OPTICAL SYSTEM AND METHOD

TECHNICAL FIELD

[ 0001 ] The present invention relates to optical systems and methods and in particular to optical systems and methods for measurement and/or for conversion of the spatial mode of an optical signal. The invention has been developed primarily for use as an optical system and methods for measurement and/or for conversion of the spatial mode of an optical signal via a nonlinear optical process and will be described hereinafter with reference to this application. However, it will be appreciated that the invention is not limited to this particular field of use.

BACKGROUND OF THE INVENTION [ 0002 ] Any discussion of the background art throughout the specification should in no way be considered as an admission that such background art is prior art, nor that such background art is widely known or forms part of the common general knowledge in the field.

[ 0003 ] Optical detection methods to measure the parameters of a particular system of interest are well known and widely used in many applications, and find particular use where the measurement environment is hostile, i.e. detection via optical methods eliminates the need for detection devices that may perish in the measurement environment, or may create further hazard such as placing electrical wiring or voltages in a flammable environment, or where contamination from foreign particles or contaminants, such as solder, maybe a problem. Generally, though, the presently known optical detection methods all utilise direct detection of an optical signal at its original frequency (or a minor variation thereof if the parameter under measurement affects the frequency of the light) and it is this frequency deviation or the magnitude of that frequency component that forms the basis for the measurement. Whilst this may be sufficient in some situations, for a vast array of applications the measurement procedure either requires a measurement of the phase of the optical signal - often requiring the use of complex detection schemes (eg. interferometric, heterodyne, homodyne etc) - or an external optical cavity for analysis of the optical signal, which all are highly susceptible to the alignment of the detection scheme and hence are not practical in situations where vibrations are significant enough to mask the effect being measured.

[ 0004 ] Similarly, optical mode-conversion techniques are commonly found in a number of incarnations, although these also require the use of complex optical systems/devices such as optical waveguides (eg. planar waveguides, optical fibres etc) or optical cavities, all of which require robust alignment systems and are highly susceptible to any misalignment of the system.

Additionally, these techniques have a finite conversion efficiency, and hence there is a large loss of the power in the conversion process.

[ 0005 ] In other applications, the intensity and frequency stabilization of lasers is a very important technique used in research and industry, e.g. ultra-sensitive spectroscopy, general laser science, gravity-wave detection, optical telecommunication and etc. Over the years, a number of techniques have been developed to stabilize the frequency of a laser by locking it to an absolute reference source such as a ultra stable cavity or an atomic transition. Most of these techniques use some form of modulation and subsequent phase sensitive detection. Consequently, there is a continuing pressure for technique to provide for increasingly more accurate measurements of a laser wavelength/frequency.

SUMMARY OF THE INVENTION

[ 0006 ] It is an object of the present invention to substantially overcome or at least ameliorate at least one of the above disadvantages, or to provide a useful alternative.

[ 0007 ] The arrangements of optical systems and methods disclosed herein provide an alternative modality for optical mode control and manipulation which has far reaching implications for many optical techniques ranging across the optical field from simple measurement devices to quantum information and/or communication devices and/or techniques.

[ 0008 ] According to a first aspect there is provided a method for converting an optical mode comprising: a) providing an input beam at a first frequency, the input beam being incident on the nonlinear material and having a high order transverse mode profile; b) providing a nonlinear material; and c) generating in the nonlinear material a frequency converted beam at a second frequency, the frequency converted beam having a transverse mode profile being the spatial convolution of the transverse profile of the input beam with itself.

[ 0009 ] The transverse mode profile of the input beam is of the form TEM _mn where w>0 and n ≥ 0 or m ≥ 0 and n ≥ 0 in media with an isotropic refractive index profile and ηE _mn , εα _mn , TE _m« and TMrø, modes in medium with an anisotropic refractive index profile. The transverse mode profile of the frequency converted beam may be comprised of a combination of even high order transverse modes up to order 2m and In that is TEM _{27n 2«} in media with an isotropic refractive index profile and HE _2m 2«, EH _{27n 2«} , TE _{2m 2n} and TM _{2m 2n} modes in medium with an anisotropic refractive index profile. The transverse mode profile of the frequency converted beam may alternately be comprised of a linear combination of even transverse mode profiles up to order 2m and 2n that is TEM _{2w 2n} in media with an isotropic refractive index profile and HE _{2OT 2«} , EH _{2m 2π} , TE _2m 2« and TM _{2m 2n} modes in medium with an anisotropic refractive index profile.

[ 0010 ] The nonlinear material is a second order nonlinear material and the frequency converted beam is generated in the nonlinear material via second harmonic generation, sum- frequency generation or difference frequency generation.

[ 0011 ] According to a second aspect, there is provided a method of measuring a desired parameter of a nonlinear material, the method comprising; a) providing an optical input beam at a first frequency and having a first high order transverse mode profile; b) directing the input beam onto the nonlinear material; c) generating a frequency converted beam in the nonlinear material at a second frequency, the frequency converted beam having a second transverse mode profile, the frequency converted beam being derived from the input beam; d) spatially detecting the frequency converted beam to determine the second transverse mode profile; and e) relating the detected second transverse mode profile to the desired phase matching parameter.

[ 0012 ] The first transverse mode profile may be of the TEM _wπ where m>0 and n≥O or m≥O and n ≥ 0 in media with an isotropic refractive index profile and HE _n ,,,, BH _mn , TE _wn and TM _mn modes in medium with an anisotropic refractive index profile. [ 0013 ] Step (e) of the method may additionally comprise a separating element for spatially isolating or separating the frequency converted beam from any residual input beam transmitted through the nonlinear material prior to detecting the second transverse mode profile.

[ 0014 ] Step (d) of the method may comprise spatially detecting the frequency converted beam at a plurality of spatially separated locations in an image plane to determine the second transverse mode profile.

[ 0015 ] According to a third aspect, there is provided a method for determining the frequency of an input beam comprising; a) providing a second order nonlinear material; b) directing the input beam onto the nonlinear material; c) generating a frequency converted beam derived from the input beam in the nonlinear material; d) detecting the transverse mode profile of the frequency converted beam; and e) relating the detected transverse mode profile to the phase matching conditions of the nonlinear material thereby to determine the frequency of the input beam.

[ 0016 ] In the case where the input beam has a transverse mode profile of the form TEMoo, step b) may comprise converting the transverse mode profile of the input beam to be of the form TEM _TOH where m>0 and n ≥ 0 or m ≥ 0 and n > 0 in media with an isotropic refractive index profile and HE _WJ! , εα _mn , TE _wn and TM _mn modes in medium with an anisotropic refractive index profile and directing the converted input beam onto the nonlinear material.

- A -

[ 0017 ] The method of the third aspect may be used for controlling the frequency of the output beam of a laser, wherein step (b) comprises directing a portion of the output beam of the laser onto the nonlinear material.

[ 0018 ] In the case where the portion of the output beam has a transverse mode profile of the form TEMoo, step (b) may comprise directing a portion of the output beam to a mode pre- converter to convert the transverse mode profile of the beam to be of the form TEM _wn where m>0 and n ≥ 0 or m ≥ 0 and n ≥ 0 in media with an isotropic refractive index profile and ηE _mn , εH _mn , υE _mn and υM _mn modes in medium with an anisotropic refractive index profile and directing the converted beam onto the nonlinear material. [ 0019 ] According to a fifth aspect, there is provided a method of controlling the frequency of a laser beam generated by a laser cavity and having a first frequency, the method comprising: a) directing a portion of the laser beam to a mode pre-converter for optionally converting the transverse mode profile of the laser beam to a high order transverse mode to obtain a high transverse order beam at the first frequency; b) directing the high transverse order beam onto nonlinear material; c) generating a frequency converted beam related to the high transverse order beam in the nonlinear material, the frequency converted beam having a high order transverse mode profile; d) relating the high order transverse mode profile of the frequency converted beam to the laser beam to determine the first frequency; and e) on the basis of the relationship of the first frequency to a desired laser beam frequency, modifying the laser cavity. [ 0020 ] According to a sixth aspect, there is provided a method for determining the optimum phase matching parameters of a nonlinear material, the method comprising; a) providing an optical input beam at a first frequency and having a first transverse mode profile; b) directing the input beam onto the nonlinear material for which the phase matching parameters are to be determined; c) generating a frequency converted beam in the nonlinear material at a second frequency, the frequency converted beam having a second transverse mode profile related to the phase matching parameters; d) spatially detecting the second transverse mode profile of the frequency converted beam; e) relating the detected second transverse mode profile to the phase matching parameters; f) adjusting the phase-matching parameters of the nonlinear crystal; and g) repeating steps e) to g to determine the optimum phase-matching parameters. [ 0021 ] The first transverse mode profile may be a transverse electro-magnetic mode of the TEM _mn where m>0 and n ≥ 0 or m ≥ 0 and n ≥ 0 in media with an isotropic refractive index profile and HE _wn , EH _mn , TE _røl and TM _WM modes in medium with an anisotropic refractive index profile. The first and second transverse mode profiles may be high order hermite-gaussian or laguerre- gaussian transverse modes where a high order mode is defined as a TEM _mn where m>0 and n ≥ 0

or m≥O and n≥O in media with an isotropic refractive index profile and BE _mn , EH _mn , TE _mπ and TM _m modes in medium with an anisotropic refractive index profile.

[ 0022 ] The phase matching parameters of the nonlinear material to be optimized and/or measured using an arrangement of the optical device disclosed herein may include the temperature of the nonlinear material, the refractive index of the nonlinear material, the mechanical stress or strain in the nonlinear material, the torque applied to the material, the position of the nonlinear material (since the fundamental focus position relative to the crystal centre is also an important parameter, especially when the Rayleigh range is small) or the amount of bending of the nonlinear material, or any other phenomenon which affects the nonlinear conversion of an optical signal in the nonlinear material such as for example an electric or magnetic field applied to the nonlinear material or radiation damage accumulated in the nonlinear material.

[ 0023 ] According to a seventh aspect, there is provided an optical mode converter comprising: an input port for receiving an input beam of a first frequency; a mode pre-converter for optionally converting the transverse mode of an input beam at the input port to a first high transverse order mode beam at the first frequency; a nonlinear material for generating a second high order transverse mode beam at a second frequency the first high transverse order mode beam, the second frequency being derived from the first frequency wherein the spatial transverse mode profile of the second high transverse order mode beam is a combination of transverse spatial modes derived from the spatial transverse mode profile of the first high transverse order mode beam.

[ 0024 ] The second high order transverse mode beam may be generated in the nonlinear material by a nonlinear process selected from the group of second harmonic generation; sum frequency generation or difference frequency generation and may be generated by a single pass of the nonlinear material. The second high order transverse mode beam may be generated in the nonlinear material by single pass second harmonic generation.

[ 0025 ] According to an eighth aspect, there is provided a measurement device comprising: a nonlinear material for receiving a input beam having a first high order transverse mode profile and generating a frequency converted beam derived from the input beam, the frequency converted beam having a second high order transverse mode profile; and a detector for detecting the frequency converted beam at a desired spatial location. The detector may detect the optical intensity (alternatively the optical power or irradiance) of the frequency converted beam at a desired spatial location.

[ 0026 ] The second high order transverse mode profile may be the spatial convolution of the first high order transverse mode profile with itself.

[ 0027 ] The measurement device may further comprise a mode pre-converter for optionally converting the spatial profile of an input beam having a fundamental order mode profile to the first high order transverse mode. The measurement device may further comprise an optical source for generation of the input beam.

[ 0028 ] The nonlinear material may be a ^ ²⁾ nonlinear material and may generate the frequency converted beam via three wave mixing. The nonlinear material may be a ^ or higher-order nonlinear material. The frequency converted beam may be generated by second harmonic generation, sum frequency generation or difference frequency generation and may be generated by a single pass of the nonlinear material.

[ 0029 ] The parameters of the nonlinear material which may affect the phase matching conditions of the frequency conversion may be held substantially fixed, and the measurement device may be adapted to measure the frequency or polarisation of the input beam. The polarisation rotation may also be measured since the conversion efficiency will depend on the polarisation of the input field. Alternatively a magnetic field may be applied to the nonlinear medium which may the birefringent axis and thus the second harmonic conversion efficiency, from which the parameters of either the magnetic field, the nonlinear material, or the input beam may be measured.. [ 0030 ] The frequency and polarisation of the input beam may be held substantially fixed and all but one of the parameters of the nonlinear material which may affect the phase matching conditions of the frequency conversion may be held substantially fixed, and the measurement device may be adapted to measure the parameter of the optical material that is not held substantially fixed. [ 0031 ] The parameter of the nonlinear material that the device may be adapted to measure may be selected from the group of the temperature of the nonlinear material, the mechanical stress in the nonlinear material, the refractive index of the nonlinear material, and the polarisability of the nonlinear material, the torque on the material, the position of the nonlinear material (using a fundamental beam with a small Raleigh range), the bending of the material or other parameter having an affect on the nonlinear conversion of an optical signal in the material such as for example an electric or magnetic field applied to the nonlinear crystal or radiation damage accumulated in the nonlinear crystal.

[ 0032 ] The nonlinear material may be susceptible to either transient or permanent changes in its optical phase matching parameters. The nonlinear material may be removable from the measurement device. The measurement device may further comprise a removable realignable mounting element adapted to mount the nonlinear material and align the nonlinear material with the input beam for measurement of a phase matching parameter of the nonlinear material.

[ 0033 ] An optical parameter of the nonlinear material may be encoded with an identifier, and the optical parameter is measurable by the measurement device to determine the identifier. The identifier may be a unique identifier. The encoded optical parameter of the nonlinear material may be the refractive index of the nonlinear crystal and the refractive index may comprise local variations in accordance with the identifier.

[ 0034 ] The measurement device may further comprise a separating element for spatially isolating or separating the frequency converted beam from any residual input beam transmitted through the nonlinear material. The separating element may selected from the group of a dispersive element, a transmissive dichroic element, or a reflective dichroic element. The transmissive/reflective dichroic element may be configured to transmit/reflect the frequency converted beam and either reflect/transmit or absorb the residual input beam.

[ 0035 ] The detector may be adapted to perform localised spatial intensity measurements of the frequency converted beam. The detector may be selected from the group of: a single spatially relocatable optical intensity detector; a plurality of spatially separated optical intensity detectors; a linear optical intensity detector array; or a two dimensional optical intensity detector array.

[ 0036 ] The detector may comprise an optical waveguide positioned at a desired spatial location for collection of a portion of the frequency converted beam and transporting the collected portion to an optical intensity detector. The detector may comprise a plurality of optical waveguides positioned at desired spatially separated locations in an image plane for collection of a corresponding plurality of portions of the frequency converted beam and transporting the plurality of collected portion to a corresponding plurality of optical intensity detectors. The optical waveguides may be optical fibres.

[ 0037 ] The optical device may be fabricated wholly using optical waveguides which may be optical fibres. For example, one arrangement of the optical device may comprise a fundamental laser source which may be a fibre laser, and may be either a single or a multimode fibre laser; an optional mode converter which may be an optical fibre mode converter and may be coupled to the fibre laser source; a nonlinear material for converting the input beam to a frequency converted beam and the nonlinear optical fibre may be coupled to the optical fibre mode

converter; and a further optical fibre coupled to the nonlinear optical fibre for propagating the frequency converted beam to a suitable detector for detection of the transverse mode profile of the frequency converted beam. Similar arrangements of the optical device may be realised using alternate optical waveguides such as planar waveguides. [ 0038 ] The frequency converted signal may be sampled at one, two , three, four or more locations along the nonlinear material and a plurality of locations may be employed for sampling of the frequency converted beam. Where the nonlinear material is a nonlinear optical waveguide (eg. a planar waveguide or optical fibre), a portion of the frequency converted beam may be coupled out of the nonlinear optical waveguide and propagated either in free-space or in a suitable corresponding waveguide to a suitable corresponding detector.

[ 0039 ] In other arrangements, one, two, three, four or more, or a plurality, of nonlinear optical materials may be arranged (eg. in series and/or parallel, or they may be cascaded), such that a frequency converted beam generated by each respective nonlinear optical material may be sampled such that each nonlinear material is able to either: provide a measurement of a number of different parameters depending on the particular environment and/or constraints and/or operating characteristics; provide a measurement of the change in the parameters of a fundamental input beam at each nonlinear material. The same fundamental input beam may be input to each of the plurality of nonlinear materials, a small portion of which may be frequency converted by each successive nonlinear material. Each frequency converted portion may then be propagated to a respective detector for determination of the transverse mode profile of the frequency converted beams and measurement of a desired parameter of either the nonlinear material or the fundamental input beam, which can then be related to the environmental parameters in which the nonlinear material resides (for example the temperature at a particular or multiple locations in a particular environment In alternate arrangements, each the nonlinear materials may be chosen such that the frequency conversion from each material is cascaded i.e. each successive cascaded nonlinear material uses the frequency converted beam from the previous nonlinear material as the input beam which is then frequency converted, For example, the optical system may comprise first and second nonlinear materials and the laser source may have a wavelength of 1064 ran. The first nonlinear material may frequency convert the 1064 ran radiation to 532 nm radiation via second harmonic generation, and the 532 nm radiation may be incident upon the second nonlinear material which may convert this 532 nm radiation to 266 nm radiation via second harmonic generation. The plurality of nonlinear materials may each be of the same material. Alternatively, the plurality of nonlinear materials may each be of different nonlinear materials having different nonlinear optical characteristics such that the frequency

conversion range of the optical system may be tailored (i.e. made either wider or narrower as desired) according to the nonlinear frequency conversion characteristics of each nonlinear material.

[ 0040 ] The nonlinear material may be selected from the group of crystalline LBO, BBO, BIBO, CBO, GCOB, YCOB, KBO, KTA, RTP, GaAs, KTP, or CLBO or periodically poled materials such as lithium niobate, KTP, KTA, or RTA or other suitable tf® (or ^ or higher) nonlinear material and may be in either bulk or waveguide form.

BRIEF DESCRIPTION OF THE DRAWINGS

[ 0041 ] A preferred embodiment of the present invention will now be described, by way of an example only, with reference to the accompanying drawings wherein:

[ 0042 ] Figure 1 is a schematic depiction of a single pass SHG device according to the present arrangements;

[ 0043 ] Figure 2 shows graphs of second harmonically generated spatial intensity profiles generated in a nonlinear material and measured in the far field in accordance with the thin crystal approximation, i.e. with TEM ₁₀ and TEM _2O collimated transverse input beam modes;

[ 0044 ] Figure 3 shows graphs of the normalized second harmonic generated efficiency as a function of the focusing parameter ξ in optimal phase matching conditions for TEMoo, TEM ₁₀ and TEM20 pump modes;

[ 0045 ] Figure 4 shows graphs of normalized second harmonic generated efficiency of the TEMoo second harmonic mode as a function of the phase matching condition Ak in optimum focusing conditions for TEM ₀₀ , TEM ₁₀ and TEM ₂₀ pump modes;

[ 0046 ] Figure 5 shows graphs of normalized second harmonic generated efficiency of the TEM ₀₀ second harmonic mode as a function of the phase matching condition Ak in optimum focusing conditions for TEMo ₀ , TEM ₁₀ and TEM ₂₀ pump modes; [ 0047 ] Figures 6A and 6B each show two dimensional depictions and corresponding cross- section graphs of SH profiles generated in the crystal far field for three phase matching temperatures with transverse mode profiles of the fundamental input beam being TEM _{1O and} TEM ₂ O respectively;

[ 0048 ] Figures 7A to 7C show graphs of the normalized second harmonic mode amplitude difference dependence, (TEMoo - TEM2 ₀ ) / (TEMoo + TEM ₂₀ ), as a function of temperature and wavelength using a pump beam having a TEM ₁₀ transverse mode profile; and

[ 0049 ] Figure 8 is a schematic depiction of an arrangement of the present device used for the frequency locking of a laser cavity.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS Introduction [ 0050 ] Second harmonic generation (SHG), also called parametric up-conversion, is based on the second order susceptibility ^ of nonlinear crystals, where a wave at frequency ω is converted into a wave at frequency 2ω via three- wave mixing (i.e. two fundamental waves at frequency ω combine to form a third wave of frequency 2ω). hi the case of continuous wave (CW) laser sources, SHG is now routinely achieved with very high conversion efficiencies. More than 80% power conversion is possible using enhancement cavities. Frequency doubling is now widely used to generate wavelengths from the far UV to the far IR. The most common SHG application is the generation of coherent green light at 532 nm by up-converting a Nd:YAG laser at 1064 nm.

[ 0051 ] Three wave mixing in χ^' nonlinear crystals can also be utilised with two input waves of different frequencies co _\ and coz to provide an third beam of frequency a*, via either sum frequency generation (SFG) where coi = ω _\ + co _∑ or difference frequency generation (DFG) where coi - co _\ + a>ι. Indeed, SHG is simply a degenerate form of three- wave mixing where

CO\ = COi.

[ 0052 ] Disclosed herein are optical systems, devices and methods for the measurement of the frequency/wavelength of a laser beam and/or measurement of parameters of a larger system or device derived from measurement of the parameters of a nonlinear optical material. The technique uses coupling of higher order transverse electromagnetic mode (TEM) coupled via a nonlinear crystal. The nonlinear optical material may be either a ^ ²⁾ or a ^p (or higher order) nonlinear material. The technique is demonstrated with a single-pass second harmonic generation pumped with a single higher order TEM beams, The generated nonlinear converted beam (eg. for example converted via second harmonic generation or some alternative nonlinear optical process) in general is a multi-mode TEM beam. The multi-mode second harmonic profile can be altered continuously by adjusting the non-linear phase matching laser wavelength, the focusing of the pump or the nonlinear medium's temperature. Thus, monitoring the evolution of the spatial mode profile of the second harmonic beam can for example enable determination of the frequency drift of the pump beam very accurately.

[ 0053 ] The requirements for ensuring efficient power conversion are high pump intensity, optimal pump focusing (SHG is relatively efficient over the order of the Rayleigh range of the

fundamental - if the focus is so tight that the Rayleigh range is much shorter than the length of the crystal then conversion efficiency will be low), high second order nonlinearity and low losses in the crystal at both wavelengths involved in the frequency doubling. Moreover, the fundamental pump and second harmonic (SH) generated fields must preserve their phase relation over the length of the nonlinear crystal. The last requirement is also known as the nonlinear phase-matching condition and is achieved experimentally by taking advantage of the birefringence of the nonlinear crystal, or to the periodic poling of the non linear material.

[ 0054 ] A full analysis of the SHG efficiency, assuming a Gaussian spatial profile, has been previously described in the work by Boyd and Kleinman [G. D. Boyd, D. A. Kleinman, J. Appl. Phys., 39, 3597 (1968)]. To date, however, a general extension of this analysis to higher order transverse mode has not been proposed. Applicants describe herein the nature of the transverse profile of the SH modes on the basis of generalized Boyd-Kleinman factors and implications of SHG with higher order transverse pump modes and the related efficiency for a vast array of applications. The predictions of the generalized model are confirmed by experimental SH modes generated by a single pass parametric up-conversion experiment, operated with a fundamental TEM _nO pump mode, where TEM _π0 refers to the higher order Hermite-Gauss (H-G) modes of order n. It will be appreciated that the following generalized model would be further applicable to other transverse mode types such as Laguerre-Gaussian (L-G) or Bessel modes with appropriate transformations. [ 0055 ] The discussion of the generalized model is first developed herein from the thin crystal approximation, which is valid experimentally as long as the beam is not too strongly focused in the crystal, i.e. as long as Z _R « 1. Next, a full description of the SH generated field beyond the thin crystal approximation is provided, i.e. taking into account the length of the type I phase- matched nonlinear crystal, hi type I phase-matching, the fundamental pump and SH fields are linearly and orthogonally polarized with respect to each other. This derivation is a generalization of the Boyd-Kleinman calculation in order to account for the SHG with a high order pump mode (i.e. high order transverse mode profile where the transverse mode profile of the beam is a H-G or L-G mode of the form TEM _n o or TEM _0n or TEM _mn where m>0 and ri>0 or equivalent Bessel mode). The theoretical predictions are compared with experimental results in the case of optimal focusing Z _R « 1.

Thin Crystal Approximation

[ 0056 ] The nonlinear interaction between fundamental transverse modes and the resulting SH modes is described with respect to the thin crystal approximation and the transverse profile of the

SH modes calculated for the scenario where the nonlinear crystal is pumped with a collimated TEM _n O beam, where n = 0, 1, 2.

[ 0057 ] The schematic depiction of the optical system as shown in Figure 1 represents the arrangement used in the present investigations. In the present example a fundamental laser beam 10 with a TEM ₀ O transverse mode profile 11 is taken from a laser source 12 (in this case a diode- pumped monolithic Nd: YAG ring laser at a wavelength of 1064 nm). A TEM _n0 pump mode is generated by misaligning a mode converting device 14 designed to prevent any transverse mode degeneracy and locked to the resonance of the TEM _n0 mode. The mode converter 14 is depicted in the present case a ring cavity, however, the mode converting device 14 may take any number of different forms as would be appreciated by the skilled addressee, or in alternate arrangements may not be required at all where the laser source 12 produces an output beam with a suitable higher-order transverse mode profile. This mode converting device thus delivers a pure transverse TEM _n0 output mode 13, which is then focused by a lens 15 into a nonlinear material 18. hi the present example, a type I lithium niobate (MgOiLiNbO ₃ ) nonlinear crystal is used although the nonlinear material may be any suitable ^ ²⁾ material such as for example crystalline LBO, BBO, BIBO, CBO, GCOB ₅ YCOB, KBO, KTA, RTP, KTP, CLBO or periodically poled materials such as lithium niobate, KTP, KTA, RTA or other suitable materials and may be in either bulk or waveguide form. In other arrangements, GaAs or ZnSe (in either bulk or semiconductor form with either phase-matched or quasi phase-matched form as appropriate) may be used as the nonlinear material. Periodically poled materials may generate SHG outputs through quasi-phase matching. In still further arrangements, the nonlinear material may comprise a suitable waveguide material, such as a nonlinear optical fibre. Still further arrangements of the optical device may be fabricated wholly using optical fibres. For example, the fundamental laser source may be a fibre laser, and may be either a single or a multimode fibre laser; the optional mode converter may be an optical fibre mode converter and may be coupled to the fibre laser source; the nonlinear material may be a nonlinear optical fibre for converting the input beam to a frequency converted beam and the nonlinear optical fibre may be coupled to the optical fibre mode converter; and the nonlinear optical fibre may be coupled to a further optical fibre for transport of the frequency converted beam to a suitable detector for detection of the transverse mode profile of the frequency converted beam.

[ 0058 ] The residual transmitted fundamental beam 20 (i.e. not converted to the SH by the nonlinear material) is filtered out from the generated SH beam 22 with a separating element 24 (which may be either a dispersive element such as a prism or grating to spatially separate the residual fundamental and the SH beams or it may alternatively be a transmissive/reflective

dichroic element is configured to transmit/reflect the frequency converted beam and either reflect/transmit or absorb the residual input beam). In still further arrangements, the separating element 24 may be a polarisation dependent device (eg. a polarising beam splitter, , Brewster plate, or a birefringent polarising beam splitter such as a Wollaston, Nicol, Rochon, Senarmont, Nomarski, Glan-Thompson, Glan-Fucault or Glan Taylor prism, or other similar device) which, since the transverse modal component of the frequency converted SH beam are orthogonal, is able to separate the individual modes in the SH beam for independent detection and analysis.

[ 0059 ] The transverse mode profile of the SH beam 22 is observed with a beam analyser / detector 26 able to provide spatial information about the transverse mode profile of the SH beam 24 in the image plane with respect to the nonlinear crystal 18, which may be in the far field of the centre of the crystal. The detector may for example be a single optical detector able to be moved across the beam profile, a plurality of single optical detectors places at strategic relevant locations in the expected transverse profile (eg. two photo-detectors that only look at the central lobe and one of the side lobes of the SH beam), a linear detector array, or a CCD array or camera from which the relevant information about the spatial profile of the beam can be mathematically or otherwise extracted.

[ 0060 ] In other arrangements, the detector may comprise a mode selective element to separate (spatially or otherwise) the individual mode components of the SH beam and subsequently detect each mode component individually. A suitable mode selective element may comprise: a Fabry- Perot cavity (in either a linear, folded or ring configuration); an optical waveguide for example planar waveguide(s) or optical fibre(s) to separate the modes; or a polarisation dependent device for example a polarising beam splitter, Brewster plate, or a birefringent polarising beam splitter such as a Wollaston, Nicol, Rochon, Senarmont, Nomarski, Glan-Thompson, Glan-Fucault or Glan Taylor prism, or other similar device. [ 0061 ] The beam focusing of the fundamental high-order pump beam into the nonlinear material, defined by the ratio 1/(2Z _R ), can be tuned experimentally by changing the focusing lens before the crystal, where z _R = πω ₀ ² jλ is the Rayleigh range of the pump beam and / is the crystal length.

Transverse profile of the generated SHG modes [ 0062 ] The SHG process combines two fundamental photons from the pump beam to generate one SH photon with twice the energy. The transverse profile of the SH mode therefore corresponds locally, i.e. in one specific crystal plane, to the decomposition of the square of the fundamental pump mode profile into the SH basis, hi effect, the transverse profile of the SH

beam is determined, for the case of second harmonic generation, by the spatial convolution of the fundamental spatial profile with itself. It will be appreciated that if there are two input beams, the nonlinear material may convert both beams via sum frequency generation (SFG) and the transverse mode profile of the SFG beam would correspond to the spatial convolution of the two input beams.

[ 0063 ] Restricting the model to a description along one dimension of the transverse plane, x, the fundamental TEM _0n pump mode basis is denoted as {«„} whose first mode has a waist of W ₀ , and the second harmonic pump mode basis as {v _n } whose first mode has a waist of ω _o /42 . In the thin crystal approximation, the normalized profile of the generated SH field for a TEM _n0 pump is therefore written as

^S _n ⁽ X ⁾ = ∑γλ-W ⁽ i ⁾

(=1 where v ₂ i denotes the even SH modes and F _m - describes the spatial overlap between the squared pump and the SHG modes in the transverse plane. Its expression is given by

where U _n denotes the fundamental modes and O _n corresponds to the normalization of the squared pump and is defined by the relation f° u _n ^A (x)dx . The generated SH components are all even since the TEM _nO pump squared profile is necessarily even. Moreover, 2« is their highest order as the squared TEM _n0 profile does not project onto higher order modes. (Similar expressions may be determined where the nonlinear material is a ^ or higher order material displaying higher-order nonlinear effects. For example, frequency tripling or odd ordered nonlinear processes would retain elements of symmetry of the first, fundamental or input field. Even ordered nonlinear processes would have similar characteristics to second harmonic generation with all odd modes missing.)

[ 0064 ] The common case of using a fundamental pump beam with a TEMQ ₀ mode profile yields Too ⁼ 1 and corresponds to a perfect spatial overlap as the profile of the generated SH mode is also a TEMoo mode (since the square of a Gaussian mode profile is also Gaussian). For non TEMoo pump modes, the overlap coefficients calculated from Equation 2 are given by

T ₁₀ = 0.58 and T ₁₂ = 0.82 for a TEM ₁₀ pump mode, and T ₂₀ = 0.47, F ₂₂ = 0.44 and T ₂₄ = 0.77 for a TEM _2O pump mode. The presence of several non-zero coefficients implies that for all cases, except a TEM ₀ Q pump mode, the generated profiles do not correspond to the pump intensity

distribution. The profiles generated in this case are shown in Figure 2 (the cross-section traces in this figure contain both the data and theory fits). In addition, ID and 2D experimental results are also shown that were obtained with collimated pump beams. Good agreement is observed between the experimental cross-sectional profile and the theoretical profile for a high order beam, particularly with respect to the temperature dependence and peak locations and heights of the features in the profiles, even though the local pump power is quite low in these operating conditions. Indeed, the pump beam waist is large (1200 μm) and the available TEM _1O and TEM _2O pump power was limited in the present investigations to 80 mW and 55 mW, respectively. The observed profiles are similar to the intensity Of TEM ₁ O and TEM2 ₀ beams, but are actually sharper, and cannot be fitted properly with a single SH mode profile. Note that both profiles are not dependent on the crystal temperature. Only the global efficiency is modified when the temperature is tuned, as the phase matching condition is not fulfilled.

[ 0065 ] In the thin crystal approximation, the SHG efficiency η _n for different TEM^o pump profiles can be directly compared by comparing the integrated power of the second order induced polarization in the transverse plane, for a given pump power. As the induced polarization is proportional to the square of the fundamental field, the comparison of a TEM _n o pump mode relative to a TEMoomode yields:

[y _n0 ( _X )(x)dx

J r- 00 M ₀ ⁴ ₀ (X) (x)dx

where u _n o(x) refers to the profile of the TEM _n O pump mode. For a given TEM ₁₀ and TEM20 pump power, we get η _\ = 3/4 and η% = 41/64.

[ 0066 ] A comparison between theoretical calculated efficiency in the thin crystal approximation and experimental results of the SHG efficiency, relative to the case of a TEMoo pump, is presented in Table I. The pump is in each case collimated to a beam diameter of approximately 1200 μm. These results are normalized to the TEMoo pump mode case, and are obtained with a collimated pump beam with identical optical power.

TABLE I

Collimated Pump TEMi ₀ TEM ₂₀

Experiment 0.73 ± 0.05 0.56 ± 0.05 Theory 0.75 0.64

_{λ £}

— 16 —

[ 0067 ] The theoretical model in the thin crystal approximation and the SHG experiment with a collimated pump beam show good agreement to within a few percent with respect to the temperature dependence and peak locations and heights of the profile features, taking into account the uncertainties in our experiments, which are mainly due to the relatively low power in the generated SH beams and temperature fluctuations in the crystal. The efficiency drops with the order of the pump mode. This is due to the lower local intensity in higher order modes. In the regime of the thin crystal approximation, the absolute conversion efficiency is very small, as the local intensity in the crystal is low for large collimated beams. Efficient SHG is achieved with a pump tightly focused into the crystal as will be seen below. Beyond The Thin Crystal Approximation

[ 0068 ] Consider now the case of a focused pump into the nonlinear crystal 18. Because of the different evolution of each H-G component of the generated SH field, a local field description and a propagation of the SH modes along the crystal is necessary.

[ 0069 ] Different H-G modes accumulate different Gouy phase shifts during propagation. Beyond the thin crystal approximation, the effect of these phase shifts is no longer negligible. Consequently, all of the SH components are not simultaneously phase matched with the pump beam 10 along the length of the nonlinear crystal 18, which is a lithium niobate crystal in the present arrangements. Since the birefringence of lithium niobate is highly temperature dependant, phase matching for each mode occurs at different crystal temperatures. Therefore a modification of the SH profile with the crystal temperature is expected. It is interesting to note that this effect, although due to mode interference in the crystal, does not need any cavity or interferometer to be observed.

[ 0070 ] To detail these aspects quantitatively, a calculation of the focalization of higher order transverse modes in a type I nonlinear crystal is derived from Boyd and Kleinman's calculation [G. D. Boyd, D. A. Kleinman, J. Appl. Phys., 39, 3597 (1968)]. In the present model calculations, three approximations have been made: firstly, losses in the crystal are considered negligible for both the fundamental and second harmonic. This approximation is just to simplify the calculations as similar losses for all transverse modes are expected, thus allowing the focus to be more specifically directed to the differences introduced by the use of higher order H-G pump modes. Secondly, the beam propagation axis is considered to correspond to the optical axis of the type I nonlinear crystal and therefore omit any walk-off effect of the beams. Finally, the waist of the input beam is considered to be centred in the nonlinear medium as this optimum case is generally adopted in the experiments.

Conversion efficiency

[ 0071 ] Because of the different evolution of each H-G mode component of the generated SH field, a local field description of the propagation of the SH modes along the crystal is necessary Starting from a TEM ₀₀ , TEM ₁₀ or TEM ₂₀ pump mode incident on the nonlinear crystal and propagating along the z axis, it can be shown that that the respective SHG intensity I ₀ , Z ₁ and J ₂ along one direction of the transverse plane denoted x can be written as

I ₀ (s,ξ,Ak) = 'c _I e- ^4s2 lH _0fi (ξ,Akf (4)

I ₀ (s,ξ,Ak) = κ _I ^-[H _uo (ξ,Ak) + (8s ² -l)H _{l 2} (ξ,Ak)J

4 ^' (5)

I ₀ (s,ξ, Ak) = where /f _/ is a constant independent of the transverse position parameter s = x Z _R Ia^ ₇ . , of the focusing parameter ξ— 112 ZR and of the phase mismatch Ak.

[ 0072 ] The generalized Boyd-Kleinman factor H, _h2P (ξ, Ak) integrates the nonlinear effects on the crystal length and takes into account the Guoy phase shift is defined as:

H _n , (ξ,Ak) =— f ^{( }} ₊ , dτ (7)

where r is defined by τ - (z — J)I Z _R , Z ^~ f corresponds to the position of the beam waist. The indexes n and 2p respectively refer to the TEM^o pump mode and to the even TEM ₂ pQ SH mode generated.

[ 0073 ] Each term in Equation 4, 5 and 6 accounts for the contribution of an even SH mode to the total SHG. Hence, the expression of the SHG intensity is very similar to the one given in Equation 1, with additional correcting terms taking into account propagation effects in the crystal. The same modes as previously described in the thin crystal approximation appear naturally here. Because of the Gouy phase shift, their propagation strongly depends on the beam focusing towards ξ, and on the crystal temperature towards Ak. These effects between the fundamental TEM _n0 and the SH TEM^o modes are all described by ^~ α _n ,2 _P (ξ, Ak). This provides a simple optical tool, for example for a wavelength meter and/or temperature sensor (or other sensor as elsewhere described) that can easily be integrated into more complicated systems. As detailed previously, only a few even modes contribute to the SH beam, and the SH beam profile always differs from the pump profile, except in the case of a TEMQ _O pump mode.

[ 0074 ] The corresponding SHG powers P ₀ , P ₁ and P ₂ can be obtained by integrating the previous intensity distribution given in Equations 4, 5 and 6 and are written:

P ₂ (ξ,Ak) _{j 2} (£δ£)f (10) where Kp is a constant independent of the focusing ξ and phase matching Ak.

[ 0075 ] To find the best SHG achievable experimentally, i.e. the optimum regime, the SHG power (i.e. P ₀ , Pi and P ₂ ) is presented as a function of the beam focusing in Figure 3, using Equations 8, 9 and 10. These curves are all normalized to the maximum achievable SHG power in the case of a TEMQ _O pump. Moreover, this representation corresponds to optimal phase matching conditions, i.e. for the value of Ak = 0 that maximizes the SHG power.

[ 0076 ] The maxima in the SHG powers P ₀ , Pi and P ₂ are respectively obtained for ξ _opt o ⁼ 2.S4, ξ _opt\ = 2.7 and ξ _oP a = 2.5. Note that ξ _opt o = 2.84 exactly corresponds to the value obtained by Boyd and Kleinman [G. D. Boyd, D. A. Kleinman, J. Appl. Phys., 39, p.3597 (1968)]. The maximum SHG efficiency achievable with a TEM ₁ O and a TEM ₂₀ pump modes, relative to the TEMoo case, for a given pump power are respectively 50% and 40%. These drops in conversion efficiency relative to the TEM ₀₀ case are easily explained by the smaller local intensity in higher order modes. The intensity being more spread out along the transverse axis, the nonlinear SH conversion is therefore reduced. Figure 3 shows a series of efficiency curves derived from equations (8), (9) and (10) to determine the optimum normalized SH conversion efficiency as a function of the focusing parameter ξ = / / (2 Z _R ), for optimal phase matching conditions, Ak = 0. The crosses and the solid lines indicate comparison of the approximate analytical and numerical integration models, respectively, for fundamental pump modes TEMo ₀ (curve 40 of Figure 3), TEM ₁₀ (curve 42 of Figure 3) and TEM ₂₀ (curve 44 of Figure 3). The maximum efficiency achievable with a TEM ₁ O and a TEM _2O pump modes, relative to the TEMoo case as determined from Figure 3 is 0.50 and 0.40 respectively.

[ 0077 ] Figures 4(a) to 4(c) show plots of the SH mode components (as a percentage of the total efficiency) for each of the curves of Figure 3: Figure 4(a) corresponding to curve 40 of

Figure 3 and being composed of only the TEM ₀₀ mode; Figure 4(b) corresponding to curve 42 of Figure 3 and being composed of a combination of the TEM ₀₀ and TEM ₀₂ modes; and Figure

4(c) corresponding to curve 44 of Figure 3 and being composed of a combination of the TEM ₀₀ , TEM ₀₂ and TEMo ₄ modes.

[ 0078 ] All the efficiency curves in Figure 3 are rather flat around the optimal value, allowing an operating range between ξ= 2 and £= 4 in which close to maximum conversion efficiency can be obtained experimentally. This corresponds to z _R values of Il A and //8. The case ξ « 1 corresponds to Z _R » I, i.e. to the thin crystal approximation. This is obviously not the optimum regime as it corresponds to a large waist ωo and therefore to a smaller local intensity. In the opposite regime corresponding to a very tight focalization, i.e. to ξ » 1, the efficiency drops because of the importance of the phase shift between fundamental and SH beams at both ends of the crystal. When this phase shift exceeds rdl, the parametric interaction is indeed partly leading to a down-conversion of the SH power. Because of the Gouy phase shift, their propagation strongly depends on the beam focusing towards ξ, this is shown in Figure 4(b), where SH mode components are presented as function of the focussing parameter ξ.

[ 0079 ] Experimentally the focusing of the pump beam into the nonlinear crystal, defined by IIIZ _R , can be tuned by changing the focusing lens 15 (either the focal length or the position of the lens) before the nonlinear material 18. The Applicants have observed experimental results for the SHG efficiency for the TEM ₁₀ and TEM ₂₀ pump mode to be 0.45 ± 0.05 and 0.34 ± 0.05, respectively. The experimental parameters used are a focusing of ZR = 1/6 and COQ = 35μm. This drop in conversion efficiency relative to the TEM ₀₀ cases is easily explained by the smaller local intensity in the TEM ₁₀ and TEM ₂₀ case - since the intensity is more spread out along the transverse axis, the nonlinear SH conversion is therefore reduced.

[ 0080 ] In the previous discussion, the influence of beam focusing has been investigated assuming perfect phase matching Ak — 0. Now consider the scenario where the phase mismatch Ak ≠ 0. Using Equation 8, 9 and 10, the normalized SHG efficiency as a function of the crystal temperature, in optimal focusing conditions, still with TEM ₀₀ , TEM ₁₀ and TEM ₂₀ pump modes, has been examined and is shown in Figures 5(a), 5(b) and 5(c) respectively.

[ 0081 ] To study the influence of a phase mismatch, the temperature of the nonlinear crystal is varied about the optimum phase-matching condition. Where the fundamental pump beam has a TEM _0O transverse spatial profile, the variation in the temperature of the nonlinear material affect the efficiency of the nonlinear conversion to the SH beam 50 according to a common sine function as shown in Figure 5(a) (the distribution is not centred in this case since the present arrangement is no longer operating in the thin crystal approximation, but rather with a focused beam).

- -

[ 0082 ] For higher order H-G modes, the different contributions of the TElVU SH components are able to be represented as a function of the crystal temperature. Different propagations along the crystal lead to different phase matching conditions for each H-G mode. The overall SHG efficiency is therefore asymmetrical because of the different behaviour of each SHG component. Figures 5(b) and 5(c) show the mode components of the second harmonic signal as a function of temperature resulting respectively from an fundamental input beam having a TEM ₁₀ and TEM ₂₀ transverse mode profile. Figure 5(b) shows the temperature efficiency curve 52 having TEM _0O and TEM ₂₀ (54 and 56 respectively) modal components. Similarly, Figure 5(b) shows the temperature efficiency curve 58 having TEM ₀₀ , TEM ₂₀ and TEM ₄₀ (60, 62, and 64 respectively) modal components.

[ 0083 ] The experimentally obtained curves in Figures 5(a) to 5(c) are in good agreement with the theoretical expectation from the above formulations. Differences mainly appear to be the smaller SH conversion efficiency in the experimental case, due to the difficulty to reach the optimal focusing regime experimentally. [ 0084 ] Note that the optimal overall phase-matching condition corresponds in each case to the generation of the SH mode of highest order in the decomposition of the SHG beam. The experimental results of the SHG efficiency in two cases are presented in Table II which shows the comparison of the results with the thin crystal approximation theory. Experimental SHG efficiency for a focused pump (Z _R — 1/6 and coo = 35 μm). These results are normalized related to the TEM ₀₀ case. Very good agreement with between the experiments and our theoretical results is seen.

TABLE II

Optimally Focused Pump TEMio TEM ₂₀

Experimental 0.45 ± 0.05 0.34 ± 0.05 Theory 0.5 0.4

Transverse profile of the generated SHG modes [ 0085 ] The following discussion presents the influence of the beam focusing and on the phase matching crystal temperature parameters on the spatial SH beam profile, for a higher order pump mode.

[ 0086 ] As seen in Figure 5, changing the crystal temperature, hence changing Ak, induces a modification in the spatial profile of the generated SH field. In the case of a TEM ₁₀ pump the SH field can be converted from a TEM _0O mode to a predominantly TEM ₂₀ profile. Changing the

crystal temperature allows reproducible control over the coefficients of the linear combination between the TEM ₀₀ and TEM ₂₀ modes. Similarly, for the TEM ₂₀ pump mode it is found that the SH field is a temperature dependant linear combination of the TEM ₀₀ , TEM ₂₀ and TEM ₄₀ modes.

[ 0087 ] The spatial profile of the SH beam at different crystal temperatures for the TEM ₁₀ pump case is shown in Figure 6A, and for the TEM ₂₀ pump case as shown in Figure 6B, and a cross-section of the transverse profile in each case has been compared with the theoretical formulations above, in particular Equations 4, 5 and 6. For all these plots optimal focusing conditions are assumed, and thus only modify the phase matching condition through the crystal temperature. [ 0088 ] To test the theoretical predictions, the arrangement of Figure 1 is utilized with a tightly focused fundamental pump beam. The generated SH profiles at 532nm, normalized to their maximum, are presented for different crystal temperatures in Figure 6A and 6B for a fundamental pump mode respectively having a TEM ₁₀ and TEM ₂₀ , transverse profile (the cross- section traces in these figures contain both the data and theory fits). From these figures it can be seen that changing the crystal temperature, hence changing Ak, induces a modification in the spatial profile of the generated SH field. In the case of a TEM ₁₀ pump the SH field can be converted from a TEMo ₀ mode to a predominantly TEM ₂₀ profile. Changing the crystal temperature allows reproducible control over the coefficients of the linear combination between the TEM ₀₀ and TEM ₂₀ modes. Similarly, for the TEM ₂₀ pump mode we find that the SH field is a temperature dependant linear combination of the TEM ₀₀ , TEM ₂₀ and TEM ₄₀ modes. For all these plots optimal focusing conditions are assumed, and are thus the only modification of the phase matching condition is through the crystal temperature. It is interesting to note that this effect, although due to mode interference in the crystal, does not need any cavity or interferometer to be observed and can be realised with only a single pass through the nonlinear material. Of course, it will be appreciated that multiple pass devices (for example where the fundamental beam undergoes a zigzag path through the nonlinear material) may also be utilised as desired.

[ 0089 ] The power of the TEM ₁₀ and TEM ₂₀ pump beams in the present experiments are 8OmW and 55mW, respectively, with a beam focusing equal to ξ _∞ p = 0.7. In order to account for the generated beam profiles with the theoretical model, the expected generated profiles with this focusing parameter have been plotted in Figure 6B and they are in very good agreement with the experiment. The main remarkable feature of these profiles being that they all go down to zero in a few points on the transverse plane in the experimental plots a feature that is very well reproduced in the theoretical plots. In the case of a TEM ₁₀ pump mode, the theoretical profiles

for ξ = 0.7 or ξ = 2.7 show identical evolution with the crystal temperature. However, in the case of a TEM ₂₀ pump mode, they show a different evolution for ξ- 0.7 and ξ= 2.5. For even higher order modes, there is an increasing mode profile sensitivity with the crystal temperature.

Approximate Analytical Model [ 0090 ] As an alternative to the above generalized formulation, an approximate numerical model has also been developed where the SHG, with diffraction and depletion, can be modelled using fast transform techniques. Beginning with the vector wave equations the equations for SHG in the paraxial approximation for a type-I nonlinear media are:

V]E ₁ - Uk _{1 (1 1}} where the subscript i = 1 indicates the fundamental and i = 2 the SH and d the nonlinearity. Going to the Fourier domain we define

[ 0091 ] Using the paraxial approximation G ₁ (s _x ,s _y ,z) , defined by

[ 0092 ] We next normalize the field to the peak of the fundamental so that

[ 0093 ] Using the normalized field we can express the spatial distribution in terms of the function G ₁ (s _x ,s _y ,z) as

f _i (x,y,z) = l[ _∞ G _i (s _x , _Sy ,z).ex _V j _π ² ^-( _s ] + _Sy ) _z .eχp[-2?g{ _Sχ x+s _y y))}is _x d _S y (15)

IZ r

[ 0094 ] where z _Q = ik _i /[ωfdμ ₀ E ₀ )andz _R = πω _Q /λ _ι . It can be shown that the coupled wave equations then take the form

(16)

Implications

- -

[ 0095 ] Presented above are results for SHG conversion of input TEM ₀₀ , TEM ₁₀ , and TEM ₂₀ modes. Nevertheless, it will be appreciated that these results may be readily extended to any transverse spatial mode profile.

[ 0096 ] Taking into account the full propagation of the modes in the crystal, a very good agreement is seen between the theoretical prediction and the experiment, in conversion efficiencies as well as in mode profiles. This implies that transverse mode coupling can be fully understood by a simple mode overlaps in the thin crystal approximation, and that a full quantitative description can be obtained with the thick crystal calculation and is termed second order nonlinear optical effect for mode selective phase-matching. [ 0097 ] The consequence of the above results is profound. It shows that mode coupling between different transverse modes is possible in a χ- ' second order nonlinear interaction, and that the ratio between the generated SH components can be adjusted continuously via changes in at least the crystal temperature, with similar results expected for all phenomena that has an effect on the refractive index of the nonlinear material. [ 0098 ] Put another way, in the approach of the arrangements of the optical systems and methods disclosed herein, higher order spatial modes are used to pump a nonlinear process in a second order nonlinear material — in standard nonlinear processes fundamental spatial mode (i.e. TEM ₀₀ modes ) are used. The use of a high order mode of the form TEM _wn where m>0 and n ≥ 0 or m ≥ 0 and n > 0 in media with an isotropic refractive index profile and HE _mn , EH _mιι , TE _n ,,, and TM _mn modes in medium with an anisotropic refractive index profile creates an spatial intensity profile in the frequency converted optical signal resulting from the nonlinear process (eg SHG) that is the square of the input spatial intensity profile and thus forms a multi-mode spatial intensity profile. For example if an input beam having a TEM ₁₀ spatial intensity profile is used as the pump then the spatial intensity profile of the SHG optical beam resulting from the nonlinear conversion with contain components of the TEMoo and TEM ₂₀ spatial transverse mode profiles. That is, the localized spatial intensity in the TEM ₀₀ to TEM ₂₀ (in the case of a TEM ₁₀ input beam) is dependent on the phase matching of the nonlinear process, hi particular, if the system is operating below the optimum phase matching condition then the local spatial intensity of the TEM ₀₀ will dominate over the local spatial intensity of the TEM ₂₀ . If on the other hand the system is operating above the optimum phase matching condition then the local spatial intensity of the TEM ₂₀ will dominate over that of the TEMo ₀ . Thus, if the ratio of the two components in measured, it can be determined whether the particular parameter of interest is above or below that of the optimal phase matching condition.

Examples

- -

[ 0099 ] It will be appreciated that the arrangements disclosed above using the phase matching of a non-linear optical process to provide unambiguous information about the state of a system may be utilized for a vast array of different applications. That is, the arrangements can be adapted to many different classes of application for example by maintaining all but one of the system parameters constant then we can gauge the value of the remaining parameter from the ratio of the spatial components in spatial intensity profile of the frequency converted output beam. For example, if all parameters, except the non-linear medium's temperature (or some other material parameter), are held constant then the ratio of the spatial intensity of the modes in the SHG field will tell us the medium's temperature. [ 0100 ] The absence of other sensors, and the absence of an optical cavity (i.e. only a single pass through the nonlinear material is necessary) opens the way to many other applications, such as a very accurate remote sensing temperature or strain sensor (or some other parameter of the nonlinear material which has an appreciable affect on either the refractive index directly or the material's phase matching properties generally). In this type of applications, the subject of the measurement would be a physical property of the nonlinear material itself wherein wavelength/frequency of the input beam would be maintained constant and the spatial profile of the SHG field monitored to give a measure of the property of the nonlinear material under test (i.e. the material's temperature).

[ 0101 ] As a specific example, mode selective phase-matching could be useful in the Laser Interferometer Space Antenna (LIS A) space mission, where temperature stability of the lasers in each module must be maintained at less than lmK@lmHz. Utilising the optical device of the present arrangements this temperature stability could be achieved without wiring the elements to other sensors, thereby greatly reducing the complexity of the control systems. A further advantage of using an arrangement according to the presently described systems is that it is extremely robust and resistant to vibrations, to which the LISA modules would be subjected to during launch. Other significant applications for the optical systems and/or methods of the present arrangements are in hostile environments for example where running a voltage is undesirable, for example near flammable materials. Indeed, at constant temperature, monitoring the spatial profile would provide a direct frequency measurement. [ 0102 ] In other applications of the present arrangements, where the interest was in the variation of the properties of the input laser beam itself, such as laser wavelength, then the present arrangements may be utilised to construct a laser wavelength meter. This in general would be achieved by maintaining the properties of the nonlinear material constant such that

variations in the SH output would be relatable to changes in the input beam itself. That is, the system could be configured as a frequency to intensity converter.

[ 0103 ] In a further application of the present arrangements, if a nonlinear material having a strong wavelength dependence for its phase matching conditions then monitoring of the spatial profile of the SH output would provide a direct frequency measurement at constant temperature (or other nonlinear optical parameter of the material); this could be used to either measure or to lock the laser wavelength without needing a cavity. This type of application is of interest whenever wavelength monitoring is needed such as for laser frequency stabilization or for laser wavelength locking to the ITU grid in the case of optical communication systems as measurement of the wavelength/frequency of a laser or determination of the temperature of a nonlinear crystal can be achieved with great accuracy.

[ 0104 ] Figures 7 A to 7C show plots of the normalized second harmonic mode amplitude difference dependence, (TEM _0O - TEM ₂₀ ) / (TEM ₀₀ + TEM ₂₀ ), as a function of temperature and wavelength using a pump beam having a TEM ₁₀ transverse mode profile. The plot has been generated using the approximate analytical model of Equations (11) to (15). The second harmonic mode dependence of the TEM ₁₀ second harmonic conversion is referenced to the optimum conversion efficiency at a temperature of 69.85 ⁰ C, wavelength of 1064 ran, with a crystal length of 20 mm and focusing parameter ξ = 2.7.

[ 0105 ] Figure 8 shows an arrangement for the stabilization and actively control of the frequency of a laser beam 70 from a laser 72 to be locked to a particular desired wavelength/frequency by the present device (the "test laser beam" 70). This is achieved by sampling of a small portion of the test laser beam, for example using a suitable beam splitter 74 and directing it into a mode-converter 76 (similar to the mode converting device 14 of Figure 1) for converting the transverse mode profile 78 of the test laser beam 70 into a mode-converted beam 80 with a high-order transverse mode profile 82. Pumping the nonlinear material with this mode-converted beam 80 will then generate a second harmonic beam 84 with a multi-transverse- mode profile 86. The residual portion 92 of the test laser beam 70, which passes through the nonlinear material 24 unconverted, is blocked by element 15 as before. The modes in the second harmonic beam 84 are detected with a detector 88 as previously described and the measured second harmonic higher order beam profile 86 gives a direct measure of the drift of the pump laser's wavelength/frequency - effectively, an "error signal" being the difference in the actual wavelength of test laser beam and the desired wavelength. This error signal is the fed back to the laser using known cavity feedback and automated adjustment techniques 90 to modify the laser cavity of laser 72 and bring the test laser beam 70 closer to the desired frequency.

[ 0106 ] To determine the accuracy of the present arrangement and thereby determine the line- width of device when used as a frequency locker, the normalized SH mode difference (TEM _0O - TEM20) / (TEMoo + TEM ₂₀ ) as a function of the temperature (i.e. Figure 7B) and the wavelength (i.e. Figure 7C) for a TEM ₁₀ pump mode is used. From the curves can is be seen that using a second harmonic beam of approximately 1 mW in power, and using detectors with 100% quantum efficiency, i.e. 1 mW = 1 mA, then the shot noise limited sensitivity is: 1.8 x 10 ^~8 Vhz . This implies a change of 1.8xlO ^~8 then T would be 4.25 x 1.8xl0 ^~8o C ~ 7.6x10 ^"8o C (from Figure 7C). If the slope (TEM ₀₀ - TEM ₂₀ ) / (TEM ₀₀ + TEM ₂₀ ) is changes by 1.8xlO ^~8 then dλ would be 14.5 x 1.8xl0 ^"8o C ~ 2.6xlO ^~7 nm ~ 265 KHz (from Figure 7B). This implies that a laser could be frequency locked and stabilized to a stability line-width of 264 KHz using these experimental parameters. Note that these results improve if the length of the nonlinear crystal is increased.

[ 0107 ] In further arrangements, the nonlinear material may be removable from the measurement device. The measurement device may further comprise a removable realignable mounting element adapted to mount the nonlinear material and align the nonlinear material with the input beam for measurement of a phase matching parameter of the nonlinear material. Where nonlinear material is susceptible to either transient or permanent changes in its optical phase matching parameters the arrangements may be used in applications such as monitoring of dangerous contaminants such as radioactivity. In this case the nonlinear material (which would be selected such that it experienced permanent changes in its nonlinear optical properties) would be taken to the danger (for example, it may held in removable mounting element and be worn by a user at a suitable location on their body). To examine the user's exposure, the mounting element would be placed in the optical system and the changes to the nonlinear properties of the material determined by examining the changes in the spatial mode profile of that particular nonlinear material. (It will be appreciated that at a facility where such exposure measurements are necessary, there would be a large number of removable devices, each with a nonlinear material held therein, with each removable device adaptable to be measured by the system, and accurate records of each individual removable device would be maintained to monitor the cumulative and relative changes to each particular individual nonlinear material used for the monitoring. These changes to the refractive index are then related back to an exposure dosage.) If we place the material in a machine that looks at the spatial mode change as a result of the refractive index change induced by the radioactivity (or other such phenomena) then the permanent cumulative exposure would be recorded.

[ 0108 ] In a still further arrangement, the removable mounting device can be used as a unique identification device such as for example a smart card. For example, an optical parameter of the nonlinear material may be encoded with a unique identifier, and the optical parameter is measurable by the measurement device to determine the identifier. The encoded optical parameter of the nonlinear material may be the refractive index of the nonlinear crystal and the refractive index may comprise local variations in accordance with the identifier.

[ 0109 ] It will be appreciated that further applications of the arrangements disclosed herein may be envisaged, and such applications may employ modulation techniques to be able to obtain additional information from the measurement of the output signals. The modulation may be applied either to the input laser beam, for example as an frequency modulation (eg. RF- modulation) or return-to-zero (RZ) or non-return-to-zero (NRZ) or any other type of modulation; or to the nonlinear material itself, for example an acousto-optical, electro-optical, magneto- optical modulation or other similar modulation scheme.

[ 0110 ] Much of the above discussion assumes that a negligible fraction of the fundamental pump/input beam is converted into the second harmonic beam. If a larger fraction is converted then saturation can occur and diffraction effects must be factored into the model. In further arrangements still of the optical systems and devices described herein, and depending on the strength of the input beam, this optical system may be configured to measure the optical intensity (interchangeably the optical power or irradiance) of the fundamental input pump beam. [ 0111 ] It will be appreciated that the methods/systems described and/or illustrated above at least substantially provide optical systems and methods for measurement and/or for conversion of the spatial mode of an optical signal.

[ 0112 ] The optical systems and methods described herein, and/or shown in the examples and/or drawings, are presented by way of example only and are not limiting as to the scope of the invention. Unless otherwise specifically stated, individual aspects and components of the optical systems and methods may be modified, or may have been substituted therefore known equivalents, or as yet unknown substitutes such as may be developed in the future or such as may be found to be acceptable substitutes in the future. The optical systems and methods may also be modified for a variety of applications while remaining within the scope and spirit of the claimed invention, since the range of potential applications is great, and since it is intended that the optical systems and methods be adaptable to many such variations.