The 45-year-old left the Hatters on Wednesday to join the Potters, who are 14th in the Championship table.

"The amount of times I've cried from text messages from players... it makes me emotional because I was married to the club," the Welshman said.

"It's a calculated gamble but it's the right step for me."

Jones became Luton boss in January 2016 and left them second in the third tier, having won promotion from League Two last season.

"We improved everything at the club and we were in a wonderful place," the former Brighton and Yeovil defender said.

Jones told BBC Radio Stoke: "It was a natural progression. There have been a number of opportunities to leave Luton but the project wasn't as good as the one I had at Luton.

"I was at a great club and it was the hardest decision of my life to leave, but when I met the board [at Stoke], there are so many similarities with Luton and Brighton.

"They are fans at heart but they have a clear vision for the club, they want success and are prepared to work tirelessly for it."

Timing a little bit unethical - Sweet

Luton chief executive Gary Sweet said Stoke acted "properly" in their approach for Jones, but did take issue with the timing of his exit.

Sweet told BBC Three Counties Radio that he felt his manager's departure during the January transfer window was "a little bit unethical" that "creates a lot of disruption in not just one, but maybe two camps".

As a result, Sweet believes there should be a rule introduced that prevents the movement of managers during transfer windows.

In an additional interview with BBC Look East, he said: "We've already done some transfer dealings - we're on the cusp of doing a couple of other little things - which are all on hold at the moment.

"What's the point of bringing a player in if the manager who identified that player with our recruitment staff isn't going to use him?"

Mick Harford has been placed in caretaker charge of Luton for Saturday's match at Sunderland.

Luton are one place and one point above the Black Cats, having played a game more than the Wearsiders.

Stoke City can be 'phenomenal club'

Stoke were among the pre-season favourites for promotion following relegation from the Premier League last term, but their hopes of an immediate return to the top flight are in the balance.

Eight wins from 26 Championship games have left the Potters eight points off the play-offs, and former Burton and Derby boss Gary Rowett was dismissed on Tuesday after less than eight months in charge.

Jones sees the potential for success, adding: "There are so many good things in place here. It can be a phenomenal club.

"I've not been given a timescale but I have my own targets. I believe if everyone buys into it what we do then we can be successful."

Having been on the coaching staff at Brighton before his step into management with Luton, Jones has described himself as "a student of the Championship".

He continued: "In the short term we can improve our position quickly but that will all depend on how well we do in the coming games.

"I like to play fast, attractive, attacking football but it's all based around hard work and doing all the small things right."

Jones has appointed two men who worked alongside him at Kenilworth Road to his staff at Stoke, with first-team coach Joaquin Gomez and head of sports science Jared Roberts-Smith following him to the Potters.