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The show has given Jim massive scope to create startling new effects involving Aladdin, his magic lamp, the genie who appears from it, the villain who becomes trapped in it and, of course, a magic carpet which flies without the use of wires. "Twenty years ago it couldn't have been done," he says enigmatically.

The show, which is being directed by Casey Nicholaw who choreographed Spamalot and directed The Book of Mormon, features many of the characters and songs from the 1992 Disney animated movie Aladdin, with more songs added. Aladdin also has three new buddies, but his pet monkey, Abu, doesn't make an appearance. Although Steinmeyer's work on the show is now done, he will be back on Broadway for opening night.

His name may not be widely known to the general public, but 55-year-old Jim Steinmeyer is something of a legend in the world of magic. Like a turbocharged Jonathan Creek, he has designed and created illusions for leading magicians, including Doug Henning, David Copperfield, Ricky Jay, Siegfried and Roy and Lance Burton.

For Copperfield he created the now-famous illusion of making the Statue of Liberty vanish, which was featured on a live television special; he has made animals disappear for Siegfried and Roy, whom he calls "the royalty of the entertainment world"; and both Copperfield and Henning have performed Origami, his illusion in which the magician's assistant steps into a box which folds into a small cube and is penetrated by three swords before being unfolded, whereupon the assistant emerges unharmed.

Alice Cooper used one of his effects on his recent tour, in which the singer was confined in a metal torture device then impaled with a rack of sharp spikes.

Steinmeyer is also a historian of magic with a particular interest in Victorian and Edwardian-era performers and has written dozens of books and papers about magic and magicians, although, he says: "It's a profession that's dying. It started dying when vaudeville died and there is a feeling we just missed the good old days."

He takes me though to his magician's cave – in reality, a bright, airy office in the garden of his home in Burbank, California, just down the road from the Disney headquarters, where he worked for a while developing theme-park attractions for their Imagineering division.

His office is lined with bookshelves containing books on magic; his many awards, including one from Britain's Magic Circle, are dotted around the room; posters of old-time magicians such as Howard Thurston, Harry Kellar and the Britons Jasper Maskelyne and Charles Morritt adorn the walls.

Intriguingly, two boxes, one metal and multicoloured and the other resembling a wooden coffin, are at one end of the room. Both have been used for sawing a woman in half, although, says Jim, "There are literally dozens of different ways to do that trick."

Jim Steinmeyer has the look of a magician, with a neatly trimmed, pointed grey beard and piercing eyes, yet he is modest and self-effacing, preferring instead to be the brains behind the stars of the shows.

"I have always had a fondness for stage magic but working with other performers has always been more interesting to me," he says. "I always felt my job was to support really good performers and I've never had a hankering to perform myself, although I've done a little bit now and again."

He first became intrigued by magic when he was seven years old, growing up in Chicago, and his older brother began dabbling in it. Jim began reading books about it and did tricks at school, performing at parties, school shows and clubs.

"I thought it would just be a hobby for me and I had no idea what I would do," he recalls. "I thought I would find a profession and go into advertising or something like that."

Then while he was at college he sent ideas for tricks to Doug Henning, who was doing television specials at the time. After several of them went well, Henning hired him.

He worked for Henning for seven years and was the magic designer for the musical Merlin, which starred Henning and was the first time magic had been incorporated into a Broadway musical. Although the show itself received mixed reviews, the magical illusions in it were spectacular. They included a levitating Henning and a horse vanishing in mid-air.

"Tricks are like songs, meaning they have tiny little plots, and when they're over you feel you've seen a complete little story," Jim says.

He went on to work on the musical Beauty and the Beast, which featured a boy transformed into a teacup and a big finale in which the Beast transforms back into the Prince. He also spent 18 months creating the magic effects for the Cameron Mackintosh-Disney co-production of Mary Poppins, which opened in Bristol before moving to the West End and on to Broadway.

In Las Vegas he created effects for Phantom of the Opera, including Raoul's "torture cage", which seemed to trap Raoul and hover in the air around him, threatening him with knife blades.

But when it comes to magic, there is very little new under the sun, he says. Nearly all magic tricks and illusions have been performed before, sometimes centuries ago. The Haunted House in Disneyland uses an illusion based on Pepper's Ghost, which was first performed in England by Dr John Pepper in 1862. And a woman has been sawn in half on stage ever since British magician PT Selbit did it in January 1921 at the Finsbury Park Empire.

"Magic doesn't modernise well," says Jim. "It's difficult to drag it into a current, fashionable era. Making a coin disappear in your hand isn't any better if you use a computer chip instead of a coin."

Although Jim Steinmeyer keeps the secrets behind many of his illusions to himself, in one of his many books, The Conjuring Anthology, he sets out 85 tricks for amateur magicians and reveals how they are done.

For the most part though, he says, "Magicians are guarding an empty box, because everything is an illusion. The reason I won't tell you how most tricks are done is because you would be so disappointed.

"You really don't want to know."

Aladdin opens at the New Amsterdam Theatre, New York, on March 20. Details: aladdinthemusical.com

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