He did say "laser scanning galvanometer", which carries a fair amount of implications on speed and accuracy - essentially you're trying to scan a video-sized vector field mechanically, and human PoV rates (depending on actual application. But think Galvo Scanners for Laser Light Show Applications - YouTube - really fast...)
(Just because we don't fully understand the application doesn't mean they haven't given a moderately good description. OTOH, if you're going to ask a "generic" community a question, it's good to phrase it in generic terms, rather than "jargon" from a particular sub-segment of the field.)"512 datapoints to form the sinewave" means 512 values in each half of the sinewave, or each whole sinewave?
In any case, an I2C 12-bit DAC like the MCP4725, running a simple driver, takes about 4 bytes for each sample. (32bits * ~1000 points/sinewave * 13 sinewaves/s is about 400kbps, which is a pretty good match for the 400kHz of "fast" I2C.
The MCP4725 also supports 3+MHz "high speed" mode, but the AVR doesn't (I don't know whether the AVR will go above 400kHz...)
But 1000 sinewaves/s * 1000 samples/sinewave * 16bits/sample (for a more optimized driver) is 16Mbps, which is completely out of the question for an AVR-class microcontroller, or an I2C DAC.
Thank you for the detailed answer. The code I used is down below.
Regarding the "precise motion" question, I still don't understand why I share that information but, it should be 0.01 degrees for the servo motor. I'm pretty sure it still means nothing to you. We are building an SD-OCT system and developing a new scanning protocol. It means nothing if I see a ladder pattern when I zoom in to a sine wave. It's because the motion must be precise to make measurements with a decent amount of error. I hope it means something now.
I just need to send the digital data as fast as I can and convert it to analog data. That was all you need to know guys. I don't understand why some of the people stuck with the other details which are out of scope. All I want to know, am I able to achieve 1 kHz sinewave using DAC? If yes, how? I need practical answers, not theoretical ones. I know how to Google it too. But I haven't seen any applications like that. The examples that I saw was not beyond 50 Hz. So that's why I said, "Any suggestions except Arduino is welcome too."
Thank you for your interest.
Code: MCP4725 Digital to Analog Converter Hookup Guide - learn.sparkfun.com