{"product_id":"arago-the-spot-of-arago-physics-t-shirt","title":"Arago – The Spot of Arago Physics T-Shirt","description":"\u003cp\u003eArago raises a round shield against the beam. Dead centre of its shadow, a star still burns.\u003c\/p\u003e\u003ch3\u003eThe Physics\u003c\/h3\u003e\u003cp\u003eLight waves bending around the edge of a round, opaque disk all arrive in step on the axis behind it. In ideal Fresnel (paraxial) theory, with a point source, the centre of the shadow is about as bright as if the disk were not there. The Fresnel number tells you which regime you are in.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eN_F = a² \/ (λ L)   and, on axis, I ≈ I₀\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eN_F: Fresnel number (dimensionless)\u003c\/li\u003e\n\u003cli\u003ea: disk radius (m)\u003c\/li\u003e\n\u003cli\u003eλ: wavelength (m)\u003c\/li\u003e\n\u003cli\u003eL: disk-to-screen distance (m)\u003c\/li\u003e\n\u003cli\u003eI₀: unobstructed intensity at the same point (W\/m²)\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eIn 1818 Fresnel submitted his diffraction memoir; Poisson, judging it, derived the \"absurd\" bright spot as an objection, Arago put a disk in the beam and saw it, and Fresnel won the 1819 prize.\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e Real disks need very round, smooth edges, hence the ≈. The spot had been seen earlier and passed unnoticed: Delisle (1715), Maraldi (1723).\u003c\/p\u003e\u003ch3\u003eOn the shirt\u003c\/h3\u003e\u003cp\u003eFront: Arago, the pixel-art character, with the title. Back: the Fresnel number, the on-axis result, the 1818–1819 prize story and the earlier sightings. Printed on a unisex tee.\u003c\/p\u003e","brand":"Printify","offers":[{"title":"S \/ White","offer_id":50936592072807,"sku":"20401383021082946417","price":29.99,"currency_code":"USD","in_stock":true},{"title":"S \/ Black","offer_id":50936592105575,"sku":"47277479269323717845","price":29.99,"currency_code":"USD","in_stock":true},{"title":"M \/ White","offer_id":50936592138343,"sku":"25288475164616047604","price":29.99,"currency_code":"USD","in_stock":true},{"title":"M \/ Black","offer_id":50936592171111,"sku":"95496969982245955389","price":29.99,"currency_code":"USD","in_stock":true},{"title":"L \/ White","offer_id":50936592203879,"sku":"14058808261680967746","price":29.99,"currency_code":"USD","in_stock":true},{"title":"L \/ Black","offer_id":50936592236647,"sku":"81673796678123092766","price":29.99,"currency_code":"USD","in_stock":true},{"title":"XL \/ White","offer_id":50936592269415,"sku":"25218675517726814267","price":29.99,"currency_code":"USD","in_stock":true},{"title":"XL \/ Black","offer_id":50936592302183,"sku":"24858537258236414650","price":29.99,"currency_code":"USD","in_stock":true},{"title":"2XL \/ White","offer_id":50936592334951,"sku":"89442341327053744171","price":29.99,"currency_code":"USD","in_stock":true},{"title":"2XL \/ Black","offer_id":50936592367719,"sku":"12804978309659561097","price":29.99,"currency_code":"USD","in_stock":true},{"title":"3XL \/ White","offer_id":50936592400487,"sku":"83719795834887047359","price":29.99,"currency_code":"USD","in_stock":true},{"title":"3XL \/ Black","offer_id":50936592433255,"sku":"11121176641507777830","price":29.99,"currency_code":"USD","in_stock":true},{"title":"4XL \/ White","offer_id":50936592466023,"sku":"20448894234835279121","price":29.99,"currency_code":"USD","in_stock":true},{"title":"4XL \/ Black","offer_id":50936592498791,"sku":"26405314291600391405","price":29.99,"currency_code":"USD","in_stock":true},{"title":"5XL \/ White","offer_id":50936592531559,"sku":"32683637599118281284","price":29.99,"currency_code":"USD","in_stock":true},{"title":"5XL \/ Black","offer_id":50936592564327,"sku":"58645432220084160775","price":29.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/arago-spot_front_16_white.png?v=1793595612","url":"https:\/\/pixelatedphysics.com\/products\/arago-the-spot-of-arago-physics-t-shirt","provider":"Pixelated Physics","version":"1.0","type":"link"}