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In laser projection applications, laser light modules are often combined with rotating diffusers in order to reduce the appearance of speckle on the projection screen. The rotation of a diffuser in a laser beam generates a beam of partially coherent light. Propagation of this light through the different optical components constituting the laser projector is thus essential when investigating the appearance of speckle. In this paper, a computationally efficient simulation model is presented to propagate partially coherent light through a homogenizing rectangular light pipe. The light pipe alters the coherence properties of the light and different consequences are discussed. The outcomes of the simulation model are experimentally verified using a reversing wavefront Michelson interferometer.
© 2013 Optical Society of America
1. Introduction
Laser light sources are currently becoming attractive for laser projection applications, due to the possibility of obtaining a wide color gamut and high luminance [1–3]. Inherently connected to the use of lasers is the appearance of speckle. A speckle pattern is created by interference of (at least partially) coherent radiation that is scattered from a random surface that is rough on the scale of the optical wavelength [4].
A common way of reducing speckle is by placing a moving diffuser in the laser beam [5]. This moving diffuser reduces the coherence area of the laser beam, resulting in a beam of partially coherent light. Simulation of partially coherent light propagation is thus essential to investigating the appearance of speckle.
Propagation of partially coherent light involves, in general, the evaluation of four-dimensional integrals, which is a tremendous computational task [6]. The computational complexity of propagating such radiation can be significantly reduced if one can represent the partially coherent field in terms of fully coherent, but mutually uncorrelated modes. Several alternative principles for such a decomposition exist, but in the context of this work the so-called shifted elementary-mode approach [7, 8] is probably the most meaningful one. In that approach, one expresses the partially coherent field as a superposition of mutually independent coherent elementary fields, which are identical but originate from different positions at the source plane. The shape of each elementary mode is fully defined by the Fourier transform of the source’s far-field intensity pattern. Remark that each mode has the same shape and hence the modes create identically-shaped far-field intensity patterns. The far-field intensity pattern after the rotating diffuser has a Gaussian shape. Hence, the elementary modes also need to have a Gaussian shape.
The interaction between partially coherent light and the complex optical components in a projector setup is of particular interest when studying the speckle in such laser projectors. In projection applications, several optical components have the sole purpose of homogenizing the light beam, such as for example a light pipe or a lenslet integrator. Both of these components exert a large influence on the propagated light. Lenslet integrators have been thoroughly investigated and described [9]. In this paper, we investigate the effect of a light pipe on the coherence properties of light. We will do this both experimentally and based on numerical simulations, where the experiments confirm the validity of the numerical model while the modelling provides a better insight into the influence of the various setup parameters.
A simulation model is presented to propagate partially coherent light through a light pipe. Propagation of partially coherent light can be investigated on the basis of field tracing [10] and on a ray-tracing basis [11]. However, interaction of partially coherent light with optical structures that perform complex optical tasks, such as homogenization, either become too cumbersome to simulate on a field tracing approach or are oversimplified in ray-tracing, i.e. many diffraction effects are neglected. The simulation model described in this paper is capable of investigating these situations and the results are experimentally verified.
In Section 2, the investigated configuration is elaborated on and the simulation model is described together with the representation of the source, the behavior of a light pipe and the evaluation of the coherence properties. The experimental verification method is discussed in Section 3. The interaction of the light with the light pipe is described in Section 4. Section 5 elaborates on the consequences and outlook, and provides a conclusion of the investigation.
2. Optical simulation model
The setup investigated in this paper is depicted in Fig. 1. The setup consists of a single-mode laser with a center wavelength of 532nm and an output power of 125mW in combination with a rotating diffuser (600 GRIT polish; diffusion angle 9°). After this rotating diffuser, a rectangular glass light pipe is placed. The purpose of this light pipe is to homogenize the incoming light through the many internal reflections of the light beam. The light pipe has an entrance facet of 8 × 15mm2 and a length of 13cm.
Fig. 1 Schematic of the investigated setup that consists of a single-mode laser in combination with a rotating diffuser, closely followed by a rectangular glass light pipe with a length of 13cm. We also denote in the figure the axes orientation used in the modelling.
2.1. Source model
Propagation of partially coherent light is simulated by means of an incoherent superposition of identical, but spatially shifted coherent elementary fields[8]. Since these elementary fields are of identical functional form, only a single two-dimensional (2-D) integration is necessary for their free-space propagation. The shape of the elementary field can be straightforwardly determined from the far-field radiant intensity pattern of the source. In the case of a single mode laser with Gaussian cross section in combination with a rotating diffuser, this radiant intensity pattern is very close to Gaussian and, therefore, also the shape of the elementary fields is Gaussian. In general, the combination of a Gaussian laser beam and a rotating diffuser leads to a Gaussian Schell-model beam in which case, not only the intensity distribution, but also the degree of spatial coherence is of a Gaussian form [12–14]. Hence in the remainder of this article, we shall consider this class of partially coherent beams only.
Gaussian Schell-model sources can be described analytically in close analogy to coherent Gaussian beams. As a limiting case, this class includes also certain quasihomogeneous sources (such as light emitting diodes), for which the radius of the effective coherence area is much smaller than the effective source area. Gaussian Schell-model sources are common in practice, e.g. in the case of various lamps, light-emitting diodes, multi-mode fibers and pulsed broad-area vertical-cavity surface-emitting lasers [15].
An illustration of the representation of the source can be found in Fig. 2. It consists of a set of uncorrelated, identical, fully coherent, transversely shifted elementary modes of finite extent. The shape of the elementary modes is Gaussian and for the diffuser used in our setup, the diameter w of these modes is determined by the diffusion angle using the near-field to far-field Gaussian distribution formulas
where θdiff is the diffusion angle and λ is the wavelength. The resulting elementary mode diameter is equal to 2.1μm, corresponding to a diffusion angle of 9°. These elementary modes are weighted by a function determined by the cross-spectral density (CSD) of the source. This weighting function can be visualized at the plane of the diffuser and is also a Gaussian profile. Approximately 50 elementary modes are utilized to perform the simulations. We have noticed that this amount of modes results in elementary mode profiles that are sufficiently overlapping at the first optical interface such that their sum is a good approximation of the source’s beam at this interface. Adding more elementary modes does not alter the outcomes and therefore, we have limited the simulation to this amount of elementary modes. The extent of the weighting function is measured and discussed in Section 4.1.Fig. 2 Intensity distribution in the transverse direction of the elementary Gaussian modes constituting the total source field. The source model consists of a Gaussian weighted linear superposition of spatially shifted but identical, fully coherent elementary Gaussian-shaped modes.
Since we consider paraxial light propagation only, and since the studied optical system does not alter the polarization properties of light, it suffices to describe light by a scalar field U. This complex scalar function must satisfy the Helmholtz wave equation, i.e. (∇2 + k2)U = 0, with k = 2π/λ as the wave number. Making use of Green’s theorem, the Rayleigh-Sommerfeld formulation is derived from the wave equation for the propagation of light in free-space [16]
where , z is the distance from the aperture (z = 0) to an observation plane and the integration is performed over the area of the aperture. This formulation is valid as long as both the propagation distance and the aperture size are greater than the wavelength of the light. As can be deducted from Eq. (2), the propagation of the partially coherent light with the Rayleigh-Sommerfeld formulation is not quite much different from a ray-tracing principle. Every point in the aperture plane influences every point in the observation plane and all these individual influences are added. In order to further reduce the computation time of our simulations, we approximated the field as a spatially separable field in x and y dimensions with U(x, y, z) = Ux(x, z)Uy(y, z). Such an approximation should not reduce the accuracy of the results significantly, as the light pipe is of a rectangular shape. Combination of propagated fields Ux and Uy provides a 2-D wavefront at the observation plane.Simulation of the complex wave front is implemented by dividing both the aperture and observation plane into gridded meshes. The Rayleigh-Sommerfeld propagation method is implemented in a fast Fourier transform (FFT) approach [17].
2.2. Simulation model of the light pipe
A light pipe is frequently used in projection applications to obtain a homogeneous light distribution at the exit facet. The homogenization is the result of various internal reflections. This can be better understood based on Fig. 3. On top of the figure, one can see that the light inside the light pipe gets internally reflected. These reflections occur for the different elementary modes independently and give rise to an irregular intensity pattern at the exit facet of the light pipe. Simulation of the light pipe can be better understood by considering the wave-field propagation of these elementary modes on a ray-based scalar approach, as earlier described in Section 2.1. In the lower half of Fig. 3, the simulation approach is explained. Every elementary mode is propagated in free-space through glass over the full length of the light pipe. This results in an intensity pattern at the exit plane of the light pipe that is still Gaussian. However, due to the small transverse extent of the light pipe, light is reflected internally. Fresnel reflections occur on the inside of the light pipe and depending on the incoming angle, corresponding Fresnel coefficients are applied. Consider for instance in Fig. 3 the blue rays of one elementary mode.
Fig. 3 Top: propagation of rays from an elementary mode in the light pipe. Bottom: equivalent propagation of the same elementary mode through free-space propagation in glass and afterwards transforming/mapping the field to incorporate the internal reflections.
They do not arrive at the surfaces S−1 and S1 at the exit plane of the light pipe, but are reflected onto the exit facet of the light pipe S0. This internal reflection can be simulated by mapping/transforming the wave-field at the exit plane of the light pipe onto the exit facet of the light pipe. Therefore, the fields of the surfaces S−1 and S1, are mirrored and added –on an amplitude basis– onto the field at the exit facet S0 of the light pipe. The fields arriving in their neighboring surfaces S−2 and S2, are mirrored twice and again added onto the exit facet of the light pipe. This mapping transformation is performed on every elementary mode and the contribution of the different elementary modes are added on an intensity basis as the modes are uncorrelated. The resulting intensity profile of the partially coherent light is exactly the same as the one shown in the top of the figure. For the setup described in Fig. 1, each elementary mode reflects on average twice inside the light pipe. The amount of reflections is dependent on the beam width at the source plane, the roughness of the diffuser –i.e. on the diffusion angle– the geometrical dimensions of the light pipe and its lateral position relative to the source.
The simulation method that is used in the remainder of this investigation is time-efficient because of the three reasons. First of all, the scalar field can be written in separable form such that only a 1-D simulation is sufficient when a light pipe with a rectangular shape is investigated. Secondly, the mapping effect of the light pipe is analytically implementable and thirdly, a shifted elementary mode model can be used to correctly represent the source. In case the first assumption is no longer valid and simulations should be performed in two dimensions, this simulation model can still be used, though the mapping function might become very complex and the simulation time will increase.
It is important to mention that this approach can be further extended to investigate other types of light pipes, such as tapered or more complex ones. Essentially, the same approach and propagation models can still be used. The main difficulty will be to correctly describe the mapping transformation of the light pipe, depending on its geometry. Therefore, the efficiency of the simulation is expected to remain high.
2.3. Calculation of the complex degree of coherence
Coherence is usually described by means of the mutual coherence function Γ. Assuming an ergodic, statistically stationary, quasi-monochromatic scalar field U, the mutual coherence function at points r1 and r2, and at times t and t + τ is given by [6]
where * denotes the complex conjugate, τ denotes the time difference and 〈〉 denotes an ensemble average [18]. The complex degree of coherence γ(r1, r2, τ) is now defined as where I(r) = Γ(r, r, 0) denotes the intensity at a point r[18]. In case |γ(r1, r2, τ)| = 1, the field is said to be fully coherent at the examined pair of points and at a time difference τ; in the case |γ(r1, r2, τ)| = 0, the field is called incoherent. In intermediate cases, i.e. 0 < |γ(r1, r2, τ)| < 1, the field is said to be partially coherent.The coherence properties of the light can be investigated by calculating the complex degree of coherence. In the case of partially coherent light consisting of a superposition of elementary modes, the (sampled) mutual coherence function at two pixels l and m in the detector plane takes on the form
and, consequently, the complex degree of coherence is Here, Ul is the field of one elementary mode at the plane of the detector at pixel l. Essentially, the field at pixel l is multiplied with the conjugate of the same field at pixel m. This is performed for every elementary beam and then the results are summed. In the remainder of this paper, we visualize the coherence properties of a field by taking the magnitude of the complex degree of coherence between the center point of the field and other points along its transverse extent, i.e. τ = 0. The resulting graph thus describes the spatial coherence properties of the light as a function of the transverse extent.3. Experimental evaluation of the simulation model
We use a reversing wavefront Michelson interferometer (RWI) [19–21] to measure the spatial coherence properties at the exit facet of the light pipe. This interferometer is illustrated in Fig. 4. The exit facet of the light pipe is imaged onto the CCD with a magnification of 31, using an aspheric lens with a focal length of 20mm. The light beam towards the CCD is split by a 50/50 beam splitter. In one arm of the interferometer, we place a gold coated mirror, whereas we place a retro reflector prism in the other arm. Both components reflect the light beam and the retro-reflector additionally rotates the light beam over 180° in the plane transverse to the propagation direction. The mirror and retro reflector are slightly misaligned to minimize distortions that appear on the CCD due to reflections at the apex of the retro-reflector. The beams of both paths are directed towards the same point on the CCD. In order to obtain equal path length in both arms, the mirror is moved laterally until a maximum fringe visibility is attained. This visibility measurement is the visualization of the magnitude of the complex degree of coherence. The movement of the mirror is achieved by a piezoelectric actuator to allow for sub-wavelength resolution changes. The interferometer thus measures the interference between the fields at r1 = (x1, y1) and r2 = (−x1, −y1), where the position (0, 0) corresponds to the center of the interferogram. Therefore, radially symmetric points with respect to the interferogram’s center are compared. If the field in r1 is coherent with the field in r2, interference fringes occur at those positions in the CCD plane that correspond to r1 and r2.
Fig. 4 Schematic of the reversing wavefront Michelson interferometer used to measure the spatial coherence area at the end of the light pipe.
Measuring the complex degree of coherence γ is possible by acquiring four images. We measure the interference pattern and the interference pattern that is obtained after shifting one of the interferometer arms by a quarter of a wavelength. We furthermore measure the intensity distribution that is obtained when the arm with the retro-reflector is blocked, and the intensity distribution that is measured while blocking the arm containing the mirror. The magnitude of the complex degree of coherence can be calculated via [20]
Note that such measurements of γ only provide a 2-D slice of the 4-D degree of coherence. A full mapping of the 4-D degree of coherence can be obtained by measuring several inter-ferograms for various shifts of the interferogram’s center. However, in the case of a rotating diffuser, the source is known to be a Schell-model source. In such a situation, the degree of coherence only depends on the difference in position vector, i.e. γ(r1, r2) ≡ γ(r1 − r2). We have experimentally verified the latter condition for the diffuser used in our setup using the reversing wavefront interferometer.
4. Comparison of the simulation results with the experiments
In this section, it is shown that the light pipe clearly changes the partial coherence properties of the light. For this purpose, two different situations are compared and discussed, namely free space propagation in air over a path length equal to that of the light pipe, and propagation through the light pipe.
4.1. Modelling of the source
As explained in Section 2.1, the elementary modes representing the partially coherent field at the diffuser plane are Gaussian beams with a diameter of 2.1μm. This size was verified making use of the interferometer setup, as referred to in Fig. 4, with the light pipe removed and the laser and diffuser translated towards the focal plane of the aspherical lens such that the diffuser plane is imaged onto the CCD. Taking into account the magnification and the pixel size, the coherent elementary mode size is measured to be 2.2μm. Therefore, one can conclude that the source model is in good correspondence with the experiments.
The extent of the source beam W at the diffuser plane is also an important parameter in the simulation model. This was first estimated by taking an image of the laser beam at the plane of the diffuser with a CCD camera and measuring its size. A diameter of 780μm was obtained. This measurement is verified making use of the interferometer setup. By measuring the diameter of the coherence area at a position in the far field, the coherence angle can be found as follows
where Dcoh is the diameter of the degree of coherence in the far-field and z is the optical path length between the diffuser and the far-field plane where Dcoh is measured. The coherence angle can then be used in in order to estimate the beam diameter. At a distance of 13cm, the coherence area diameter is found to be Dcoh = 114 μm. Solving Eqs. (8)–(9) for this situation results in an extent of the source beam at the diffuser plane of 780μm. This leads us to conclude that this measured value for the extent of the source beam is accurate.4.2. Free space propagation
The effect of a light pipe on the partially coherent light beam is investigated by comparing propagation through the light pipe with free space propagation over an equivalent distance in air, namely LLP/n = 13cm/1.5 = 8.67cm. Remark that this distance does not correspond with the optical path length but that we simply want to correlate this situation with the propagation through the glass light pipe.
The degree of coherence as a function of the transverse direction, obtained from the simulation model, is depicted in Fig. 5. The blue marks represent the simulation model and the green curve represents the theoretical value of the magnitude of the complex degree of coherence for a Gaussian Schell-model beam propagated over the same distance [18]. One can see that the simulation model corresponds well with the theoretical function. The coherence diameter is defined as the distance over which the degree of coherence has dropped to the value 1/e2. In this situation, the coherence diameter is 75μm. Therefore, points in the observation plane that are separated by a distance larger than 75μm are considered uncorrelated.
Fig. 5 Degree of coherence |γ[(0, 0), (x, 0), 0]| between the beam’s center position and a transverse point at a distance x from the center (for a single mode laser beam passing through a rotating diffuser) for free-space propagation over a distance of 8.67cm after the diffuser. Blue marks: simulation, green curve: analytical value. The profile of the degree of coherence is Gaussian and the coherence diameter is 75μm.
The complex degree of coherence after free-space propagation is also measured using the interferometer setup depicted in Fig. 4. The degree of coherence is shown in Fig. 6. Taking into account the magnification and pixel size, we measure a diameter for the coherence area of 75μm. From Fig. 6, it can also be seen that the coherence function has a Gaussian shape. Note that the interferometer setup uses a retro-reflector and thus it measures the correlation γ[−r1, r1, 0] between points that are radially symmetric with respect to the correlation measurement’s center (which is the center of Fig. 6). The diameter of the interference pattern in Fig. 6 thus corresponds to the diameter of the coherence area in Fig. 5. There are still some small variations in the pattern of Fig. 6 because the image was not measured exactly at a phase shift of 90° with respect to as was described in Eq. (7). Furthermore, the “lines” at the angles ±45° are the result of using a retro-reflector. The light from the beam splitter is reflected towards this retro-reflector and is incident very close to its center. As a result, some distortion arises.
Fig. 6 Intensity map of the degree of coherence |γ[(−x, −y), (+x, +y), 0]| measured at the CCD of the interferometer setup for free-space propagation over 8.67cm. The coherence diameter is found to be 75μm.
4.3. Influence of the light pipe on the complex degree of coherence
Simulations using the model described in Section 2 of the situation described in Fig. 1 results in a degree of coherence as depicted in Fig. 7. The blue curve represents the degree of coherence obtained from the simulations of the light pipe. The green curve represents the degree of coherence for free-space propagation over the equivalent length. The degree of coherence in the situation of propagation through a light pipe no longer exhibits a Gaussian profile, but is rather a profile of equidistant peaks with a slowly-varying Gaussian envelope. These peaks are the result of internal reflections in the light pipe. The width of the envelope function is the same as that of the free-space propagation, namely 75μm.
Fig. 7 The degree of coherence for the light of our partially coherent source that has propagated through the 13cm long light pipe. It exhibits a spiked profile with a slowly-varying envelope of width 75μm.
In Fig. 8(a), one can see the measured degree of coherence at the end of the light pipe, captured with the CCD camera. We obtain the same profile as in the simulation model, namely a spiked intensity pattern underneath a Gaussian distribution with a width of 75μm. The good correspondence between measured and simulated complex degree of coherence shows the validity of the model described in Section 2. From the simulations, we can obtain more insight in the effect of the light pipe. Because the source is quasi-homogeneous, the far-field coherence function FFcoh is defined by the Fourier transform ℱ of the near-field intensity. The near field intensity distribution at the rotating diffuser NFint can be described as the convolution between a Gaussian profile (originating from the Gaussian single-mode beam impinging at the diffuser) and a spatial comb (as a result of the internal reflection inside the light pipe), i.e. in 1-D this can be written as
where exp(−ax2) is a Gaussian function with a width equal to the width of the single-mode beam impinging on the diffuser, D is the transverse size of the light pipe and x is the transverse coordinate in the near field. As a result, the profile of the complex degree of coherence in Fig 7 is the product between a Gaussian envelope and a comb, where ξ is the spatial frequency in the far-field. Consequently, the far-field grid spacing xgrid is given by where λ is the wavelength, z is the longitudinal propagation distance from near field to the exit facet of the light pipe. As a result, if the width of the light pipe is reduced, the distance between the peaks in Fig. 7 will increase. Note that the factor 1/2 is the result of the fact that the reversing wavefront interferometer measures the coherence of radially symmetric points (cfr. Section 3).Fig. 8 The magnitude of the degree of coherence at the exit facet of the light pipe, measured at the detector of the interferometer setup (a) and a cross-section taken at the center (b).
In Fig. 8(a), one can see that the two-dimensional interference pattern has a slightly different period in the vertical and horizontal direction. This is due to the rectangular shape of the light pipe, i.e. the width of the light pipe is different from its height. This is experimentally detailed by measuring the period of the far-field interference pattern in Fig. 8. The results are shown in Table 1.
Table 1. Experimental verification of the far-field coherence function period in the case a light pipe is used.
It is clear from the aspect ratio values in Table 1 that the experimental verification of the period corresponds well with Eq. (13). Also, the experimentally determined horizontal and vertical period of the grid in the far field coherence function corresponds reasonably well with the theoretical prediction by Eq. (12) where the difference between the measurements and the analytical formula can largely be attributed to inaccuracy when measure the magnification in the interferometer. Recall that the footprint of the exit facet of the light pipe is 8 × 15mm2, so the interference pattern in both directions has a different period. A cross-section profile of the degree of coherence is taken in the horizontal direction and is depicted in Fig. 8(b).
One has to remark that in theory, the degree of coherence should reach a zero value between two consecutive peaks of the grid pattern for a source with a low degree of coherence. This is not obtained in practice because in order to calculate Fig. 8(a), using Eq. (7), two interference patterns are measured for which the only difference between them is a lateral translation in one of the interferometer arms with a distance of λ/4. The measurements are not taken instantly, so some fluctuations arise resulting in a broadening of the grid pattern. Additionally, the grid-like pattern will be somewhat blurred due to the finite size of the CCD pixels and due to optical imperfections in the interferometer. The sum of these external factors results in a fringe visibility that does not reach zero for a low degree of coherence.
As a conclusion, it is clear that the light pipe influences the coherence properties of the light and we are able to model this behavior accurately with a fast and efficient simulation model. In comparison with the degree of coherence in case of free-space propagation, the degree of coherence in case of a light pipe is rather spiked (see Fig. 7). Certain neighboring points inside the original coherence area appear to have no correlation with each other. Thus, they are not able to interfere. This result is the combined effect of the light pipe and the rotating diffuser. The rotation of the diffuser is necessary to provide an important reduction of the spatial coherence of the incident field. Owing to the rather complicated propagation of light through the light pipe, the light at certain neighboring points at the exit plane of the light pipe originate from distant points at the input plane. Since the fields at these distant points are only weakly correlated, the same can also be said about neighboring points at the exit plane. For a fully coherent laser beam, neighboring points at the exit plane are still fully correlated and the grid pattern will not appear in the complex degree of coherence at the exit plane of the light pipe.
5. Summary & Conclusion
The influence of a light pipe on the coherence properties of partially coherent light is investigated by comparison to a free-space propagation over the same equivalent path length. A fast and efficient simulation model is designed for the propagation of partially coherent light, in which the light pipe is incorporated as a mapping transformation of the wave field. The coherence properties in both situations are clearly different and the consequences are discussed. As speckle is a coherence effect, observed as a granular pattern on the projection screen, we will as a future task employ our simulation model to investigate the effect of micro-structured projection screens to reduce the speckle [4]. We can conclude that the usage of a light pipe does change the coherence properties of the light. These outcomes have also been experimentally verified making use of a reversing wavefront Michelson interferometer. A light pipe alters the degree of coherence of the partially coherent light in such a way that certain neighboring points inside the original coherence area are no longer correlated with one another. This means that the field at these points can not interfere anymore and could reduce the speckle appearance on the screen. This aspect will be investigated in future work together with the interaction of partially coherent light with micro-optical components. The simulation model is currently implemented as a 1-D propagation method, but this does not yet pose any shortcomings. Furthermore, this investigation method can also be applied to homogenizing light pipes that are tapered or more complex.
Acknowledgments
The authors would like to acknowledge financial support from the Agency for Innovation by Science and Technology in Flanders (IWT), the Fund for Scientific Research (FWO), the Industrial Research Funding (IOF), Methusalem VUB-GOA, IAP BELSPO VI-10 under grant IAP P7-35 “photonics@be”, the OZR of the Vrije Universiteit Brussel, and the Academy of Finland (grant 118951).
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