We propose a method based on quantitative theoretical analysis for achieving speckle contrast of 1% or less in images created by a full-frame laser projection display system. The method employs a stationary multimode optical fiber to achieve the effect of using a rapidly moving diffuser, but without moving the fiber or any other system component. When a suitably large projector lens is used, low-speckle illumination light delivered through the fiber acts in conjunction with wavelength diversity at the projection screen to achieve speckle contrast of 1% in viewed images. We describe in detail how the proposed method might be used with most types of high-power visible lasers being considered for large-venue displays. When used with visible laser diodes, the method may also be suitable for use in laser-based television.
©2012 Optical Society of America
Speckle is the random modulation of light intensity that occurs when a rough surface, such as a projection screen, is illuminated with coherent or partially coherent light resulting in projected images with a granular appearance. The intensity modulation is a result of random constructive and destructive interference that occurs between light rays from different points on the projection screen as they reach the detector (e.g., the viewer’s eye).
Laser speckle can be regarded as spatial noise that degrades information content of laser projected images. It also induces fatigue when viewing speckle-filled images for prolonged periods of time. Applications that tolerate little or no speckle contrast include laser cinema and laser projector-based simulation / training systems.
The numerical value of speckle contrast, C, in a projected image is given by:Equation (1) indicates that speckle contrast is the standard deviation of the pixel intensity divided by the mean value of the intensity. Although some laser projection applications may tolerate speckle contrast of 5%, the most demanding system developers are striving for contrast of 1% and less.
2. Speckle reduction theory
Trisnadi has analyzed the problem of speckle reduction in laser projection display systems . The speckle-reduction factor for a projection system, R, is the product of three factors:
In Eq. (2), Rλ is the reduction factor due to wavelength diversity, Rσ is the factor due to polarization diversity, and RΩ the factor due to angle diversity. The speckle contrast value resulting from all speckle-reduction methods combined is C = 1/R.
The speckle reduction factor due to wavelength diversity, Rλ, is:Equation (4) involves the ratio of 2Δd to the coherence length of the light source, Lc , and can be rewritten as:
It is well known that speckle contrast reduces as 1/√N where N is the number of incoherent and uncorrelated emitters (of equal intensity) having their light combined at the detector. One way to interpret Eq. (5) is that Rλ is equal to the square-root of some effective number of emitters, Neff, determined by the statistical distance spread induced by screen roughness divided by coherence length of the light source. (Note: In our scheme, we achieve low speckle contrast at the end of a highly multimode optical fiber by relying on intermodal dispersion to induce a statistical spread in distances through the fiber that is many times longer than the coherence length of the light being delivered. However, as we describe later, use of a multimode fiber in this manner determines the RΩ factor in Eq. (2), not Rλ).
3. Kohler’s and related speckle reduction methods
Our proposed speckle reduction method was inspired by Kohler’s work first reported in 1974. Kohler et al. used a Q-switched ruby laser with a carbon disulfide (CS2) cell and a flexible multi-fiber bundle to produce low-speckle pulsed-laser photographs . The ruby laser’s collimated beam (150 mJ, 8 ns, 694 nm, 3 mm diameter, 0.008 nm bandwidth) was sent through a 30-cm-long, liquid carbon disulfide cell that broadened spectral bandwidth to 0.1 nm as a result of self-phase modulation. Use of the fiber bundle alone produced noticeable speckle reduction, but the combination of disulfide cell and fiber bundle resulted in a more significant reduction. Figure 1 shows qualitatively the speckle reduction that was achieved, but Kohler did not report quantitative values for speckle contrast.
A number of other groups have since reported on the use of a multimode fiber to reduce speckle contrast in recorded images when using laser illumination [3–9]. However, most of these works provide only a qualitative assessment of speckle contrast reduction, and none provide a quantitative recipe for achieving any specific speckle contrast value, let alone 1% or less, in viewed or recorded images. Accordingly, a prime objective of this work is to analyze quantitatively the speckle contrast reduction one can expect by injecting spectrally broadened light into a multimode fiber, and how this fiber-delivered light can be used to create a projected / viewed image with 1% speckle contrast or better.
4. Speckle reduction for cinematic laser displays
Laser projection displays include full-frame, scanned-line, and scanned point (raster-scan) projectors. Although it may prove useful for all, our method is primarily intended as a speckle reduction method that can be applied in practical fashion with full-frame image projectors as are commonly used in movie theaters. The method is specifically intended for projectors that employ lasers as a direct replacement for filtered arc-lamp light sources. Present lamp-based cinema projectors can accommodate numerical apertures as high as 0.8 or 0.9 for illumination light, but, to improve projector efficiency and contrast ratio, numerical apertures in the range of 0.2 to 0.4 may be desired in the future .
Lasers being considered for laser cinema and related applications included high-power (10W and more) continuous-wave lasers, including fiber lasers and laser diode arrays, nanosecond-pulse lasers, picosecond-pulse, and sub-picosecond (e.g., 100 femtosecond) pulse lasers. We will discuss how our method might be used with all of these.
5. General description of the method
We seek a speckle reduction method that can reduce speckle contrast to 1% or less in projected images, and preferably without relying on moving fibers, diffusers, screens or other moving components in the projection system. We also want to achieve 1% contrast without relying on polarization diversity. In other words, we seek Rλ RΩ products in Eq. (2) of 100.
Our basic strategy for achieving projected images having speckle contrast of 1% or less is to deliver low-speckle laser illumination light to the image projector through a highly multimode optical fiber. The multimode fiber is designed considering details of the laser light spectrum injected into the fiber, the diameter of the lens used in the projector, and the effective roughness of the screen used to display projected images.
Figure 2 shows the general aspects of the proposed low-speckle laser light source in the context of using semiconductor laser diodes. Emission from one or more lasers is coupled into the proximal end of a multimode optical fiber. If the light spectrum is broad enough, and the multimode fiber is designed and configured properly, then light delivered at the fiber’s distal end will have low speckle contrast. An optional mode scrambler may be placed near the input end of the fiber to equalize power distribution among modes of the multimode fiber, thereby reducing the fiber length needed to achieve a desired speckle contrast. Light delivered by the multimode fiber is used as an illumination source in a projection display system that projects 2D (or 3D-capable) images onto a projection screen with magnification.
Laser diode sources that might be used include edge-emitter lasers, vertical-cavity surface-emitters (VCSELs), and vertical external-cavity surface-emitter lasers (VECSELs). Such lasers might be employed as single-emitter devices, 1D array “bars”, or 2D laser diode arrays, including 2D bar stacks and monolithic 2D arrays of surface-emitter lasers.
Light is coupled into the multimode optical fiber using any of several possible multiplexing methods, including angle- or geometric multiplexing, fiber-array multiplexing, and wavelength-beam-combining. Multimode fibers that can be used include conventional step-index and gradient-index silica fibers, photonic crystal and photonic bandgap fibers, and specialty optical fibers that employ materials other than silica glass. A single-core multimode fiber is preferred, but a fiber bundle or liquid light guide may also be used.
The preferred system configuration employs a full-frame 2D or 3D-capable projector that includes an imaging light-valve device such as a digital mirror device or liquid-crystal-on-silicon chip, but a line-scan or raster-scan projector might also be used in some situations. The preferred projection screen is a reflection-mode, surface-scattering, stationary screen. Other screens that may be used include transmission screens, volume-scattering screens, and screens that are vibrated or otherwise moved (although we seek to avoid the need for moving screens). The system detector is the retina of the viewer’s eye or a suitable electronic camera/detector.
A key aspect of our method is to achieve speckle contrast of about 1% (range 0.005 to 0.05) at the end of the multimode delivery fiber by designing the fiber according to the injected light spectrum. Considering typical large-venue display parameters, this might allow RΩ values in the range of 20 to 135, depending on the diameter of the projection lens. For example, and as will be discussed below, a “K value” of (137)2 = 18,800 is achieved when using a 100 mm diameter projector lens (see Eq. (22)-(24) below). If speckle contrast at the end of the fiber is 0.7% or less, then it should be possible to achieve RΩ > 100, and speckle contrast of 1% in the viewed image, without relying on wavelength diversity at all.
However, when RΩ is less than 100, and perhaps as low as 20 (corresponding to 5% speckle contrast at the end of the multimode fiber), it should in many instances be possible to use the wavelength diversity aspect of our method to achieve Rλ > 5 (range 1.2 to 10) such that the product RλRΩ is greater than 100. Speckle contrast of 1% or less in projected images would then still be achievable.
6. Compatibility with laser projection displays
Advantages of laser-based light sources for projection display applications include high color saturation, very wide color gamut, low etendue (better light collimation for improved projector efficiency and higher contrast ratio), ability to adjust and stabilize color balance, and polarized or unpolarized emission, as needed. Our speckle reduction method preserves these laser advantages, but with some caveats.
Existing lamp-based projectors can accommodate NAs as large as 0.8 (large etendue), but, there is interest to reduce light source NA as a way to improve projector efficiency and contrast ratio. Therefore, when using our method, one will likely want to use multimode fibers with NAs of 0.4 or less.
Delivery of polarized laser light through a length of standard multimode fiber generally depolarizes the light, but, considering that present lamp-based projectors do not employ polarized light sources, this should not be a major liability. Liquid-crystal on silicon (LCoS) projectors require plane-polarized light, but projectors based on Texas Instruments’ digital-mirror (DLP tm) imaging chip or Silicon Light Machines’ Grating Light Valve (GLV tm) do not. When polarized light is needed, delivery of light through a small-core multimode fiber, as in our scheme, may enable new and efficient polarization-recovery schemes at the delivery end of the multimode fiber that are compatible with existing projector designs. Alternatively, one might use a polarization-maintaining highly multimode fiber if such is available.
Considering the above, expected advantages of the proposed speckle-reduction method for large-venue projection display applications include:
- - Potential for reducing speckle contrast below 1% with simple hardware
- - Preserves desirable features of using lasers
- - No moving parts
- - Compact and robust (the multimode fiber can be coiled, 10 to 20 cm coil diameter)
- - Low cost (multimode fiber is inexpensive, even when properly terminated)
- - Compatible with existing projectors
- - Compatible with high laser power delivery
- - Fiber optic delivery enables light source placement outside the projection room
7. Multimode fiber technology
In our method, speckle contrast at the end of the multimode fiber is reduced by exploiting intermodal dispersion in a fiber length much shorter than the fiber’s “mode-coupling length” such that little or no redistribution of power among fiber modes occurs. Present fibers exhibit transmission losses of 10 to 15 dB/km at green wavelengths, somewhat higher losses at blue wavelengths, and lower losses at red wavelengths. A 100 meter-long fiber with 15 dB/km loss would transmit only 70%, and one with 10 dB/km loss would transmit 80%. While such transmission efficiencies may be tolerable, they are not desirable considering the relatively high laser power levels involved in large-venue projectors. We would like to achieve 1% speckle contrast using 30 meters of fiber or less, and, ideally, less than 10 meters. Transmission efficiency through the multimode fiber should then be 90% or better not considering Fresnel reflection losses at the fiber end facets (which can be eliminated using anti-reflection coatings on properly terminated fibers).
Glass-clad silica fibers have longer mode coupling lengths (see below) and are easier to use at higher power levels than are hard polymer-clad (hard-clad) silica fibers. However, at present, glass-clad silica fibers have numerical apertures of 0.22 or less, whereas hard-clad silica fibers can achieve NAs as large as 0.47. With enough development effort, both types of fibers can be used successfully with our method. All-glass photonic crystal fibers that achieve NAs as large as 0.65 at visible wavelengths appear feasible , but they are not yet available commercially as far as we know.
8. Intermodal dispersion in multimode fibers
Dispersion mechanisms that operate in multimode optical fibers include material, waveguide, and intermodal dispersion. Only intermodal dispersion is capable of achieving 1% speckle contrast at the end of the fiber considering spectra of light sources we intend to use. Intermodal dispersion is substantially larger in step-index fibers than in gradient-index (GRIN) fibers, and step-index fibers are preferred for this reason. GRIN fibers can be used practically if spectral bandwidth of the light source is large enough.
Intermodal dispersion (or modal dispersion) in a step-index multimode fiber results in temporal pulse spreading as given by :Eq. (6) can be rewritten in an easier-to-use form:
Equation (7) indicates that pulse (temporal) spreading in a step-index fiber is proportional to fiber length and the square of fiber NA. As an example, a 0.2 NA fiber (nominal index of 1.5) would exhibit temporal spreading of 0.44 ps per cm of fiber length, or 44 ps per meter of fiber. Equation (7) applies only if the length of step-index fiber being used is much shorter than the fiber’s “mode coupling length”, Lc. For fiber lengths much longer than Lc, such that mode mixing establishes an equilibrium power distribution among fiber modes, temporal spreading due to modal dispersion then scales with the square root of fiber length, according to :
Therefore, to minimize transmission losses and otherwise maintain a manageable length for the multimode delivery fiber, it is desirable that the fiber used in our scheme be (much) shorter than its corresponding mode coupling length.
A detailed discussion of mode coupling is beyond the scope of this discussion. Suffice it to say that mode coupling length is inversely proportional to the fiber’s mode coupling coefficient which depends on details of the fiber’s design and manufacturing process. This coefficient is determined by fiber flaws introduced during manufacturing, including impurities, inhomogeneities, microbends, core-cladding irregularities, and refractive index fluctuations. The mode coupling coefficient is thought to scale in proportion to λ2 and inverse proportion to the square of the fiber core radius, a2 .
Silica fibers with core sizes in the 50- to 500-micron range are of primary interest (to us) for handling high laser power levels while providing good flexibility for coiling. Hard-clad silica-core fibers have numerical apertures as high as 0.47 and can have mode coupling lengths longer than 500 meters . Glass-clad silica fibers typically have numerical apertures of 0.11 to 0.22 and mode coupling lengths longer than 1 kilometer . Accordingly, Eq. (7) should apply for glass-clad and hard-clad multimode fiber lengths as long as 100 meters. Plastic-core optical fibers (POF) have very short mode coupling lengths on the order of 10 meters, and cannot reliably handle high laser power levels. Plastic optical fibers are not considered further.
Equation (7) indicates that modal dispersion depends on fiber length and NA, but not on fiber core size (if fiber length is much shorter than mode coupling length). This situation provides freedom to increase fiber core size as needed to prevent unwanted nonlinear effects in the multimode fiber, or to prevent fiber damage at high power levels. Nonlinear effects and laser-induced damage are important considerations when high-peak-power pulsed laser emission is involved, but they are usually less of an issue when broadband and continuous-wave laser emission is being delivered through the multimode fiber.
Equations (7) and (8) refer to the maximum time delay through fiber; i.e., time delay between the first-to-arrive and last-to-arrive light. In general, the average time delay, which considers the specific distribution of power among modes of the multimode fiber, will be less. Therefore, Eq. (7) provides an estimate of the shortest length of multimode fiber needed to achieve the desired speckle contrast at the end of the fiber. Actual required fiber lengths will depend on average time delay and will likely be longer.
The average time delay is determined by the impulse response of the multimode fiber, which, in turn, depends on fiber NA, fiber length, and, for fiber lengths much shorter than the mode coupling length, on launch conditions. Impulse response may also depend on how tightly the fiber is coiled. The actual impulse response of a given multimode fiber configuration can be measured [16,17], and it may be possible to model it with reasonable accuracy in some cases [18,19].
Considering that delivery of light through a multimode fiber results in a statistical spread of path lengths by which light reaches the end of the fiber, one might reasonably expect that speckle contrast at the end of the fiber is reduced in a fashion similar to that indicated in Eq. (5). This in fact happens, and we now analyze in detail the expected contrast reduction.
9. Quantitative analysis of speckle contrast at distal fiber end
To arrive at a quantitative assessment of expected speckle contrast at the end of the multimode fiber, an analytical framework is needed that considers detailed shapes of light-source spectra and fiber impulse response functions. Ideally, this analytical framework should address arbitrary real-world shapes as well as ideal shapes commonly used in theoretical analyses (Gaussian, Lorentzian, etc). Such a framework has been developed analyzing “modal noise” in optical telecommunications links that employ multimode fibers [20,21].
According to Ref. 20, given the light source’s power-density spectrum, P(ν), and the fiber’s impulse response function, h(t), speckle contrast at the output end of the multimode fiber is given by :
In Eq. (10), < I > is the total ensemble average intensity summed over all fiber modes and all wavelengths. The main assumptions made in deriving Eq. (9) are that (a) the modal dispersion constant of the fiber does not vary significantly over the range of wavelengths being delivered through the fiber, and (b) that the number of fiber modes is very large.
Equation (9) provides a quantitative result for the expected speckle contrast at the end of the multimode fiber. The formalism can be used given arbitrary shapes for the light spectrum and impulse response function of the fiber. The light spectrum and fiber impulse response can be measured, digitized, and processed numerically on a computer to arrive at the speckle contrast value indicated by Eq. (9). It may also be possible to model the impulse response function of the multimode fiber length, and then numerically calculate speckle contrast.
In the discussions below, we apply Eqs. (9)-(12) assuming somewhat idealized spectral profiles as would apply to laser spectra of interest so that closed-form analytical expressions can be obtained for speckle contrast at the end of the fiber.
10. Source with a Gaussian spectrum
The formalism of Eqs. (9)-(12) is first used to calculate speckle contrast at the end of the multimode fiber when the light source has a Gaussian spectral profile with 1/e half-width, Δν, and the spectrum has no sub-structure. It is assumed that the fiber’s impulse response function does not change shape with fiber length and that width of the impulse response function increases linearly with fiber length. Speckle contrast at the end of the multimode fiber is then given by :Eq. (7). Note that the definition of T’g accounts for the fact that the average time delay through the fiber is less than δτ, which is the maximum time difference between first-to-arrive and last-to-arrive light. The derivation of this equation assumes that the fiber’s (magnitude squared) incoherent transfer function is a sinc2(ωδτ /2) function, which can be approximated as exp –(ωTg’/2)2. In cases of interest to us, the second term in brackets in Eq. (13) is typically much greater than 1, so that C(L)2 scales inversely with fiber length L and contrast C(L) scales inversely with L1/2. Note that, for a long enough fiber such that the second term in Eq. (13) dominates, contrast at the end of the fiber is determined by the spread in distances through the fiber induced by intermodal dispersion (c T’g) divided by coherence length of the injected light (λ2/Δλ).
Table 1 shows calculated lengths of multimode fiber needed to achieve 1% speckle contrast at the end of the fiber. The values in Table 1 apply when the Gaussian spectrum has no sub-structure, or when the spectrum has “unresolvable” substructure with features much smaller in frequency/wavelength space than the width of the fiber’s incoherent transfer function. For a fiber length of 10 meters, and fiber NA = 0.22, the minimum resolvable element is about 10−3 nm (1 pm) at a wavelength of 500 nm. If, in this case, the light spectrum consists of many narrow lines spaced by 1 pm or less, but the envelope function that describes the relative intensities of the lines is a Gaussian, then Eq. (13) and Table 1 can be applied directly. Figure 3 shows that speckle contrast decreases monotonically with fiber length.
11. Speckle contrast with “rippled” emission spectra
While many laser and non-laser light sources have a near-Gaussian spectral profile, it is often the case that spectra have some “ripple” or modulation that makes the spectral profile deviate from a true Gaussian shape. The formalism of Eqs. (9)-(12) provides the means to quantify the expected increase in speckle contrast for an arbitrary spectral profile. The detailed spectral profile and impulse response of the fiber can be digitized and speckle contrast calculated by doing the indicated integrations numerically on a computer.
12. Speckle contrast with multi-line spectra
The case of a multi-line (or multimode) spectrum has been addressed by Hlubina who arrived at a closed-form analytical expression for speckle contrast at the end of a multimode fiber . In the expression that follows, C(L) is the speckle contrast at the end of a multimode fiber of length, L. The individual lines in the spectrum are assumed to have Gaussian profiles with 1/e half-widths = Δν, center-to-center spacing between lines = Δνs, and a Gaussian envelope function with 1/e half-width = Δνe . The number of discrete modes in the spectrum is N, and the indicated sums are taken from –(N-1)/2 to + (N-1)/2. This equation can be evaluated on a computer using software such as Mathcad.
Using the relation Δλ = λ2Δν/c, Eq. (14) is transformed into a more useful form for our needs:
Equation (15) is quite general and useful. The number of modes N can be adjusted independently of other parameters, and, in particular, independently of Δλe. The ratio of the individual linewidths to the line spacing, Δλ /Δλs, can also be adjusted to any value. Many different types of laser spectra encountered in practice can be modeled as a result.
It was shown earlier that speckle contrast decreases monotonically with fiber length when the source spectrum has a Gaussian profile and no substructure. In the multi-line case of Eq. (15), and when the spectral linewidth, Δλ, is much smaller than the line spacing, Δλs, a more complicated “S-shaped” curve describes how speckle contrast decreases with fiber length.
In Fig. 4, Y = 2πΔνT’g = (2πΔνL NA2) / (2√3n1c) and is plotted on a log scale. An S-shaped curve applies as L is increased. Instead of decreasing monotonically, speckle contrast exhibits a “plateau region” in which contrast remains constant as fiber length is increased. The presence of the plateau region increases the length of fiber needed to achieve a desired speckle contrast for given Δν and fiber NA values.
The curve in Fig. 4 is a direct result of application of Eqs. (9)-(12). The Fourier transform of the multimode fiber’s impulse response function (i.e., its incoherent transfer function) gets narrower in frequency or wavelength space as fiber length increases. The autocorrelation of the multi-line source spectrum is another multi-line spectrum with features similar to that of the source spectrum. So, when the (magnitude squared) incoherent transfer function is multiplied by the autocorrelation function of the source spectrum, and integrated as indicated in Eq. (9), the result is different depending on which of three different operating regimes applies, as discussed below.
In Regime I, the fiber’s (magnitude squared) incoherent transfer function is much wider than the frequency / wavelength spacing between spectral lines of the source spectrum, and Y is much less than 1. Equation (15) reduces to:Eq. (13) but with Δν replaced by Δνe. Speckle contrast decreases with fiber length according to the width of spectrum’s envelope function, ignoring spectral substructure.
In Regime II, or the plateau region, the fiber’s incoherent transfer function is less than the frequency spacing between spectral lines, Δλs, but wider than the individual linewidths, Δλ. When the integration in Eq. (9) is performed, the incoherent transfer function “picks out” the centermost peak (at ω shift = 0) of the autocorrelation function of the power density spectrum, which means that m = n in Eq. (15). In this regime, Eq. (15) reduces to :
When the number of equally spaced modes in the spectrum, N, is large enough that they span a total range about 25% larger than 2Δλe (or larger), then Eq. (17) can be approximated as
For arbitrary relative power levels of the individual modes, Pn, (arbitrary envelope function), speckle contrast in Regime II is given by :
For N modes having equal power, CII(L) = 1/√N as expected. If the modes are not spaced equally, then, for Eq. (19) to apply strictly, the fiber has to be long enough that the incoherent transfer function is narrower than the smallest mode spacing in the spectrum. When in the plateau region, speckle contrast at the end of the fiber can be no less than 1/√Neff . If only 100 modes appear in the spectrum, speckle contrast can be no better than 0.10 and this value applies only if all modes have equal power.
In Regime III, the (squared magnitude of the) fiber’s incoherent transfer function is narrower than the widths of the individual lines in the multi-line spectrum, Δν (or Δλ). Speckle contrast is reduced from that in Regime II, as indicated in Eq. (20), and can be reduced to 1% or less using an adequately long fiber :
In general, it is desirable to eliminate the plateau region of Regime II so speckle contrast decreases monotonically with length, thereby minimizing the fiber length needed to achieve a desired speckle contrast. The analysis in Ref. 21 indicates that this situation is achieved when the ratio Δλ / Δλs = 0.25, and that this is true regardless of the value of the ratio Δλs / Δλe.
Parameters for a multimode laser diode might be Δλ = 0.01 nm, Δλs = 0.2 nm, and Δλe = 2 nm. Figure 5 shows speckle contrast versus fiber length for four different Δλ / Δλs ratios, and for Δλe / Δλs = 10. Figure 6 is a similar plot, but for the ratio Δλe / Δλs = 2.5. Equation (15) was used to generate the data for both figures. (Fiber NA is 0.22). Figure 6 looks different because speckle contrast is higher in the plateau region, since fewer modes are involved. The plateau region is absent in both figures when the ratio Δλ / Δλs = 0.25.
13. Speckle contrast in projected images
Having achieved nominal 1% speckle contrast at the end of the fiber, the fiber end acts as a spatially and temporally incoherent (or partially coherent) illumination source for the projector. As a result of light propagating in the many different modes of the multimode fiber, with different time/phase delays corresponding to the various fiber modes, light emerging from the fiber behaves as if it had passed through a random phase-diffuser plate moving at a very high speed - even though there is in fact no actual diffuser and nothing is moving physically. The projector includes a lens that produces magnified images at the screen.
Speckle contrast for the moving-diffuser situation has been analyzed by Goodman in the case of a full-frame projection system . Speckle contrast in the projected image is:Eq. (2) and does not include any speckle reduction that might occur as a result of polarization or wavelength diversity.
The parameter M is more generally the number of “degrees of freedom” associated with temporally integrated speckle. In our case, M is the number of coherence times (associated with the light source spectrum) included within a time interval equal to the average time delay of light passing through the multimode fiber. Numerically, M is equal to (1/Cend)2 where Cend is speckle contrast achieved at the end of the multimode fiber. If Cend = 0.01, then M = 10,000.
The parameter K, which is the number of degrees of freedom associated with spatially integrated speckle, is essentially the number of projector lens resolution areas at the projection screen contained within one eye resolution area, also measured at the projection screen. If distances from the screen to projector lens, and screen to eye pupil, are the same, then K ≈(NAprojector / NAeye)2 where NAprojector is the numerical aperture of the projector lens measured from the screen looking back at the projector lens, and NAeye is the numerical aperture of eye pupil measured from the screen.
M and K will always be much greater than 1 in our scheme. The expected contrast in the viewed image due to illuminating with the multimode fiber, not considering system wavelength diversity or polarization diversity, is then given by:Eq. (21), but the K value used in Eq. (22) is different according to whether the projector lens is overfilled or not. Goodman has derived  expressions for K based on Westheimer’s experimental eye measurement data :
In Eq. (23) for an overfilled projector lens, D is the diameter of the projection lens, λ is the light wavelength, ze is screen-to-eye distance, and zp is the screen to projector lens distance. For a 100 mm diameter lens, equidistant pupil and lens, and 500 nm wavelength, K ≈18,800, and 1/√K = 1/137 = 0.007. For a 60 mm diameter lens, K ≈6768 and 1/√K = 1/82 = 0.012. These contrast values might be reduced by moving the eye farther from the screen than the projector, considering that K scales as (ze / zp)2.
In Eq. (24) (projector lens not overfilled), b is the edge dimension of a square diffuser plate “phase cell” measured at the diffuser plate, and m is the magnification of the projection imaging system. In our case, b is approximately the width of a spatial coherence cell at the end of the fiber. As a rough approximation, b2≈ (core area)/(number of fiber modes propagating) = πa2 / (4V 2/π2) = πλ2 / (16 NA2), where we have used V = 2πa NA / λ. If we are using a 0.22 NA fiber, then at 500 nm, b2 would be approximately equal to 1 x 10−12 m2 and b would be about 1 μm. In Eq. (23), if ze = 6 m and m = 300, then K would be ≈520 in this case.
One can get higher K values by over-filling the projection lens, but with the possible tradeoff of somewhat lower projector efficiency and having to deal with diffraction effects. When under-filling the lens, even if K = 500 (1/√K = 0.04 to 0.05), one can still achieve 1% contrast in projected images by ensuring that M > 2000 (or so) and that Rλ in Eq. (2) is > 5.
Interestingly, Eq. (22) indicates that, even if speckle contrast at the end of the fiber is much less than 1% (M >> 10,000), speckle contrast in the projected image can be larger than 1% if K is too small. When illuminated with low-contrast light from the fiber, the projector lens aperture literally restores partial spatial coherence over an area corresponding to the area of the lens’s point spread function. This situation causes residual speckle contrast to clamp at a minimum value of 1/√K as M increases. This explains why speckle can sometimes be observed with broadband white-light sources.
The value for C given by Eq. (22) is the reciprocal of the RΩ factor in Eq. (2). One can conclude that, in some instances of practical interest, it should be possible to achieve 1% speckle contrast in viewed images (“at the screen”) by achieving RΩ >100. However, when RΩ < 100, one can still exploit wavelength diversity at the projection screen to achieve Rλ such that Rλ RΩ > 100. In fact, we can sometime tolerate RΩ values as low as 20 (or 1/RΩ as large as 0.05) and still achieve 1% speckle contrast at the screen using our method.
14. Speckle reduction due to wavelength diversity
In our scheme, a light source having large enough spectral bandwidth, and therefore adequate wavelength diversity, is needed to achieve speckle contrast of 1% to 5% at the end of the multimode fiber. Since the projection screen has surface roughness, this wavelength diversity can be exploited to further reduce speckle contrast in the projected image.
Given an arbitrary light source spectrum, speckle contrast in the projected image due only to wavelength (frequency) diversity is given by :Eq. (9)). Normal incidence and observation angles, relative to screen normal, have been assumed. Equation (25) also assumes that light is fully plane polarized before and after reflection from the screen; that is, it ignores any possible polarization diversity effects. This value for C is the reciprocal of Rλ in Eq. (2).
If the light source spectrum has a true Gaussian profile (no modulation or substructure) with 1/e full-width = δν, then Eq. (25) becomes:
In this case, the contrast reduction due to wavelength diversity at the screen is determined by the ratio of σh divided by coherence length of the illumination light, λ2/Δλ. If δλ = 10 nm, λ = 500 nm, and σh = 250 μm, then C ≈ 0.15 and Rλ ≈6.7. Therefore, when used in combination with angle diversity, Rλ RΩ products > 100 might be achieved with RΩ as low as 15, which corresponds to a projection lens diameter of about 45 mm. Since lens diameters of 60 to 100 mm are typical of cinema-grade projectors, it should be possible with most laser light sources being considered here to achieve speckle contrast of 1%.
When the light source spectrum has significant modulation or spectral substructure that makes it depart from a true Gaussian profile, then the integration indicated in Eq. (25) must be implemented, and perhaps numerically on a computer. However, many of the light source spectra we will consider can be modeled as a multi-line spectrum having multiple Gaussian-profile lines or peaks, each having 1/e half-width = Δλ, center-to-center spacing = Δλs, and a Gaussian envelope function (with 1/e half-width = Δλe). In this case, an analytical expression can be derived for speckle contrast due to wavelength diversity that is completely analogous to Eq. (15) by substituting 8π Δλ σh / λ2 for 2π c Tg’Δλ / λ2 in Eq. (15):
It can be shown that this equation reduces to Eq. (27), with Δλ replaced by Δλe, when σh is adequately small, or when the spectrum is a Gaussian envelope with negligible substructure (Δλs / Δλe << 1 or Δλ / Δλs ≈1).
We now have an analytical framework for calculating expected speckle contrast at the end of the multimode fiber, and therefore M in Eq. (22), and speckle contrast in the viewed image considering angle and wavelength diversity. Next, we consider spectra that may be encountered when using specific laser-based light sources, and how the multimode fiber might be designed to achieve 1% contrast in viewed images using such sources.
15. Speckle reduction for semiconductor laser diode sources
Semiconductor laser diode (laser diode, LD) sources generate high power with adequate beam quality, are relatively low-cost, and have overall emission bandwidths ranging from several nanometers to more than 10 nm. As such, laser diodes are well-suited to making low-speckle light sources using our scheme. This discussion primarily considers the use of broad-stripe, Fabry-Perot, edge-emitter laser diodes and arrays, but we also discuss use of VCSELs and related surface-emitter laser diodes.
15.1 Single-emitter Fabry-Perot laser diodes
Fabry-Perot (FP) edge-emitter laser diodes include gain-guided and index-guided versions. High-power FP diode lasers tend to be gain-guided devices, but our discussion applies to both types. Single-emitter FP laser diodes typically have multi-longitudinal-mode line spectra. Spacing between the longitudinal modes is λ2/2nLcav,where n is the refractive index of the semiconductor laser medium (roughly = 3) and Lcav is the length of the laser diode cavity (usually 300 to 500 microns for lower power devices, and 1 to 2 mm for high-power emitters). Accordingly, mode spacing values, Δλs, are in the range of 0.02 nm to 0.2 nm at 500 nm center wavelength. A high-power free-running FP laser with output power in the 0.1 to 1W range (typical for visible-wavelength devices) might have a Gaussian envelope with 1/e half-width = 1 nm and Δλs = 0.1 nm, or about 25 longitudinal modes in the spectrum. Higher and lower numbers of modes are possible; the actual number of modes lasing simultaneously and the width of the Gaussian envelope function depend on device design and operating conditions.
Linewidths of individual longitudinal modes in the spectrum also depend on various design and operating factors, and can vary from <10 MHz to as much as 10 GHz. At 500 nm center wavelength, this range corresponds to Δλ values of 10−5 nm to 0.01 nm. For linewidths at the low end of this range, the best one can hope to do is operate in Regime II of the multimode fiber (the plateau region).
When operating in the plateau region of the multimode fiber, simultaneous lasing on 25 equal-power modes would achieve speckle contrast at the end of the multimode fiber of about 0.20 (20%). For a Gaussian envelope function, the effective number of peaks, Neff, used for calculating speckle contrast in the plateau region ( = 1/√Neff) may be substantially less than the actual number of peaks lasing within the Gaussian envelope. Therefore, in this scenario, achieving 1% contrast at the end of the fiber is possible only if operation in Regime III is possible (long enough fiber with adequately broad mode linewidths).
The width of the Gaussian envelope function, and therefore Neff, will change as operating current is changed. The 1/e half-width of the envelope function often gets smaller as operating current is increased, thereby reducing Neff. Mode-locking significantly increases the number of modes lasing simultaneously, compared to free-running operation of the same laser diode, and would reduce speckle contrast when operating in Regime II.
As suggested by Fig. 5, it may be possible to operate in Regime III and achieve speckle contrast of 1% if linewidths of the longitudinal modes are broad enough (0.01 nm or larger). Possible techniques for increasing mode linewidth include current modulation  and inducing optical feedback into the diode laser so the longitudinal mode wavelengths “scan” erratically on a time scale much shorter than the integration time of the retina or other detector used in the imaging system .
15.2 Multiplexed arrays of Fabry-Perot laser diodes
Until high-power, single-emitter LDs become available that can provide 10W at desired red, green, and blue wavelengths, it will be necessary to combine the emissions of multiple LD emitters. A straightforward way to achieve high power and broad spectra, for our purposes, is to employ one of the various “laser multiplexing” techniques that have been developed to combine emissions from multiple incoherent emitters into a multimode fiber. Such methods include fiber-array multiplexing (arrays of fibers mated to arrays of LD emitters, and then combining outputs of the multiple fibers into a single multimode fiber), angle- or geometric multiplexing (using a macro-lens to focus multiple collimated LD beams into a multimode fiber), and wavelength beam combining (see below). In principle, the emissions of hundreds or thousands of incoherent emitters can be combined into a fiber with a core diameter in the 100- to 500-μm range and numerical aperture (NA) in the 0.1 to 0.4 range.
15.3 Multiplexed arrays, good spectral overlap
One can envision combining multiple laser diode emitters into the same fiber such that the combined spectrum has a quasi-continuous, quasi-Gaussian profile with FWHM of 10 to 20 nm. For example, one might select devices having slightly different center wavelengths so the combined emission spans a 10 to 20 nm range. Alternatively, one can operate different devices at slightly different temperatures in order to stagger the spectra and span a wavelength range of several nanometers . Such spectra might be achieved using geometric/angle multiplexing or using fiber-array multiplexing methods. Fiber-array methods are particularly convenient for operating LDs at different temperatures as needed to shift wavelengths and tailor the combined spectrum.
If enough laser diode emitters are combined, and the center wavelengths of their spectra are staggered appropriately, the spectral substructure corresponding to multi-longitudinal mode operation may be effectively washed out so that Eq. (13) (or Eq. (16)) can be applied. For example, if the combined laser spectrum is a reasonably good quasi-Gaussian profile with 1/e half-width = 1 nm, then Eq. (13) can be used to estimate that a 60 meter, 0.22 NA fiber would achieve 1% contrast at the end of the fiber, as would a 15 meter, 0.44 NA fiber. If M = 10,000 and K = 6770 (Eq. (22) with a 60 mm diameter projection lens and 500 nm wavelength), then speckle contrast in the projected image would be about 0.016. Using Eq. (28) to calculate 1/Rλ = 0.25 in this situation (assuming σh = 250 μm), then 1/RλRΩ = 0.016 x 0.25 = 0.004. Speckle contrast well under 1% might then be achieved in the projected image.
15.4 Multiplexed arrays, moderate spectral overlap
If the spectra are combined in a quasi-random (not well-controlled) fashion, then one might achieve a combined spectrum such that the ratios, Δλ/Δλs, vary statistically over some range. As in the earlier discussion, if the smallest values of Δλ/Δλs, are in the range of 0.25, then the spectral substructure might be considered effectively washed out such that Eq. (13) can be applied. If the total number of combined longitudinal laser modes is high enough, and one is willing to use longer multimode fibers in the 50 to 100 meter range, then Δλ/Δλs as small as 0.1 might enable speckle contrast of 1%.
As a way to estimate speckle contrast in this case, we assume, for example, that 10 diode emitters, each having 25 longitudinal modes spaced by Δλs = 0.1 nm, are combined into the fiber, and the spectra are offset or “staggered” in wavelength by 0.01 nm. Therefore, the 250 modes would span a total range of 2.5 nm, which implies a Gaussian envelope function with 1/e half-width of 1 nm, as in the good-overlap case described above. We also assume that the average spectral overlap ratio Δλ/Δλs is equal to 0.1. Using Eq. (15), one calculates that a 60 meter, 0.44 NA fiber would achieve 1% speckle contrast at the end of the fiber and therefore 0.004 contrast in the projected image, as before. If one were to use the same 60 meter, 0.22 NA fiber used earlier, then speckle contrast at the end of the fiber would be 0.022 and 1/RΩ would be 0.025. Since 1/Rλ = 0.25, according to Eq. (28), 1/RΩRλ at the screen would be 0.006 or still less than 1%.
15.5 Multiplexed arrays, no spectral overlap
Another case that may apply to multiplexed laser diode emitters is that in which the individual multi-line spectra are not overlapped well enough to adequately wash out spectral substructure. The combined spectrum consists of many distinct narrow longitudinal modes and, in general, the spacing between modes is not well-controlled.
For the moment, consider the case where the multiple lines in the combined spectrum are evenly spaced, but the individual linewidths are a small fraction of the line spacing. Referring back to the previous example, if 250 modes are spaced by 0.01 nm, but the individual linewidths are a very narrow 0.00001 nm, then Eq. (15) predicts speckle contrast at the end of fiber = 0.07 for a 2 meter, 0.22 NA fiber. This is about equal to 1/√250 as expected when operating in the plateau region. However, because the linewidths are so narrow, one remains in the plateau region (and speckle contrast at the end of the fiber remains = 1/√250) for any practical fiber length longer than 2 meters or fiber NA higher than 0.22. Since 1/Rλ = 0.25 would still pertain here, speckle contrast in the projected image would be estimated at 0.25 x 0.07 = 0.018. When in the plateau region, the only way to reduce speckle contrast below this value is to increase the number of lines in the spectrum. If one maintains the same spacing as lines are added, then speckle contrast in the projected image is reduced not only as a result of a lower contrast at the end of the fiber, but also as a result of increased wavelength diversity at the projection screen. Even further reduction of speckle contrast can be achieved if the fiber can be made long enough (considering fiber NA) to get into Regime III, but doing so with a reasonable fiber length may require broadening the individual linewidths.
When the lines in the combined spectrum are very narrow and randomly spaced, the (magnitude squared of the) fiber’s incoherent transfer function must, in principle, be narrower than the smallest center-to-center line spacing in the combined spectrum in order to achieve the Regime II speckle contrast value cited above that assumed evenly-spaced modes. In practice, a somewhat shorter fiber may be tolerable depending on actual details of the combined spectrum. In a worst-case situation, if a very large number of longitudinal modes are being randomly combined in the fiber, and the narrow lines are not overlapped, the fiber may have to be so long that the width of the fiber’s incoherent transfer function approaches that of the individual mode linewidths in order to achieve the Regime II speckle contrast value.
15.6 Wavelength-beam-combined FP laser diodes
Spectral beam combining (SBC) or wavelength beam combining (WBC) methods can couple emissions from multiple incoherent emitters, each having a slightly different wavelength, into a multimode fiber [28–30]. Although other configurations are possible, a common WBC setup includes a grating, a Fourier transform lens, an array of laser emitters (in this case, a 1D diode array “bar”) and an output coupler mirror. See Fig. 7 . The Fourier transform lens, grating, and output coupler mirror constitute an external cavity for emitters in the diode laser array, and this external cavity forces each of the diode emitters to operate at a different wavelength as determined by grating dispersion and the Fourier transform lens acting together. (The diode bar and grating are each located a distance f from the transform lens having focal length f).
Emissions from the multiple LD emitters are forced by the external cavity to each have a different wavelength. The wavelengths are separated by Δλs, which is determined by spatial separation of emitters in the focal plane of the Fourier transform lens, the focal length f, and grating dispersion, according to:
A typical wavelength-beam-combined spectrum is shown in Fig. 8 . When grating dispersion occurs along the array’s slow-axis direction, the widths of the individual peaks, Δλ, are determined by the “stripe widths” of the individual diode emitters which, in a typical 1D laser diode bar, are all identical. Therefore, the ratio of spectral peak width / spectral peak spacing is determined by (but is not necessarily the same numerical value as) the ratio of diode emitter width / diode emitter spacing in the array. In other words, the ratio Δλ / Δλs is related to the fill-factor of the laser diode array in this WBC configuration. The WBC spectrum typically has an overall width of 10 to 20 nm, which is ideal for our speckle reduction method and narrow enough to maintain high color saturation. The fill-factor of the bar can be made high enough that the ratio Δλ / Δλs is in the 0.25 to 0.50 range, and one WBC scheme may achieve ratios as high as 0.8 . As discussed earlier, a high fill-factor helps in our speckle reduction scheme by eliminating the plateau region of the multimode fiber.
When using wavelength beam combining methods, the combined beams are usually collinear and spatially overlapped (coaxial). This means one can easily couple the beam into the multimode fiber such that all wavelengths are coupled into all fiber modes. A mode scrambler may not be needed.
For all of the above reasons, WBC laser diode arrays appear to be particularly well-suited to application of our speckle reduction method. A single laser diode bar combined into a relatively short multimode fiber could provide enough power for large-venue display applications and achieve 1% speckle contrast. If more power is needed, multiple laser diode bars (2D array stacks) can be wavelength-beam-combined into the same multimode fiber [29,30], or fibers from several WBC bars (one fiber per WBC bar) can be bundled together inside the projector.
As an example, consider a WBC situation where Δλ = 0.25 nm, Δλs = 1 nm, Δλe = 10 nm, and a total of 20 emitters are combined. Equation (15) predicts that speckle contrast = 0.007 might be achieved at the end of a 25 meter / 0.22 NA fiber, or 6 meter / 0.44 NA fiber. If a 100 mm diameter projection lens is used such that 1/√K = 0.007, then 1% speckle contrast might be achieved in the projected image without relying at all on wavelength diversity at the projection screen (Rλ = 1). Alternatively, if one exploits wavelength diversity at the screen (Rλ = 9.4 by Eq. (28), 1/Rλ = 0.11), then one could achieve 8% contrast at the end of the fiber with a very short 20 cm / 0.22 NA fiber, and less than 1% contrast in the viewed image.
The longitudinal modes of the external cavity are not shown in Fig. 8. If external cavity length is 50 cm, and wavelength is 500 nm, spacing between external-cavity longitudinal modes would be about 0.25 pm (0.00025 nm). This spacing is small enough that, in most cases of practical interest for using WBC systems with our method, existence of the external-cavity substructure can be ignored.
15.7 Speckle reduction for VCSEL and VECSEL laser diodes
Vertical cavity surface emitter lasers (VCSELs) offer many prospective advantages for use in our scheme. Emission bandwidths are in the 0.1 to 1 nm range (and can be larger) and emission spectra have virtually no sub-structure. VCSEL emission wavelength is also inherently more stable versus changes in operating temperature, compared to edge-emitter FP laser diodes, which is important for maintaining color balance in laser projectors. Present VCSELs are relatively low power lasers (100 mW), but, owing to their high beam quality, emissions from very many VCSELs can be multiplexed into a small-core, small-NA multimode fiber. High-power 2D arrays of VCSELs can be made and efficiently coupled into a multimode fiber. As an example, a 500 nm VCSEL array having a spectrum such that Δλ = 0.5 nm would achieve 3% contrast at the end of a 15 meter / 0.22 NA fiber, and 1% at the screen by virtue of 1/Rλ = 0.033. Unfortunately, VCSELs having red, green, and blue emission wavelengths of interest for projection display applications are not yet commercially available.
Vertical-external-cavity surface-emitter lasers (VECSELs) offer watt-level output power at visible wavelengths useful for projection displays. The external-cavity feature of such lasers allows a nonlinear optical crystal to be placed inside the laser resonator as needed to generate second harmonic light at red, green, and blue wavelengths. Such lasers are being commercialized under the NECSELtm brand name (NECSEL Intellectual Property, Inc.) and 1D arrays of NECSELs can produce multi-watt power levels at red, green, and blue wavelengths. However, these frequency-doubled VECSEL lasers have significantly different emission spectra, for our needs, than a VCSEL laser which directly produces visible light without nonlinear conversion. Frequency-doubled VECSEL spectra are usually multi-longitudinal-mode spectra having an envelope function only a few tenths of a nanometer wide, and very narrow longitudinal mode linewidths. As a result, frequency-doubled VECSELs would most likely be used in Regime II (the plateau region) in our scheme, and many devices would have to be combined into the multimode fiber to achieve 1% speckle contrast at the end of the fiber. As an example, assume a frequency-doubled VECSEL array of 20 emitters such that each emitter has 12 longitudinal modes, and that modes are spaced by 0.05 nm (Δλe = 0.25 nm). If 100 such emitters (5 arrays, 1200 total modes) are combined into the fiber with an average spacing between modes of 0.0005 nm (Δλe still = 0.25 nm), then speckle contrast = 0.035 would be achieved at the end of a 30 meter / 0.22 NA fiber, and contrast at the screen would be about 2% (with 1/Rλ = 0.54). Lower contrast might be achieved, for example, by multiplexing more emitters into the fiber or by using a larger wavelength spacing to achieve a larger Rλ factor.
16. Speckle reduction for femtosecond lasers
Ultrafast lasers are usually mode-locked lasers having a multi-line spectrum and a nominally Gaussian envelope function. The individual longitudinal modes are locked in phase relative to each other, thereby resulting in ultrafast pulse emission, and spectral widths of the longitudinal modes are much narrower than the separation between modes. Although the laser modes are locked in relative phase, the autocorrelation of the power density spectrum, Cp(ν), which enters into Eq. (9) includes no phase information. Therefore, Cp(ν) is identical to that of a non-mode-locked multi-longitudinal-mode laser except that the mode-locked spectrum typically has many more modes.
The spectral bandwidth of a transform-limited 100-fs (FWHM) mode-locked laser, operating at 500 nm center wavelength, is about 3.7 nm FWHM (for the Gaussian envelope). Cavity length for a typical mode-locked laser is in the 100 cm range, which implies a longitudinal mode spacing of 10−4 nm. Therefore, using Eq. (15), a fiber length of about 20 meters (fiber NA = 0.22) might achieve speckle contrast of 1% operating in Regime I. Operation in Regime II, with a longer and/or higher-NA fiber, might also achieve 1% contrast considering that more than 10,000 modes would be lasing simultaneously. If pulse duration is shorter than 100 fs, then even shorter multimode fibers would achieve 1% contrast at the end of the fiber. Since the width of the envelope function is nanometers wide, a large Rλ factor readily enables 1% contrast at the screen.
The peak power of a femtosecond laser is high enough to incur (usually unwanted) nonlinear effects in the multimode delivery fiber that can change the spectrum of light passing through it. Such nonlinear effects may be controlled to some extent by adjusting the core diameter of the multimode fiber. Cost, size, reliability, and ease-of-use are the main problems with today’s high-power femtosecond lasers that might otherwise be suitable for large-venue display applications. The technology is advancing rapidly, so this could change.
17. Speckle reduction for picosecond lasers
High-power red, green, and blue picosecond lasers have been developed for large-venue display applications . The spectral bandwidth of a transform-limited 1-picosecond (FWHM) laser operating at a 500 nm center wavelength is about 0.4 nm FWHM (1/e half-width approximately 0.5 nm for the Gaussian envelope). If cavity length is 1 meter and longitudinal mode spacing is 10−4 nm, then Eq. (15) predicts 0.012 speckle contrast at the end of a 100 meter / 0.22 NA fiber, or 25 meter / 0.44 NA fiber. Since Eq. (28) predicts 1/Rλ = 0.33 for 250 μm screen roughness, speckle contrast for the projected image could be well under 1%.
If the transform-limited pulse duration is 10 ps (0.04 nm FWHM bandwidth), then a 40 meter / 0.44 NA fiber would at best achieve about 0.03 speckle contrast at the end of the fiber, which, since 1/Rλ is 0.87 in this case, is not good enough to achieve <1% contrast in the projected image.
A straightforward way to broaden the spectrum of a picosecond laser is to inject light into a small-core (<100 μm) optical fiber which broadens the spectrum via self phase modulation (SPM). SPM theory indicates that observed spectral broadening depends on laser intensity (W/cm2), pulse duration, and propagation distance in the fiber according to :
As an example of what’s possible, Wang et al. reported using a 100-micron-core silica glass fiber to spectrally broaden 527 nm picosecond laser pulses . Input bandwidth was 1.5 nm at 527 nm and peak power density in the fiber was 3.2 GW/cm2 (2 μJ pulse energy, 8 ps pulse duration). Different lengths of 100-micron-core fiber were used with the same input pulse energy. The observed spectral bandwidths were: 5.8 nm FWHM for a 22 cm fiber length (factor of 3.9 increase), 7.7 nm for a 42 cm length (factor of 5.1 increase), and 13 nm for an 84 cm length (factor of 8.7 increase). The result for the 84 cm fiber included some broadening due to stimulated Raman scattering. Spectral profiles are shown in Fig. 9 .
Wang’s results were achieved using rather high pulse energy (2 μJ), but the fiber core size was also quite large (100 μm). It should be possible to achieve similar results using lower pulse energies in the 0.1 to 0.5 μJ range using somewhat smaller-core fibers. Since one only needs broadening into the 0.1 to 1 nm range in our scheme, use of SPM broadening with picosecond pulses seems feasible.
18. Speckle reduction for fiber lasers
High-power fiber lasers emitting at visible wavelengths have great potential for use with the proposed speckle reduction method. Prospective devices include up-conversion fiber lasers pumped with infrared laser diodes, down-conversion fiber lasers pumped with blue laser diodes, and infrared fiber lasers converted to visible wavelengths using nonlinear conversion; e.g., frequency-doubled fiber lasers.
Fiber lasers have potentially wide emission bandwidths and long cavity lengths such that despeckling can be achieved using a multimode fiber operating in Regime I. For example, 500-femtosecond fiber-based lasers have been frequency-doubled to generate high average power at 520 nm with a 2 nm FWHM emission bandwidth, and having a longitudinal mode spacing of 10−4 nm . With such a laser, 1% speckle contrast might be achieved using a 60-meter, 0.22 NA fiber or a 15 meter, 0.44 NA fiber.
A continuous-wave fiber laser pumped with blue (GaN) laser diodes has been reported that produced 600 mW of cw power at 522 nm with an emission bandwidth of 1.2 nm FWHM . Fiber laser cavity length was only 4 cm, which corresponds to a longitudinal mode spacing of about 0.002 nm. Equation (15) predicts speckle contrast of 0.034 at the end of a 10 meter, 0.22 NA fiber operating in Regime II. The corresponding 1/RΩ value would also be about 0.034. Considering that 1/Rλ = 0.30 for such laser (assuming 250 μm screen roughness), speckle contrast of 1% should be achievable at the screen using a 60 mm projection lens.
19. Speckle reduction for broadband OPO / OPA sources
Optical parametric oscillators (OPOs) and amplifiers (OPAs) are nonlinear optical devices that can generate broadband visible light when optically pumped with a narrowband pump laser. Such devices can be continuously tunable in that they generate any wavelength of interest in the visible range, starting with a fixed-wavelength pump laser, simply by adjusting some operating parameter of the nonlinear crystals being used. Continuous tunability renders OPO/OPA devices potentially attractive for 3D laser displays based on six- and other multi-wavelength schemes. Average power levels in the 10W range can be achieved at red, green, and blue wavelengths. If the spectra are not deliberately narrowed, OPO and OPA emission spectra are usually at least several nanometers wide and can be tens of nanometers wide.
OPOs intended for projection displays typically employ a nanosecond-pulse laser to pump nonlinear crystals placed in an optical resonator. Historically, the nonlinear crystals have been operated in a high temperature oven (e.g., 150°C), but recent advances in magnesium-oxide-doped nonlinear materials may allow reliable operation of nanosecond OPOs at or near room temperature. OPO cavity lengths are in the 5 to 10 cm range. It should be straightforward to achieve 1% speckle contrast using such a source as described in the previous section for the visible fiber laser having a 4 cm cavity length.
OPAs pumped by ultrafast lasers (picosecond or femtosecond) are attractive because the need to operate nonlinear crystals in a high-temperature oven is often eliminated. OPAs pumped by nanosecond lasers may also operate reliably near room temperature when magnesium-oxide-doped, periodically-poled nonlinear crystals are used. An optical resonator is not required in an OPA device, which means that the associated spectra do not have longitudinal mode substructure. Speckle reduction to 1% should be possible using a multimode fiber operating in Regime I.
20. Speckle reduction for nanosecond lasers
Nanosecond-pulse RGB lasers have been developed for projection displays that provide average power levels in the 5 to 10W range at pulse rates of 25 kHz, and with pulse energies of a few hundred microjoules . Red, green, and blue wavelengths are generated from a 1-μm-wavelength pump laser using nonlinear frequency conversion methods. The green wavelength has a very narrow spectral bandwidth (< 0.05 nm), but the red and blue emission spectra have bandwidths of a few nanometers since these wavelengths are generated using a green-pumped optical parametric oscillator (OPO). The OPO’s infrared “signal” and “idler” emission wavelengths are frequency-doubled to produce relatively broadband red and blue emission.
Considering only the green (non-OPO) light in such systems, Eq. (30) indicates that it is much more difficult to achieve spectral broadening due solely to self phase modulation (SPM) when using nanosecond-pulse lasers. With 200 μJ of pulse energy in a 10 ns pulse, Eq. (30) implies that, using the same 100-μm silica fiber as in Fig. 9, one would need a 10,000 times longer fiber (10 km) to achieve SPM-broadening similar to Wang’s results.
On the other hand, not much spectral broadening is actually needed in our scheme. A spectral bandwidth of 0.1 nm might achieve 1% speckle contrast at the end of a 80 meter, 0.66 NA fiber if the spectrum has no substructure. Such fibers are not commercially available today, but could be in the near future. Note that, since 1/Rλ would be 0.86 (250 μm screen roughness), one would not get much help from wavelength diversity at the projection screen to reduce speckle contrast. If the spectrum can be broadened to 0.5 nm, without substructure, then a 36 meter, 0.44 NA fiber would achieve 1% speckle contrast at fiber’s end and 1/Rλ would be 0.47. Speckle contrast at the screen would then be less than 1%.
Various approaches can be considered for broadening nanosecond-pulse light into the 0.1 to 1 nm range. One might use a much smaller-core fiber than Wang’s fiber, but coupling high-average power into a 10-micron-core fiber, for example, with high efficiency and good reliability is problematic. One might use a highly nonlinear fiber (with increased n2 in Eq. (30)) to reduce the required length of broadening fiber or enable larger core sizes. However, the value of n2 in Ge-doped silica fibers is typically no more than a factor of two or three higher than for pure silica fibers, and absorption losses increase as the Ge doping level is increased. Lead silicate glass and related specialty fibers have n2 values about 10 times larger than that of pure silica, and bismuth oxide-based glasses have n2 values more than 50 times that of pure silica, but these fibers absorb strongly at visible wavelengths.
Another option may be to use a hollow-core (e.g., photonic bandgap) fiber filled with a suitable nonlinear medium. Media that have been investigated in hollow-core fibers include ethanol , water , nitrobenzene or carbon disulfide , and noble gases such as xenon . Kohler achieved a useful amount of spectral broadening with a nanosecond-pulse laser when using SPM in a 30-cm-long carbon disulfide (CS2) free-space cell (not a hollow fiber), and using a large beam diameter of 3 mm . Using Eq. (30) to extrapolate Kohler’s results, a CS2-filled, 100-μm, hollow-core fiber only 1 meter long might achieve spectral broadening into the 0.1 to 1 nm range using 500 μJ, 10 ns (50 KW peak power) pulses. The main problem with this approach is that transmission through meter-long lengths of filled fiber tend to be low (e.g., 50%), but this could change as new advances are made.
One might also use a free-space CS2 cell to achieve spectral broadening, as Kohler did. Equation (30) suggests broadening to 0.1 nm could be achieved using 500 μJ pulses (at 500 nm wavelength) focused to a beam diameter of about 400 μm in a 30-cm-long CS2 cell, if laser pulse duration is in the 4 to 5 ns range.
Spectral broadening due to stimulated Raman scattering (SRS) can be achieved readily with nanosecond pulses in silica glass fibers of reasonable core size and length . Some system developers have been able to exploit SRS to achieve very good speckle reduction in laser projection displays using a multimode delivery fiber . However, spectral broadening due to SRS might be considered hard-to-control and may not always be compatible with the color space scheme (color gamut) one is trying to achieve. Some system designers may prefer a spectral broadening method that does not generate the multiple Stokes wavelengths typical of SRS spectra in optical fibers.
It is possible to broaden nanosecond-pulse spectra by injecting light into a fiber having a zero-dispersion wavelength (ZDW) shorter than the input wavelength such that propagation occurs in the fiber’s anomalous dispersion regime. One might then expect spectral broadening due to soliton fission and super-continuum generation. However, at present, fibers with zero-dispersion-wavelengths shorter than 670 nm are not available as far as we know.
One way around this problem is to mix 1-micron-wavelength infrared light and its corresponding frequency-doubled green light in the same fiber, using a fiber with a zero-dispersion wavelength between the infrared and frequency-doubled wavelengths. Infrared light experiences anomalous dispersion as it propagates and breaks up into ultrafast pulses as a result of soliton fission. The infrared soliton light spectrally broadens the visible light co-propagating in the same fiber by the process of cross-phase modulation (XPM) and other nonlinear processes. Such broadening has been demonstrated, for example, by injecting 1064 nm and 532 nm emission from a nanosecond-pulse, frequency-doubled microchip laser into a fiber having a zero-dispersion wavelength slightly shorter than 1064 nm . In theory, one might limit spectral broadening to a nominal 10 nm bandwidth by properly controlling input peak-power density, core diameter, and/or fiber length. The main problem with this approach, for our needs, is that ZDW-shifted fibers tend to be single-mode fibers that cannot reliably handle high average power.
21. Speckle reduction for narrowband CW lasers
Continuous-wave (cw) diode-pumped solid-state lasers and fiber lasers are commercially available that provide more than 10W at green wavelengths. These lasers would be well-suited for projection displays were it not for severe speckle contrast resulting from very narrow emission bandwidth. These lasers are typically single-frequency lasers having a spectrum that consists of a single longitudinal mode with a width on the order of 1 MHz (<10−6 nm at 500 nm).
As was discussed for nanosecond lasers, the single-mode cw spectrum must be broadened to at least 0.1 nm to use our speckle reduction method with any practical length of multimode fiber. One possible solution may be to mix the cw green light with cw 1064 nm light in a photonic crystal fiber having a zero-dispersion wavelength between 1064 nm and 532 nm, as described above for nanosecond-pulse lasers - but this may not work for cw lasers if emission bandwidth is too small. Reliable implementation of this technique at high average power, and with a single-mode fiber, may not be feasible even when using a cw laser.
Another option may be to modulate the single-frequency cw laser emission using a bulk external phase modulator, thereby adding sidebands to create a multi-line spectrum. (A bulk modulator would probably be needed to handle high average laser power). Commercially available resonant phase modulators can achieve a modulation frequency as high as 10 GHz. If one can generate 50 sidebands with 10 GHz spacing, then the overall width of the spectrum would be 500 GHz or about 0.4 nm at 500 nm. In this case, Eq. (28) indicates 1/Rλ = 0.52 if screen roughness is 250 μm. One would then need a 1/RΩ factor of 0.02 or less to achieve 1% speckle contrast at the screen. However, in the present example, operating the multimode fiber in Regime II (plateau region) would achieve 1/RΩ = 1/√50 = 0.14, which is not good enough.
A second phase modulator might be used in tandem with the 10 GHz modulator to enable operation in Regime III of the multimode fiber. For example, if the second phase modulator has a modulation frequency of 100 MHz, it would add multiple sidebands spaced by 100 MHz to each of the sidebands generated by the first modulator, thereby broadening each of the 10 GHz-spaced sidebands. If each of the 50 sidebands from the 10 GHz modulator can be broadened to 4 GHz (40 sidebands spaced by 100 MHz), then Eq. (15) indicates that speckle contrast of 0.013 might be achieved at the end of a 100 meter, 0.44 NA fiber. Assuming a 60 mm diameter projector lens, 1/RΩ would be about 0.018, and 1/RλRΩ would be less than 1% at the screen.
Although bulk phase modulators that achieve more than 50 sidebands have been reported in the literature [44,45], it’s not clear that generation of this many sidebands can be achieved easily with commercially available and low-cost phase-modulator devices.
Phase-locked sideband generation is basically what happens when a laser is actively mode-locked (placing the modulator inside the laser resonator) except that many more modes or sidebands lase simultaneously in a mode-locked laser. As discussed earlier, mode-locking to generate picosecond or femtosecond pulses enables 1% speckle contrast to be achieved in straightforward fashion using our scheme. Accordingly, it may be preferable to mode-lock the laser rather than modulate it externally.
22. High-brightness LEDs
Great progress has been made in recent years to increase output power and improve the beam quality of visible light-emitting diodes (LEDs). High-brightness LEDs are now available at red, green, and blue wavelengths such that multiple LED emitters can be multiplexed into a fiber-optic bundle, or liquid light guide, to make high-power light sources potentially suitable for large-venue displays.
Liquid light guides are attractive for delivery of LED light in that they are readily available with “core” diameters as large as 10 mm, and in lengths of 1 to 2 meters and longer. They are also relatively inexpensive. Transmission efficiency at visible wavelengths is in the 70% to 90% range for a nominal 1-meter-long guide.
Numerical apertures of liquid light guides are in the 0.5 to 0.7 range. Since LED emission bandwidths are several tens of nanometers wide, and LED spectra have virtually no substructure, such high NAs are well-suited for application of our speckle reduction method with relatively short light-guide lengths.
As an example, if LED bandwidth is 30 nm, Eq. (28) indicates Rλ≈23 for screen roughness of 250 μm, or Rλ≈15 for 100 μm roughness. If light-guide NA is 0.5 to 0.7, then any length longer than about 30 cm will result in a situation where M in Eq. (22) is greater than 10,000 so that RΩ is dominated by the applicable K value. Even for projector lens diameters as small as 25 to 60 mm (1/RΩ ≈1/√K ≈.01 to 0.03), speckle contrast in the viewed image could be much less than 1% due to the large Rλ factors that would apply.
A potential issue is that, because the light-guide core diameter is already quite large (3 to 10 mm), as needed to efficiently couple LED emission into the guide, one cannot rely as much on up-collimation (as when using small-core fibers) to reduce beam divergence so that projector efficiency does not suffer. There is probably a tradeoff to be made regarding power delivered through the light guide versus projector throughput. This issue may be especially important considering the trend toward higher F# (lower etendue) as a way to improve projector contrast ratio and efficiency. This situation may change as further improvements are made in LED power and beam quality.
23. Extended method
To achieve 1% contrast, it may be necessary in some situations to include additional speckle reduction measures resulting in what we refer to as the “extended method.” Although more complicated, the extended scheme does not involve any moving components and preserves the basic intended aspects of our speckle-reduction method.
Figure 10 shows the layout of the extended scheme system, which is related to Kohler illumination as commonly used in optical microscopes. Several multimode fibers, each of which is coupled to a separate laser light source, are configured into a 1D or 2D fiber array pattern at the delivery end of the fibers. The laser sources are incoherent relative to each other. Partially de-speckled light emerging from each of the multimode fibers is up-collimated by the collimator lens to a diameter (say, 10 mm) that fills the dimension of the light-valve image-forming device (e.g., digital mirror device, liquid-crystal-on-silicon chip) being used in the projector. The delivery end of the fiber array is positioned one focal length from the collimator lens having focal length, f. A fixed diffuser plate is placed a distance, f, on the other side of the collimator lens. The diffused light is then used to illuminate the area of the projector’s imaging device chip, and the image is projected onto the screen using a projector lens with focal length, F, such that F > f. It is assumed that, to achieve large magnification of the projected image, the imaging device is placed about one focal length F from the projection lens and that projector magnification at the screen is mproj.
Speckle contrast, not considering wavelength diversity or angle diversity of the illumination light, is again given by Eq. (22), which we reproduce here:
As before, the parameter M = 1/C2end where C2end is speckle contrast at the end of each multimode fiber in the array, and K is the number of projector lens resolution areas/elements at the projection screen contained within one eye resolution area or element. K is given by Eq. (23) or (24) depending on whether or not the projector lens is overfilled. Speckle contrast in the extended scheme of Fig. 10 is:
Equation (31) indicates that, when certain measures are taken to decorrelate the emissions from the multiple fibers (see below), speckle contrast in the viewed image is reduced by a factor 1/√Nfiber. Equation (31) also anticipates that, in some cases, it may be possible to increase K to K’ when NAcollimator is greater than NAprojector. The extended method can reduce the length of multimode fiber needed to achieve 1% contrast in the viewed image, and may reduce the projector lens diameter consistent with 1% contrast.
For Nfiber to be the value that applies in Eq. (31), the multiple fiber ends must be arranged with a large enough spacing that the speckle patterns produced by each fiber are fully decorrelated at the projection screen via angle diversity. The required angular separation is that which shifts the speckle patterns from adjacent fibers by one eye resolution element at the screen. In a typical setup, as in Fig. 10, the minimum required spacing, Δx, is given by:
If θeye = 5 x10−4 radians, mproj = 300, and f = focal length of collimator lens = 10 to 25 mm, then the required Δx would be about 1.5 to 3.8 mm, which, considering the small fiber diameters that would be used (500 microns or less), could be implemented easily in practice.
Since a small multimode fiber diameter would be used, beam divergence after the collimator lens when using a single fiber would be low (20 milliradians or less, assuming up-collimation to 10 mm diameter) even when using a fiber NA as high as 0.44. Therefore, most of the collimated light should reach and pass though the projector lens aperture. However, when using a multi-fiber array, beam divergence after the collimator lens would increase and would be determined by the lateral extent of the fiber array and the collimator focal length, f. Light collection by the projector lens might then be reduced as a result. The number of fibers in the array is limited practically by the spatial separation needed to decorrelate the speckle patterns and the tolerable reduction in projector throughput.
The enhanced value of K = K’ should be regarded as a potential enhancement resulting from an increased number of lens resolution areas/elements being included within one eye resolution element at the screen. The reasoning is similar to that presented in Goodman’s analysis of speckle contrast in Kohler illumination schemes . When NAcollimator ≥ NAprojector, there may be as many as (NAcollimator / NAprojector)2 collimator lens resolution areas/elements, each of which contributes an independent speckle pattern at the screen, within each projector lens resolution element. Experiments are needed to determine if and under what conditions such enhancement is achieved.
24. Summary and conclusions
We believe we have identified a practical and versatile method for achieving 1% speckle contrast (and less) in full-frame laser projection images. The method exploits intermodal dispersion in a multimode fiber to create a low-speckle light source that can be used to illuminate the imaging light valve in a projector. No moving diffusers or other moving components are needed. The method can be used with many types of visible laser sources.Our scheme is compatible with existing cinema-style projectors and potentially enables low-speckle laser sources to replace arc lamps with only minor projector modifications. When used with visible laser diodes, our speckle reduction method could be robust, compact, and low-cost enough for use in laser-based television. The method may also be useful in other applications that require low-speckle UV, visible, or infrared illumination, and that can tolerate delivery through a multimode fiber. We will soon conduct experiments intended to validate our theory using various laser light sources of interest for display applications.
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