λ is the grating period. The efficiency has a well-defined dependence on polarization. In particular, it is possible to obtain theoretical 100% diffraction efficiency with one linear polarization at any angle, or to obtain 100% efficiency with unpolarized light at specific angles. In the latter case, high efficiency is the result of aligning the peaks of the s- and p-polarization efficiency-versus-thickness curves. The first of these "s-p-phased gratings" for astronomy is in use with the 6dF spectrograph. Consideration of polarization is particularly important for high spectral resolution, which requires large incidence angles. We also discuss the possibility of separating polarization states for improved throughput along the entire optical train of a spectrograph.

109 citations


Journal ArticleDOI
TL;DR: In this article, the authors developed a phenomenological model of grating formation that incorporates the photophysics and photochemistry of the initiator dye, reaction-diffusion kinetics of the monomer-polymer system, phase separation of the liquid crystal, nematic order evolution of liquid-crystal droplets, and volume shrinkage of the polymer.
Abstract: The real time formation of anisotropic volume holographic reflection gratings in a liquid-crystal/photopolymer mixture is studied. We develop a phenomenological model of grating formation that incorporates the photophysics and photochemistry of the initiator dye, reaction-diffusion kinetics of the monomer-polymer system, phase separation of the liquid crystal, nematic order evolution of liquid-crystal droplets, and volume shrinkage of the polymer. We then test this model by experimentally monitoring the diffraction efficiency for s and p polarization, Bragg wavelength, and laser scattering in real time as the grating is formed. The model yields good agreement with experimental data for different recording intensities and exposure times. We discuss the physics of the system as it evolves in time and explain the major features of anisotropic grating formation in acrylate-based holographic polymer-dispersed liquid crystals.

88 citations


Patent
12 Nov 2004
TL;DR: In this article, an apparatus and a method of printing a diffraction grating was described. But the present paper relates to diffraction gratings applied to a substrate (1), such as a hologram.
Abstract: There is provided an apparatus and a method of printing a diffraction grating. In particular, the present invention relates to diffraction gratings applied to a substrate (1), such as a hologram.

82 citations


Journal ArticleDOI
TL;DR: An analysis of the diffraction efficiency of diffractive lenses displayed on spatial light modulators that depends on the modulation response of the display is presented and it is demonstrated that the application of a minimum Euclidean projection principle leads to high modulation Diffraction efficiency even if the phase-modulation depth is much less than 2pi.
Abstract: We present an analysis of the diffraction efficiency of diffractive lenses displayed on spatial light modulators that depends on the modulation response of the display. An ideal display would produce continuous phase-only modulation, reaching a maximum phase-modulation depth of 2π. We introduce the concept of modulation diffraction efficiency that accounts for the effect of nonlinearities only in the phase modulation of the display. We review a diffractive model with which to evaluate this modulation efficiency, including modulation defects such as nonlinear phase modulation, coupled amplitude modulation, phase quantization, and a limited modulation depth. We apply this diffractive model to Fresnel lenses and show that these modulation defects produce a lens multiplex effect. Finally we demonstrate that the application of a minimum Euclidean projection principle leads to high modulation diffraction efficiency even if the phase-modulation depth is much less than 2π. We demonstrate that the modulation efficiency can exceed 90% for a modulation depth of 1.4π and can exceed 40% (the equivalent for a binary phase element) for a modulation depth of only 0.7π. Experimental results from use of a twisted nematic liquid-crystal display are presented to confirm these conclusions.

76 citations


Journal ArticleDOI
TL;DR: In this paper, a grating-embedded multilayer structure with (2M /spl times 2M) dynamic and propagation matrices in both transverse electric and transverse magnetic polarizations is presented.
Abstract: Using a rigorous coupled-wave analysis, a (2 /spl times/ 2) matrix method in homogenous layered media is generalized for a grating-embedded multilayer structure with (2M /spl times/ 2M) dynamic and propagation matrices in both transverse-electric and transverse-magnetic polarizations, where M is the number of harmonics retained in the field expansion. A numerically stable algorithm to implement the method is described. Numerical results are presented for a diffraction-based optoacoustic sensor that consists of a reflective movable membrane and fixed grating fingers on a transparent quartz substrate and are compared with the scalar diffraction method to illustrate the limited applicability of the scalar approach.

Patent
12 Aug 2004
TL;DR: In this article, a photopolymerizable material was proposed for single-step, fast recording of volume holograms with properties that can be electrically controlled, which can be used for switchable optical coupling and reconfigurable optical interconnects.
Abstract: A new photopolymerizable material allows single-step, fast recording of volume holograms with properties that can be electrically controlled. Polymer-dispersed liquid crystals (PDLCs) in accordance with the invention preferably comprise a homogeneous mixture of a nematic liquid crystal and a multifunctional pentaacrylate monomer in combination with photoinitiator, coinitiator and cross-linking agent. Optionally, a surfactant such as octancic acid may also be added. The PDLC material is exposed to coherent light to produce an interference pattern inside the material. Photopolymerization of the new PDLC material produces a hologram of clearly separated liquid crystal domains and cured polymer domains. Volume transmission gratings made with the new PDLC material can be electrically switched between nearly 100% diffraction efficiency and nearly 0% diffraction efficiency. By increasing the frequency of the switching voltage, switching voltages in the range of 50 Vrms can be achieved. The optional use of a surfactant allows low switching voltages at lower frequencies than without a surfactant. In an alternative embodiment, a PDLC material in accordance with the invention can be utilized to form reflection gratings, including switchable reflection gratings. In still further embodiments, a PDLC material in accordance with the invention can be used to form switchable subwavelength gratings. By further processing, static transmission, reflection, and subwavelength PDLC materials can be formed. In addition, PDLC materials in accordance with the present invention can be used to form switchable slanted transmission gratings suitable for switchable optical coupling and reconfigurable optical interconnects.

Journal ArticleDOI
TL;DR: The large-scale VPHG research facility set up at the Center Spatial de Liege enables production of VPHGs up to 380 mm in diameter, with fringe frequencies from 315 to 3300 lp/mm.
Abstract: Volume phase holographic gratings (VPHGs) possess unique properties that make them attractive for numerous applications. After reviewing major VPHG characteristics through theory, we discuss some aspects of the dichromated gelatin recording material and the holo- graphic recording process. The large-scale VPHG research facility set up at the Center Spatial de Liege enables production of VPHGs up to 380 mm in diameter, with fringe frequencies from 315 to 3300 lp/mm. We describe the work that has been undertaken in our laboratory to remove the last limitations inherent in VPHGs. © 2004 Society of Photo-Optical Instru- mentation Engineers. (DOI: 10.1117/1.1803557)

Journal ArticleDOI
TL;DR: A flexible setup for holographic steering of laser tweezers in microscopy using a high resolution spatial light modulator (SLM) and hologram read-out is done in the off-axis Fresnel regime rather than in the typically used on-axis Fourier regime.
Abstract: We demonstrate a flexible setup for holographic steering of laser tweezers in microscopy using a high resolution spatial light modulator (SLM). In contrast to other methods, hologram read-out is done in the off-axis Fresnel regime rather than in the typically used on-axis Fourier regime. The diffractive structure is calculated as a Fresnel hologram, such that after reflection at the SLM only the desired first diffraction order is guided to the input of an optical microscope, where it generates a tailored optical tweezers field. We demonstrate some advantageous features of this setup, i.e. undesired diffraction orders are suppressed, the optical traps can be easily steered in real-time by just "mouse-dragging" a hologram window at the SLM display, and a number of independently steerable optical traps can be generated simultaneously in a three-dimensional arrangement by displaying a corresponding number of adjacent hologram windows at the SLM screen.

Journal ArticleDOI
TL;DR: A telecentric confocal optical arrangement is presented that greatly reduces the diffraction aberrations of the acousto-optic tunable filter (AOTF) and substantial improvement in the potential spatial resolution is predicted with confocal optics.
Abstract: A telecentric confocal optical arrangement is presented that greatly reduces the diffraction aberrations of the acousto-optic tunable filter (AOTF). Analytical expressions for the aberrations were identified based on the fundamental properties of Bragg diffraction, and additional aberrations due to focusing through the AOTF were also included. The analysis was verified by use of a geometrical ray trace optical code, and an experimental AOTF system was analyzed. Considerable improvement in the potential spatial resolution is predicted with confocal optics, which could accommodate large pixel-limited image fields of greater than 106 pixels. When the image quality of the experimental system was assessed, the resolution was found to be improved by the confocal optics and was diffraction limited. Higher resolution could have been obtained with the use of larger optics to increase the throughput before being limited by the aberrations.

Journal ArticleDOI
TL;DR: This work uses high dispersion of artificial materials to synthesize diffractive optical elements that are blazed over a broad spectral range or for two different wavelengths.
Abstract: The efficiency of conventional diffractive optical elements with echelette-type profiles drops rapidly as the illumination wavelength departs from the blaze wavelength. We use high dispersion of artificial materials to synthesize diffractive optical elements that are blazed over a broad spectral range (∼1 octave) or for two different wavelengths.

Journal ArticleDOI
TL;DR: A diffraction biosensor to monitor the dynamic interaction of biological molecules in a label-free way was realized by the binding of an anti-biotin antibody to the biotin-functionalized region of a periodically patterned surface, which generated significant optical contrast to diffract the surface plasmon field.
Abstract: Surface plasmon enhanced evanescent field at a (noble) metal/dielectric interface can be employed to enhance the diffraction efficiency of surface grating structure composed of biomolecules. Based on a Kretschmann configuration, we realized a diffraction biosensor to monitor the dynamic interaction of biological molecules in a label-free way. It was demonstrated by the binding of an anti-biotin antibody to the biotin-functionalized region of a periodically patterned surface, which generated significant optical contrast to diffract the surface plasmon field. With the aid of the synchronic surface plasmon resonance signal, a quadratic dependence of diffraction signal on the amount of bound antibody was found, which coincides with the theoretical expectation. Time-dependent measurements were conducted to estimate the density of biotin thiols on the functional region.

Journal ArticleDOI
TL;DR: This work presents a comprehensive analysis, based on the leading term and the linear superposition principle, of the focusing performances of various Fresnel zone plates and most of the obtained results are also applicable to multilevel diffractive lenses.
Abstract: A series-form expression for the individual diffracted field of a general annular ring is derived from the Rayleigh-Sommerfeld diffraction integral. It can be used for the accurate and fast simulation of any diffractive focusing element composed of concentric transparent rings. We present a comprehensive analysis, based on the leading term and the linear superposition principle, of the focusing performances of various Fresnel zone plates. Many problems, such as the equivalent aperture function, the diffraction efficiency, the focal spot pattern, the suppression of higher orders and the appearance of "fractional orders," and the explanation for the appearance of Fraunhofer diffraction patterns, are analytically investigated in detail. Because of the great similarity between Fresnel zone plates and multilevel diffractive lenses, most of the obtained results are also applicable to multilevel diffractive lenses.

Journal ArticleDOI
TL;DR: A novel scheme of an iterative Fourier transform algorithm with regularization to obtain the optimum trade-off between uniformity and diffraction efficiency in the design of diffractive optical elements is proposed.
Abstract: There is a trade-off between uniformity and diffraction efficiency in the design of diffractive optical elements. It is caused by the inherent ill-posedness of the design problem itself. For the optimal design, the optimum trade-off needs to be obtained. The trade-off between uniformity and diffraction efficiency in the design of diffractive optical elements is theoretically investigated based on the Tikhonov regularization theory. A novel scheme of an iterative Fourier transform algorithm with regularization to obtain the optimum trade-off is proposed.

Journal ArticleDOI
TL;DR: In this article, the authors performed a characterization of the diffraction efficiency of Bragg gratings written in liquid-crystalline composite materials by the interference pattern of two curing beams.
Abstract: We have performed a first characterization of the diffraction efficiency of gratings written in liquid-crystalline composite materials by the interference pattern of two curing beams. The grating fringes consist of polymer slices separated by films of continuous nematic phase. The dependence of the diffraction efficiency on temperature reveals a nonmonotonic behavior, with several maxima and minima. The shapes of curves are dependent on slight changes in the initial concentration of the nematic component of the mixture; the number of extrema increases with an increase of this concentration. The dependence of the diffraction efficiency on an applied external voltage also appears to be nonmonotonic: The shape depends on the sample’s temperature. Both switch-on and switch-off responses have been observed. The behavior of our gratings can be explained in the framework of the conventional Kogelnik theory for the diffraction efficiency of Bragg gratings.

Journal ArticleDOI
TL;DR: In this article, a time-compensated monochromator with toroidal gratings used in conical diffraction is presented, which is shown to be very effective in a broad spectral region, ranging from VUV to soft X-rays, with much higher efficiency than the classical diffraction mounting.
Abstract: The use of a grating monochromator for the selection of one or more high-order laser harmonics produced by a femtosecond pulse interacting with a gas jet may alter the duration of the pulse itself. This is due to the differences in the optical paths of the rays caused by ordinary diffraction when a grating is used. The time stretching can be almost eliminated by using two gratings in time-compensated configurations. Unfortunately, the classical diffraction mounting has low efficiency, overall in the extreme-ultraviolet region. High broadband efficiency can be obtained by using the conical diffraction mounting. A time-compensated monochromator with toroidal gratings used in conical diffraction is here presented. It is shown that the time compensation is very effective in a broad spectral region, ranging from VUV to soft X-rays, with much higher efficiency than the classical diffraction mounting.

Patent
07 Sep 2004
TL;DR: In this paper, a volume phase grating and a prism coupling structure are used to diffract the various wavelengths through angles associated with improving the coupling efficiency of each wavelength into the silicon waveguide, and the diffractive optical element may be formed as a separate element or formed as an integral part of the coupling facet of the prism coupler.
Abstract: A coupling arrangement for allowing multiple wavelengths to be coupled into and out of a relatively thin silicon optical waveguide layer utilizes a diffractive optical element, in the form of a volume phase grating, in combination with a prism coupling structure. The diffractive optical element is formed to comprise a predetermined modulation index sufficient to diffract the various wavelengths through angles associated with improving the coupling efficiency of each wavelength into the silicon waveguide. The diffractive optical element may be formed as a separate element, or formed as an integral part of the coupling facet of the prism coupler.

Journal ArticleDOI
TL;DR: A new class of high-dispersion immersed diffraction gratings for which the reflective nature of the diffraction is provided by the phenomenon of total internal reflection regardless of grating tooth shape is reported, suitable for many laser-based technologies.
Abstract: We report a new class of high-dispersion immersed diffraction gratings for which the reflective nature of the diffraction is provided by the phenomenon of total internal reflection (TIR) regardless of grating tooth shape. Thus, the component can be fabricated from a single dielectric material and requires no metallic or dielectric film layers for high reflection diffraction efficiency. With the absence of metallic absorption, diffraction efficiencies of these TIR gratings can reach more than 99% for 15–20-nm spectral bandwidths, making them suitable for many laser-based technologies.

Journal ArticleDOI
TL;DR: In this paper, a micromechanical grating array is used as a configurable optical filter for low-cost and compact visible and near-infrared spectrometric sensors.
Abstract: We present a micromechanical grating array that acts as a configurable optical filter for low-cost and compact visible and near-infrared spectrometric sensors. We show how the grating array can be used either as a fast scanning monochromator or as a diffractive filter, and that the expected signal-to-noise ratio is approximately equal for the two modes of measurement. The free spectral range of the filter can be matched to a defined spectral measurement region, so that we can optimize the relationship between spectral resolution and electromechanical complexity. We have numerically studied diffraction efficiency and errors in filter shape. Finally, we fabricated a small configurable grating array and present measurement results that demonstrate electrostatic filter modulation.

Patent
12 Mar 2004
TL;DR: A diffraction element having a large wavelength separating effect with a high diffraction efficiency despite its excellent mass-productivity using a simple process is described in this paper, where one cycle of a lattice is equal to or smaller than the wavelength of an incident light.
Abstract: A diffraction element having a large wavelength separating effect with a high diffraction efficiency despite its excellent mass-productivity using a simple process. A diffraction element (30) formed on a transparent substrate (301), and being protruded and recessed in section with protruded portions forming symmetrical rectangular lattices (302), wherein one cycle of a lattice is equal to or smaller than the wavelength of an incident light, and the diffraction element is used with a light entering obliquely the lattice formed surface of the diffraction element (30).

Journal ArticleDOI
TL;DR: In this article, a photorefractive polymer composite sensitized at 1550 nm through direct two-photon absorption has been developed, which shows an external diffraction efficiency of 3% in four-wave mixing experiments and performs holographic reconstruction of distorted images utilizing thin-film devices made of this polymer composite.
Abstract: A photorefractive polymer composite sensitized at 1550 nm through direct two-photon absorption has been developed. We show an external diffraction efficiency of 3% in four-wave-mixing experiments and perform holographic reconstruction of distorted images utilizing thin-film devices made of this polymer composite. Amongst other potential applications, the demonstration of accurate, dynamic aberration correction through holography in this all-organic photorefractive device presents an alternative to complex adaptive optics systems currently employed in through-air optical communication links.

Journal ArticleDOI
TL;DR: In this article, the most promising applications of diffractive beam splitting are discussed, including spot array generation, multiple and multifocal imaging, matched filtering, laser beam mode selection and simultaneous contour shaping.
Abstract: Recent advances in diffractive optics theory and technology have made it possible to design programmable, multichannel optical systems based on diffraction gratings, lenslet arrays and digital holography algorithms. Spot array generation, multiple and multifocal imaging, matched filtering, laser beam mode selection and simultaneous contour shaping are among the most promising applications of diffractive beam splitting.

Journal ArticleDOI
TL;DR: The diffraction properties at wavelengths of 632.8 and 1550 nm for volume transmission gratings made of a sequence of continuously aligned nematic liquid-crystal layers separated by isotropic polymer slices are reported.
Abstract: We report the diffraction properties at wavelengths of 632.8 and 1550 nm for volume transmission gratings made of a sequence of continuously aligned nematic liquid-crystal layers separated by isotropic polymer slices. The gratings are generated by holographically curing a solution of liquid crystal diluted in an isotropic prepolymer by means of a laser beam at a wavelength of 352 nm with a total intensity of approximately 10 mW/cm2. A diffraction efficiency of 98% was measured, and an electric field as low as 5 V/µm switches off the phase grating. Measured angular spectra are fitted by use of the modified coupled-mode theory including the effects of grating birefringence.

Journal ArticleDOI
TL;DR: It is revealed that the first-order diffraction efficiency reaches the maximum value (theoretically 100%, experimentally approximately 90%) at an optimum applied voltage when the phase difference between the extraordinary and ordinary rays agrees with one-half the wavelength.
Abstract: We propose a novel formation method of arbitrary phase profiles of circular light by controlling azimuthal angles of liquid-crystal directors; its principle is described theoretically. A new liquid-crystal blazed grating is demonstrated by use of the proposed method. It is revealed that the first-order diffraction efficiency reaches the maximum value (theoretically 100%, experimentally approximately 90%) at an optimum applied voltage when the phase difference between the extraordinary and ordinary rays agrees with one-half the wavelength. Furthermore, the polarization states of the diffracted light beams are analyzed by Stokes parameter measurements, and unique polarization-splitting properties are revealed.

Patent
09 Feb 2004
TL;DR: In this article, a first diffractive optical element pattern with a pattern pitch that is no greater than a wavelength of incident light is formed on a first main surface of substrate such as a glass plate.
Abstract: A first diffractive optical element pattern with a pattern pitch that is no greater than a wavelength of incident light is formed on a first main surface of substrate such as a glass plate. Second diffractive optical element patterns are formed at positions that are respectively incident to positive first-order diffracted light and negative first-order diffracted light produced by the first diffractive optical element pattern. Negative first-order diffracted light produced by each second diffractive optical element pattern is incident upon a boundary face of the substrate at an angle that is smaller than the critical angle, and so exits the substrate.