The manipulation of optical beams through a scattering medium using the measurement of its transmission matrix (TM) offers an exciting range of opportunities, from fundamental understanding of disordered media to information processing and optical imaging. The use of the TM to create a focus through a scattering medium is a key element to produce nonlinear signals for in-depth imaging in biological media. Measuring a TM by wavefront shaping and self-reference interferometry in a broadband spectral regime (typically from a 100 fs pulsed laser) permits moreover to coherently gate the incident beam so that the pulse stays inherently short enough to produce nonlinear signals. The TM knowledge moreover permits to access point scanning modality for imaging large field of views. It is still however very challenging to perform nonlinear imaging by frequency mixing through a scattering medium using this method: first because scanning a refocus exhibits non homogeneous features of the reference speckle, and second because the TM is valid only for a limited range of wavelengths which is well below the wavelength differences used in sum frequency generation, four wave mixing or Coherent Anti Stokes Raman Scattering (CARS). In this work, we exploit the angular and spectral correlation properties of the TM to produce fast sum frequency mixing imaging of large field of views through up to millimeter thick highly scattering biological tissues, such as mice spinal cords. This work opens new prospective for chemically specific imaging modalities in tissues such as CARS.
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