We study the field-of-view (FOV) of an upconversion imaging system that employs an Amplified Spontaneous Emission (ASE) fiber source to illuminate a transmission target. As an intermediate case between narrowband laser and thermal illumination, an ASE fiber source allows for higher spectral intensity than thermal illumination and still keeps a broad wavelength spectrum to take advantage of an increased non-collinear phase-matching angle acceptance that enlarges the FOV of the upconversion system when compared to using narrowband laser illumination. A model is presented to predict the angular acceptance of the upconverter in terms of focusing and ASE spectral width and allocation. The model is experimentally checked in case of 1550-630 nm upconversion.
© 2016 Optical Society of America
Image frequency upconversion of a full unscanned 2D image in a nonlinear (NL) optical crystal is becoming an attractive alternative to image in the visible spectral region objects illuminated with (or radiating at) wavelengths in the infrared (IR), thus taking advantage of the better characteristics of image sensors in the visible in terms of noise, speed, resolution, or uncooled operation as compared to existing imaging sensors in different spectral ranges of the infrared and in the THz region [1,2]. This technique already proposed in the early days of nonlinear optics  has been of limited application since then due to the lower effective nonlinear coefficients available for birefringent phase matching with typical nonlinear crystals, the high power density required, and was restricted to systems based on high peak power pulsed lasers. There has been a noticeable recent progress in this field due to the use of the high second-order nonlinear effective coefficients available in poled ferroelectric crystals and by exploiting the high power density that can be achieved in CW lasers in intracavity nonlinear optical mixing . The most appealing technique for image up-conversion is the second-order nonlinear process of sum-frequency mixing, although other alternatives based on third-order nonlinear processes have been explored .
A typical configuration used in nonlinear image upconversion systems based on sum-frequency mixing (SFM) is a modified paraxial 4f-system [Fig. 1(a)] with a NL crystal placed in the Fourier plane . As illumination, an IR beam is directed to the object (target) to be imaged and after being modulated by the spatial information of the object (either transmitted through or reflected from the object), the resultant beam is focused by a first lens (L1) into the NL crystal (i.e. a quasi-phase-matched (QPM) crystal). A second beam acting as a pump (which does not travel through the focusing lens of the 4-f system), is coupled into the system by means of a dichroic beamsplitter (BS) and interacts in the NL crystal with the focused IR beam in a SFM process. Then, the upconverted radiation is collected by the second lens (L2) and imaged onto an image sensor (Focal plane sensor, FPS) after filtering (F) the remaining IR, pump and other possible harmonics. In general, focal lengths f1 and f2 need not to be equal. This configuration allows independent control of the waist of the interacting beams. For resolution purposes, the pump beam, which acts as a soft aperture with a gaussian spatial profile in the Fourier plane (spatial filter), should be as wide as possible, in comparison to the IR beam, in order to permit a high number of IR spatial frequencies to pass.
For a given pump beam size, the resolution of upconverted images is also increased as the IR illumination beam size is more tightly focused in the Fourier plane (allocated in the center of the nonlinear medium) [7,8]. This means, according to the invariance of the waist-divergence product of focused Gaussian beams, that the IR beam will exhibit far more divergence than the pump beam and, therefore, the pump beam can be regarded as a nearly collimated beam (i.e. an on-axis wave-vector) and the IR illuminating beam as a set of different angles (i.e. on-axis and off-axis wave-vectors) [Fig. 1(b)]. As a result, 2-D image upconversion requires simultaneous upconversion of a set of different incoming angles, and thus non-collinear QPM (NC-QPM) between kp (SFM pump wave-vector), kIR (IR illumination wave-vectors), kG (QPM grating reciprocal vector) and kup (upconverted wave-vector) [Fig. 1(c)]. With the use of Type 0 QPM, no spatial walk-off effects need to be taken into consideration. Hence, there is a trade-off between focusing and the area of the object that can be upconverted (i.e. the field-of-view of the upconverter).
Figure 2(a) shows how different parts of the object to be upconverted are mapped to different angles by lens L1 of the 4-f system. The IR illumination beam diameter and the focal length of L1 will determine the angular spectrum of the illumination. For a given input illumination beam diameter on L1, the shorter the focal length of L1 the wider the angular spectrum to be upconverted [Fig. 2(b)]. The upconverted angles will be determined by the product of the incoming angular IR spectrum I(θ) and the the angular acceptance (i.e. the incoming angles that can be upconverd with a noticeable efficiency) of the NL crystal η(θ). In order to upconvert as much as possible of the IR illuminated area, the IR angular spectrum and the NL crystal angular acceptance must be of the same width. When a monochromatic beam tuned for perfect collinear QPM is used as the illumination source (i.e. laser illumination), the QPM angular acceptance is defined by the collinear wave-vector conservation relation, thus providing a rather narrow angular acceptance. In case of tight focusing of the IR beam, the angular acceptance of the PPLN crystal could be smaller than the angular content of the incoming light [Fig. 2(c)]. As a consequence, only object points near to the system axis (corresponding to wave-vector directions close to collinear interaction) can be upconverted, yielding therefore a reduction in the FOV of the upconverted system.
However, the upconverted angles in non-collinear interactions are wavelength-sensitive and, therefore, a spectrally broad illumination beam would allow the phase-matching condition to be extend to a wider set of incoming angles, and thus, object points far from the system axis could be efficiently upconverted at different wavelengths [Fig. 2(d)]. Thus, as compared with narrowband laser illumination, the use of an ASE illumination source would provide a richer set of wavelengths participating in a non-collinear QPM interaction and, consequently, it would allow enhancing the FOV of the upconverted images by providing an effective angular acceptance wider than that of single-wavelength illumination.
In practical situations, illumination with ASE sources may be of interest in applications where long interaction lengths are needed (i.e. nonlinear crystal of a few centimeters), due to the inverse relation between the nonlinear crystal angular acceptance and the crystal length, the angular acceptance of the crystal becomes small (reduced) and correspondingly the FOV of the upconverter. It is also advantageous for potential compact solid-state laser devices (monolithic/semimonolithic) that would need either short NL crystal lengths or short focal length lenses. Although short length NL crystals would benefit from broad angular acceptance (at the expense of efficiency), the use of lenses with reduced focal length could lead to overfill the wide angular acceptance of short length crystals, and then, limiting the upconverted FOV.
Throughout this work, we will regard an ASE source as comprising a seed ASE emitter, a spectral shaping stage and a booster amplifier [Fig. 3]. In contrast to thermal sources, if the seed source is adequately bandwidth limited the amplifier gain can be concentrated in a narrower band, which increases spectral brightness (intensity) thus yielding higher level of upconversion (given a fixed upconverter efficiency) and longer working distances. A model is presented for fiber amplified ASE sources. These sources are regarded as spatially coherent and spectrally incoherent.
In  we found that the FOV in an image upconversion system might be increased, due to NC-QPM, if a set of illumination wavelengths is employed. In this work we show the way in which the FOV of an ASE-illuminated upconverting system depends upon the upconverter optical configuration (focusing conditions, crystal length), the spectral bandwidth and the spectral allocation of the IR ASE illumination, and how the spectral characteristics of the ASE source may be tailored to increase the FOV for a given upconverting system. This allows to optimize the spectral intensity of the IR illumination source for a targeted FOV, thus avoiding the waste of optical power associated to those spectral components which are far from reaching phase-matching and cannot be upconverted.
Despite experimental data in the following sections were obtained for upconversion of illuminating wavelengths around 1550 nm, the techniques and model presented can be extended to conversion of other spectral bands in the IR.
2. Contribution of NC-QPM to the FOV of upconverted images
In this section, we will show briefly the enhancement in the upconverted image area owing to NC-QPM. As aforementioned, when using a multiple wavelength source for illumination in an upconversion imaging system, the upconverted image area can be regarded as the contribution of the different upconverted illumination wavelengths. In Fig. 4 we show upconverted images of a square pattern when laser illumination in the SWIR is tuned from λQPM = 1544.5 nm to longer wavelengths (where λQPM is the wavelength at which the collinear upconversion efficiency of the nonlinear crystal is maximum) using the upconverter described in  (i.e. mixing of a 1064 nm pump wave with SWIR illumination providing upconverted images around 630 nm). As it can be seen, different wavelengths show different upconverted annular-like patterns due to NC-QPM and the sum of the isolated upconverted images at different wavelengths resembles to that of an ASE illuminated system. Therefore, if a multiple wavelength source is employed for illumination (i.e. an ASE source) the area of the upconverted image (field-of-view) increases. The total upconverted area will depend upon the number of incoming wavelengths, relative amplitudes andupconverter parameters.
3. Angular acceptance of the image upconverter
Here we present the upconversion efficiency as a function of the incoming angular and wavelength spectrum of the IR beam.
When IR illumination is comprised of a single-wavelength at λQPM, only incoming angles near to collinear interaction can be upconverted and the maximum upconversion efficiency is obtained at normal incidence [Fig. 2(b)]. The set of the accepted incoming angles is limited by the angular acceptance of the upconversion efficiency. In addition, the NL crystal angular tolerance (Full-Width at Half Maximum (FWHM) of the angular acceptance) depends inversely upon the nonlinear crystal length. For incoming IR wavelengths other than λQPM, the incoming angular acceptance is determined by non-collinear interaction (and the crystal length as well) and the maximum of the upconversion efficiency is shifted away from 0°. As a result, it can be drawn that a multiple-wavelength illumination source effectively increases the angular acceptance of the upconversion process [Fig. 2(d)].
In Figs. 5(a) and 5(b) we plot a numerical calculation of the normalized efficiency of a SFM process (1064 nm pump wave and an IR illumination signal around 1545 nm yielding an upconverted wave around 630 nm for a QPM period Λ = 11.785 μm and temperature of 22°C) as a function of both IR illumination wavelength and incoming angle for differentfocusing conditions and two different PPLN crystal lengths L (L = 5 mm in Fig. 5 left column and L = 25 mm in Fig. 5 right column) by solving the non-collinear phase-matching equation. It can be seen that for any incoming IR illumination direction, there is a wavelength that can be non-collinearly phase-matched, mainly by longer wavelengths than λQPM. This means that only longer wavelengths than that of QPM contribute to increase the FOV with a good efficiency. The crystal length influences in either broadening (figures on left side) or sharpening (figures on right side) the angular response of the nonlinear crystal.
For a given input aperture, the angular content of the IR incoming beam is governed by the focusing lens. Tighter focusing implies a wider angle span of the incoming IR wave-vectors (i.e. broader angle spectrum). From a practical perspective, it is also important to take into account the transverse light distribution of the illuminated beam. The actual angular spectrum impinging on the nonlinear crystal will be the transverse light distribution transformed by the focusing lens (i.e. typically a focused gaussian beam). If the IR illumination angular dependence is considered, and assuming gaussian illumination [Figs. 5(c)-5(f)], only a finite incoming angular spectrum (Δθ) is allowed and it imposes a reduction in the number of wavelengths (Δλ) that can be non-collinearly phase-matched. We can see the influence of focusing for IR illumination waists of 10 µm [Figs. 5(c) and 5(d)], 20 µm [Figs. 5(e)and 5(f)] and 30 µm [Figs. 5(g) and 5(h)] respectively. The less focused the gaussian beam, the narrower the wavelength band which effectively contributes to the upconversion process.
Because of the aforementioned points, the span of the accepted IR angle spectrum will depend upon the focusing lens (L1) and the nonlinear crystal length (i.e. the product of the incoming angular spectrum and the angular acceptance of the nonlinear crystal). And, in turn, the accepted angles of the focused IR signal will determine the maximum number of IR illumination wavelengths contributing to the FOV. In order to obtain the complete angular response for a given focus and crystal length, the total angular acceptance will be the weighted sum of the angular acceptance for every single wavelength within the illumination spectrum. In the next section we will present a model for calculating the normalized upconverted angular spectrum for a given upconverter configuration (lens focal and crystal lengths) and an arbitrary IR illumination wavelength spectrum.
4. Modeling the upconverted angular acceptance
To derive a model for calculating the upconverted transverse profile as a function of the IR illumination incoming angle we will assume gaussian beams for both the IR illumination (i.e. fundamental mode LP01 in an optical fiber can be approximated by a gaussian beam) and the SFM pump beam (i.e. TEM00 beam profile for the pump cavity mode). We will also assume that the focused spot (mode waist) of both the illumination and the pump wave is placed in the center of the nonlinear crystal. From the perspective of image upconversion, it is of particular interest the case where the pump wave area is wider than the illumination wave. This enables an increased image resolution if compared to the situation where both beams have the same focusing conditions. Thus, we consider the Rayleigh length of the pump wave is much longer than the nonlinear crystal length. Under this assumption, we can approximate the pump as a plane wave (i.e. with a constant phase). In contrast, the IR illumination will be a tightly focused Gaussian beam, thus showing a position dependant phase (non-constant radius of curvature and Gouy phase-shift). Then, a non-perfect matching of interacting wavefronts arises from the phase-difference between the pump and the IR illumination and depends upon the degree of focusing and the crystal length. This wavefront phase-difference will sum an additional term to the nonlinear process phase-mismatch thus affecting the overall upconversion efficiency. When both waves have the same degree of focusing, the phase-difference is constant and the overall upconversion is corrected by the well-known Boyd-Kleinman factor . For any other situation, the phase-mismatch will be position dependant and it will mean a penalty in the overall efficiency. We have neglected phase terms in our analysis for simplicity and we mostly concentrate on the angular response of the upconverter.
The radiated field of the upconverted signal is obtained after integration along the crystal of the nonlinear second-order polarization induced by the interaction of two electrical fields of different frequency (Epump and EIR). Under paraxial wave, undepleted pump and slowly varying amplitude approximations the upconverted field can be written as :
In our analysis, we are particularly interested in the angular spectrum of the IR illumination accepted for upconversion. For this purpose, the incoming field (with a gaussian spatial profile) at focus is decomposed in a basis of plane waves using its Fourier transform in the k-space, where k is the wave-vector associated to every plane wave:Fig. 2(b)].
By converting Eq. (1) into its transform coordinates and integrating along the nonlinear crystal (z-coordinate), the upconverted field can be obtained:Eq. (3) FT is used for Fourier transform in both space and time and * represents a two-dimensional convolution. In the following sections, we will regard the pump wave as comprised of a single wavelength and the IR illumination (ASE) as a set of wavelengths enclosed in a given spectral band. For every IR incoming angle and for a given pump wavelength, the IR wavelength which simultaneously fulfills the energy (λup−1 = λpump−1 + λIR−1) and momentum conservation (Δk) condition will be upconverted, thus generating a set of upconverted wavelengths. The upconverted image angular spectrum shall therfore be obtained by integrating Eq. (3) within the upconverted wavelength spectrum, which, in turn, is the wavelength-shifted IR illumination spectrum (Eup(θ) = ∫Eup(θ,λup)dλup). From this point and on, we will make use of the normalized upconverted intensity (E0up(θ) = Eup(θ)/A) and will be given by:Eq. (2) that the angular spectrum is a function of θ but also of wavelength (through wavevector k). For a narrow spectral bandwidth, we can consider the angular spectrum is the same for every wavelength comprising the amplitude spectrum of the interacting fields (either pump or IR illumination). Consequently, the interacting fields may be considered separable functions for angle and wavelength and the normalized upconverted field can be written as follows:
Following a similar analysis to that of , the phase mismatch for non-collinear interaction under the approximation of small angles can be written as:
The normalized upconverted intensity as a function of the incoming IR illumination angle (i.e. the field-of-view of the upconverter) is therefore the sum of the angular acceptance of the nonlinear crystal for every spectral component within the IR illumination bandwidth weighted by the spectral intensity of the illumination GIR(λIR) and the incoming IR angular spectrum FIR(θ). Equation (9) could be extended to a double integral to take into account different wavelengths for the pump wave as well. For a given illumination angular spectrum FIR(θ) and upconverter configuration η(θ, λIR) (lens L1, QPM crystal period/length and pump wave), the upconverted intensity can be tailored by shaping of the IR illumination spectral intensity. This model could also be employed to obtain the FOV of other illumination sources such as multiple-wavelength laser sources, femtosecond-lasers or spectrally filtered supercontinuum sources. The upconverted intensity can be also be obtained as a function of the upconverted angular spectrum by just applying a (de)magnification factor M (angular reduction/expansion due to upconversion and lenses of different focal length) to the illumination angular coordinate.
In Fig. 6 we used the presented model [Eq. (9)] for predicting the upconverted transverse pattern of single-wavelength IR illumination from λQPM = 1544.5 nm to λ = 1551 nm. Left column of Fig. 6 represents the experimentally measured patterns and right column of Fig. 6 the simulated patterns. Both the pump and the IR signal power were adjusted for good visibility of the annular-like patterns in the experimental measurements. The simulation was performed for parameters very close to that in the experimental system (PPLN crystal length of L = 5 mm and period of Λ = 11.785 μm, laboratory temperature of T = 22° C and IR illumination focused to a waist of ω0 = 22 µm). As it can be seen, the predicted patterns match well with the image measurements and every wavelength has a different contribution to the FOV. The differences with the annular patterns found in other works [12, 13] are related to a shorter PPLN crystal and different illumination angular spectrum.
5. FOV vs ASE spectral allocation and bandwidth
In this section, we use our model to calculate the FOV obtained when either the ASE bandwidth or allocation is changed. In Fig. 7 we show the evolution of the FOV for different ASE bandwidths as the spectral allocation is scanned around the QPM condition (λQPM). The different sets of curves are plotted in terms of the ASE bandwidth central wavelength (λ0). The FOV for laser illumination is also added for comparison (red curve). The parameters chosen in the calculation are similar to those available in our experimental system (i.e. IR beam focused to a waist radius of 22 µm, a room temperature of 22°C, Λ = 11.785 μm and λQPM = 1544.5 nm). Calculations are obtained for two nonlinear crystals of length 5 mm [Fig. 7(a)] and 25 mm [Fig. 7(b)] respectively under the assumption of a flat spectral intensity throughout the whole ASE bandwidth. It is worth to note that here we do not take into account potential diffraction effects that could appear when using strongly focused Gaussian beams. Because of the high divergence of a strongly focused gaussian beam, a long crystal shows higher diffraction loss than a short one for a fixed crystal section. We suppose that the NL crystal section is large enough to avoid diffraction effects regardless of the NL crystal length. For completeness, an aperture function could be inserted in our model once all crystal dimensions are known.
Situations in which most of the ASE bandwidth is below λQPM are of no practical interest since there is not a noticeable improvement of the FOV and upconversion efficiency is rather low [Fig. 5]. Optimum situations appear to be those situations where the central wavelength of the ASE bandwidth is placed at a longer wavelength than λQPM, but if it is placed far from λQPM the FOV may suffer from distortion(i.e. the center of the image shows lower intensity than the edges) for short enough bandwidths. From Fig. 7 we can draw that given an ASE spectral allocation, there is an optimum ASE bandwidth at which the FOV can be maximized showing good efficiency and no FOV distortion. As it can be seen, a distinct behavior can be noticed for nonlinear crystals of different length. Therefore, the optimum spectrum allocation and bandwidth depends on the incoming angular spectrum and the length of the crystal as well. The key of FOV optimization relies on properly weighting wavelengths with an upconversion efficiency with a maximum near 0° and wavelengths which upconversion maxima are shifted to other angles [Fig. 5]. For a flat spectral intensity, FOV can only be controlled by ASE spectral allocation and bandwidth selection. In practice, the ASE could be spectrally shaped to take into account the incoming angular spectrum in order to get the desired weighting and thus the targeted FOV.
6. Experimental verification of FOV vs ASE spectrum
To verify we measured and calculated the upconverted intensity for different ASE spectra. We used the experimental setup in  for the upconverter. For the ASE illumination we cascaded a pumped Er-doped fiber (i.e. ASE seed), a free-space tunable Fabry-Perot (FP) filter and an Er-doped fiber amplifier. The seed ASE spectrum was filtered and tuned by rotating the FP filter to three different spectral positions. The set of pictures of Fig. 8(a) represents the recorded images after upconverting the different spectra shown in Fig. 8(b). The IR illumination power was adjusted to upconvert the same intensity at the center of every image. A dotted color line at each image of Fig. 8(a) is added for properly association to curves in Figs. 8(b)-8(d). Figure 8(c) represents the upconverted intensity along the dotted lines of Fig. 8(a), and its corresponding upconverted intensity calculated using the measured spectral intensity is shown in Fig. 8(d). Here, both the measurement and the calculation of the upconverted intensity are plotted as a function of the horizontal pixel number of the CCD camera.
We have investigated the impact of the illumination spectrum on the field-of-view in a nonlinear image upconverter. The focusing lens imposes the full angular spectrum to be upconverted. In cases in which tight focusing of the illumination beam is required, it translates into a broad angular spectrum. The nonlinear crystal poling period and length impose the angular acceptance of the upconverter (which is typically narrow if the illumination beam is comprised of a single-wavelength), clipping the incoming angular spectrum and, thus the upconverted area. The angular acceptance of the nonlinear crystal can be broadened if a multiple-wavelength source is employed as illumination (for a fixed pump wavelength). By properly controlling the bandwidth, allocation and power spectral density of the ASE illumination spectrum, the FOV can be effectively increased. We have derived a model to predict the FOV for a given ASE illumination spectrum. For a fixed upconverter configuration (focusing lens, nonlinear crystal length and poling period) it may allow for optimization of the ASE spectrum illumination (total bandwidth, allocation and relative amplitudes of every spectral component) of the upconverter. The presented results could be applied to predict the FOV of additional broadband illumination sources (i.e. femtosecond or spectrally filtered supercontinuum lasers) or the FOV under multiple-wavelength upconversion pump beam.
The authors would like to acknowledge financial support from the Government of Spain through projects TEC2011-26842 and TEC2014-60084-R.
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