A self-imaging resonator can be simultaneously resonant for many transverse modes and therefore allows cavity build-up for images of various shapes. The stability properties of such a cavity are reviewed. We have used this device for the first time to enhance the efficiency of second harmonic generation of weak images. We characterize the global and local efficiency of the second harmonic generation, and discuss its limitation due to the spatial bandwidth of the cavity and the diffraction along the crystal length.
© 2010 Optical Society of America
Shortly after the first laser was built, second harmonic generation (SHG) was proposed  and demonstrated  using crystalline quartz. Since then, frequency doubling laser beams has found many applications both in research and industry.
As a nonlinear process, Second Harmonic Generation efficiency depends on the fundamental pump intensity, and can be improved, for instance, by using high peak power pulsed sources or by focusing a beam on the nonlinear material. In the single pass configuration, there is in principle no spatial dependence of the process and the transverse shape of the pump beam can be transfered to the second harmonic beam. Still, a perfect reproduction of the input image on the second harmonic field is hampered by diffraction and phase matching condition effects .
In the continuous wave regime, the power inside a resonator can be, depending on the finesse of the cavity, much larger than the input pump beam. Resonant doubling cavities have allowed up to 85% conversion efficiency [4, 5]. However, the cavity acts as a spatial filter on the impinging laser beam, and only the fraction of the beam corresponding to the resonating Hermite-Gauss eigenmodes of the cavity is transmitted, the remaining intensity being reflected back. Such a filtering cavity prevents the transmission of any complex pattern and its conversion into a frequency doubled image.
The issue of the transverse properties of second harmonic generation regained interest recently, in particular thanks to the progress of quantum information and quantum imaging. The need to handle simultaneously and at the quantum level many transverse modes of the light has emerged . Multiplexing quantum communications requires to individually address all of the spatial parameters of the light, and efficient frequency doubling of images is a first step toward this goal. In the single pass configuration, studies have been performed to address the issue of efficient coupling between higher order Gaussian modes . We propose here to demonstrate how transverse degenerate second harmonic generation can be achieved in a multimode cavity configuration, and combine the higher intra-cavity power for many transverse modes simultaneously.
In this paper, we first present the empty self-imaging cavity and discuss its number of resonant transverse modes. Experimental results on the transmission of multimode beams are presented. In a second step, we introduce inside the self-imaging cavity a nonlinear χ(2) crystal and transfer images in the fundamental wavelength into images in the second harmonic field. Experimental results are presented and discussed.
In the following, we define an image in a given plane as a transverse distribution of the electromagnetic field at this plane. Any image can be obtained through the superposition of gaussian modes. We define the near-field plane as the plane for which the original spatial distribution of the image phase and amplitude can be recovered. The far-field plane corresponds to the plane for which the spatial Fourier transform of the original image is obtained.
2. The linear self-imaging cavity
An optical cavity filters both the longitudinal and the transverse shape of a laser beam. A high finesse cavity will for example transmit only a narrow frequency bandwidth. As for transverse modes, it is well known that, once fixed the geometry, cavities define a set of spatial eigenmodes the resonances of which can be sustained. When a monochromatic light beam impinges on a cavity, its transverse pattern is projected onto this set of eigenmodes (e.g. TEMmn Hermite-Gauss modes), and only the projected resonating modes can be transmitted through the cavity, the other modes being reflected back. This spatial filtering feature of the cavity is obviously detrimental to the transmission of generic images, preventing the use of a resonator to enhance the field intensity for image-related non-linear processes.
More specifically, in a non-degenerate cavity, a set of Hermite-Gauss modes TEMmn form a base of the transverse modes and they are resonant at different cavity lengths, due to a Gouy phase term, taking into account the diffraction along the axis of the cavity. For a given frequency ν, the resonance of the mode TEMmn is obtained for a cavity length L satisfying:
where α is the Gouy phase shift of the TEM00 over one cavity length , p an integer, and ν the laser frequency. For a fixed cavity length, is also the minimum frequency difference between two resonant transverse modes divided by the free spectral range. In the following, we will define as this normalized frequency difference. Cavities show a partial degeneracy if α is a fraction of 2π . The confocal cavity, for instance, corresponds to α = π, and allows the resonance of TEMmn modes for either m + n = 0 or 1 mod(2). Complete degeneracy is achieved for α = 0 mod(2π). In this case, all of the TEMmn modes, and thus all linear combinations of them, will be simultaneously resonant.
Arnaud described  such a fully degenerate cavity. From a geometrical point of view, in such a cavity, any optical ray retraces its own path after one round trip. Its ABCD matrix  after one round trip is strictly equal to identity.
The linear cavity, described in Fig. 1, consisting of a plane mirror, a lens of focal length f and a spherical mirror of radius of curvature R, with distances L 1 and L 2 between them, is a self-imaging cavity when L 1 and L 2 verify the following conditions:
For L 1 and L 2 close to this value, the stability condition of the cavity can be calculated using the ABCD matrix of the cavity after one round trip. Expressed as a function of the deviation from degeneracy ε = L 1 - L 1,deg and δ = L 2 - L 2,deg, the stability region corresponds to the non-white region of Fig. 2, where experimental values of f and R have been used. In the same figure, colors represent the normalized frequency difference between transverse modes as a function of the lengths ε and δ. The full transverse degeneracy can be obtained only for ε and δ equal to 0. However, in a real cavity with finite losses, we can take into account the width of the resonance peaks for a given cavity length which is equal to the free spectral range divided by the Finesse ℱ. When , several transverse modes of the same frequency can be very close to resonance. Since we consider transmission of complex images, which require both amplitude and phase conservation of several modes through the cavity, we set an arbitrary threshold for the resonance of a mode if its transmission is over 98 % of the maximal transmission allowed by the cavity, we find approximately simultaneously resonant transverse modes of the same frequency. To compute this, we took into account the natural degeneracy of the TEMmn with equal values of m + n. This number diverges to infinity when the cavity reaches full degeneracy. But this is not physically possible and the main limitation is then the transverse size of the optical elements (numerical aperture), and the precision with which optical elements are aligned.
3. Image transmission through the cavity
The experiment that we present here, and described in Fig. 3, uses a 1064 nm, 200 mW Nd:YAG laser injected into the self-imaging cavity described in the previous section. The entrance mirror is highly reflecting (HR) at 532 nm and its reflectivity R is equal to 90 % at 1064 nm. The output mirror is anti-reflective coated (AR) at 532nmandi R = 1% at 1064 nm. The finesse of the cavity is approximately 60. Once the cavity is aligned, it is locked at resonance using a Pound-Drever-Hall technique with an electro-optic modulator (EOM) inducing a 6 MHz phase modulation. For second harmonic generation, we use a 2mm × 1mm × 10mm PPKTP crystal, AR coated for both 1064 nm and 532 nm wavelengths, inserted near the plane mirror of the cavity.
We use a f = 50mm lens and a R = 38mm mirror. The optimal distance between the plane mirror and the lens is therefore L 1deg = 115.8mm, and the one between the lens and the spherical mirror is L 2deg = 88mm. Setting up this cavity near the degeneracy point requires special care. The methods to align such a cavity can be found in details in 
Once transverse degeneracy is reached, image transmission has to be demonstrated for the fundamental wavelength infrared images. The laser is focused on the plane mirror, with a size of 300μm, corresponding to a Rayleigh length of 20cm. The laser beam is intercepted by a resolution mask of various transverse shape (see Fig. 3), on a conjugate plane of the plane mirror. We call it a near-field plane (NF).
By removing the plane mirror of the cavity or the crystal, we can measure the quality of the transmitted images by the whole optical system (the imaging of the mask on the CCD being preserved as the entrance mirror has no curvature). We test the limitations due to the insertion of the crystal without taking into account the cavity resonance influence. The results, presented in Fig. 4 on the left, show that the insertion of the crystal without a cavity does not change the quality of the image. When the entrance mirror is replaced and the cavity is aligned the image transmission degrades because the cavity is not perfectly at degeneracy and acts as a filter. Its transverse degeneracy is highly sensitive to any small misalignment and variation of length. Therefore, a key to achieve image transmission through the crystal and the cavity has been to optimize the size of the image in the cavity itself. These aspects will be discussed in section 5.
The locking of the cavity is very stable and allows us to go continuously from one image to another by changing the transverse position of the mask, while keeping the cavity locked. From the recorded images, we can evaluate a lower bound of the number of resonating modes inside the cavity for each transverse dimension as the ratio between the total area of the image (0.6 mm), and the area of the smallest detail we can distinguish within it (60 μm). In our case, we get a total of about a hundred modes. As detailed in the previous section, the number of modes resonating inside the cavity is mainly limited by the alignement precision of both length L 1 and L 2. In our case, in order to obtain 100 resonant modes, one must assure α < 5 × 10-4, which leads to L 1 - L 1,deg < 20μm and L 2 - L 2,deg > 20μm. This value is coherent with the maximum precision we could get with the standard micrometric translation stages we used.
4. Frequency doubled images
Conversion of images to another wavelength can be obtained by frequency doubling inside a nonlinear crystal but the single pass configuration has a very low efficiency, typically 10-4 for a 100 mW cw beam intensity. The self-imaging cavity induces degenerate pump resonance inside the cavity and increases the doubling efficiency since this value is proportionnal to the pump intensity.
The longitudinal overlap between the fundamental and the second harmonic fields is studied, and expressed usually as the phase matching condition for plane waves. As for images, the transverse overlap along the crystal length has also to be taken into account. Diffraction along the crystal axis is different for the two wavelengths, and this will degrade the quality of the generated image. We can calculate the expected profile of the second harmonic field using Fresnel diffraction integrals along the crystal length in the case where the phase-matching condition is fulfilled for plane waves, and any walk-off effect neglected. Let ρ be the transverse coordinate, A(ρ) and B(ρ) respectively the transverse profiles of the images at the fundamental and the second harmonic wavelengths λ0 and in the center plane of the cristal. lc is the crystal length, and n its index of refraction at these wavelengths. The second harmonic field is then:
This equation is similar to the transverse coupling in parametric down-conversion studied in , the full development leading to this result is presented in . It reveals that the coherence length l coh is the relevant parameter to decribe the transverse properties of the frequency doubling. In the case of a 10 mm long crystal of index 1.8, l coh = 22 μm. Numerical calculations show that below 5lcoh, details within the image will be significantly distorded in the second harmonic generation process. It is interesting to compare this new result to the optimal waist for frequency doubling given by the Boyd and Kleinman condition , and which is of the order of 20μm for our crystal. Even though this number comes from a very different optimization, we see that imaging capacity are consistent with this condition. This can be interpreted in a more general framework of a complete multimode description of non linear processes, which can be found in .
Using the PPKTP crystal inside the self-imaging cavity, we are able to perform the second harmonic generation of images, from 1064nm to 532nm. The produced green images are separated from the infrared using a dichroic mirror, and recorded with a camera in a conjugate plane of the crystal. The results are shown in Fig. 5. Figure 6 (Media 1) shows a movie of the green image, while moving the mask and therefore changing the initial image before the cavity. The stability of our locking system allows us to keep the cavity locked for various transverse pump shapes, even when its power is very low, as shown in the movie.
In our case, the pump intra-cavity power is about 10 times higher than the input power, and therefore the green frequency doubled beam can be expected 100 times more powerful in comparison to the single-pass configuration, which justify the use of such a cavity for the frequency doubling of images.
5. Local doubling efficiency and spatial bandwidth of the cavity
The doubling efficiency can be defined as the ratio between the output green power and the input infrared power. In order to caracterize our frequency doubling setup, we first measured this doubling efficiency versus the fundamental power, for a given transverse mode. Figure 7 represents both the output green power, and the doubling efficiency as a function of the input infrared power. We observe a quadratic dependance of the green power (and therefore a linear dependance of the doubling efficiency) as it is the case in single-mode second harmonic generation.
We also measured the doubling efficiency versus the size of the transverse mode in the following way: we illuminated horizontal and vertical slits with a gaussian mode of large waist (1.2mm), and controlled the slit widths in the crystal plane. The results for the transmitted infrared power and the doubling efficiency are presented on Fig. 8(a) and Fig. 8(b).
As expected, the transmitted infrared power is increasing linearly until the slit width is larger than the gaussian waist (1.2mm) of the beam in the horizontal case. Things are different in the vertical case due to the smaller size of the crystal in this direction (1mm vs 2mm in the horizontal direction). The vertical slit image suffers more losses inside the cavity when its width is comparable to the crystal size which degrades the transmitted power for slit width larger than 0.7mm.
The doubling efficiency has a more complex dependance on the slit width. First, when the slit is almost closed, the doubling efficiency increases almost linearly until a threshold: 200μm for the vertical slit, and 300μm for the horizontal one. Then, the doubling efficiency is almost constant, even though the infrared power is increasing inside the cavity. This proves that the doubling efficiency depends only on the local power of the fundamental beam. The slowly decreasing slope in this region comes from the gaussian shape of the pump, which has lower local power away from the center. This curve represents the spatial bandwidth of the second harmonic setup.
To understand where the threshold in the doubling efficiency comes from, we performed analytical calculations of the doubling efficiency of top hat functions using equation 5 without any spatial filtering from the cavity, which would correspond to a slit illuminated by a plane wave. The results are shown on Fig. 8(c). In this case the threshold is reached for a slit width of 200μm approximately (ten times the coherence length l coh). The adequacy between experimental curves and theoritical calculations for small slit widths proves that the main limitation in the threshold of the doubling efficiency does not come from any spatial filtering for the cavity, but rather from the diffraction along the crystal length.
In this article, we demonstrated experimentally image transmission and frequency doubling of images in a self-imaging resonator, with a PPKTP crystal. Images transmission show that the cavity is completely degenerate and filters only small spatial details. As expected, the doubling efficiency is linear with respect to the fundamental wavelength power. On the other side, this efficiency is constant with respect to the beam shape, within the spatial bandwidth of the cavity, which only depends on the crystal longitudinal and transverse sizes. This device can find many applications. Because of its great stability, one could use it as a spatial mode discriminator (), or for optical image processing (). We are now investigating the behaviour of a self-imaging optical parametric oscillator (OPO) to create multimode quantum states of light. Indeed, the broad use of OPO’s as a source for quantum stated of light is limited to single-mode states of light. A self-imaging OPO produces fully multimode squeezed light, enabling the multiplexing of quantum information .
We would like to thank Hans Bachor for lending us the PPKTP crystal. Laboratoire Kastler-Brossel, of the Ecole Normale Supérieure and the Université Pierre et Marie Curie - Paris 6, is associated with the Centre National de la Recherche Scientifique. We acknowledge the financial support of the Future and Emerging Technologies (FET) programme within the Seventh Framework Programme for Research of the European Commission, under the FET-Open grant agreement HIDEAS, number FP7-ICT-221906
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