Yes, it is perfectly possible and it's one of the most sought-after options for creating truly unique products. The wood coloring acts as the base finish of the piece, upon which we can apply other personalization techniques. However, the visual result and the behavior of the material vary depending on the chosen technique:
UV Printing: The printing is done directly on the already colored wood. The colors of your design (whether a logo, names, or graphics) will always interact with the base color they are printed on, which may cause them to deviate from the original file.
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Laser Engraving: The laser beam burns the wood deeply or superficially. In this process, when engraving directly on birch, the grain of the lower layers may become darker or lighter (respectively).
Superficial :
Copyright: @elenamurfotografia
Deep:
Copyright: @fotografie.kay