Because the minister, Shinjiro Koizumi, is married to Christel Takigawa, a celebrity television presenter, a  media frenzy ensued. He remarked that the fuss about his decision – with both supporters and critics weighing in – indicated that Japan was "rigid" and "old-fashioned".

According to a ranking by the Organisation for Economic Cooperation and Development, laws in Japan and South Korea grant fathers more time off for childcare than any of the world's other wealthy countries, including those in Scandinavia.

Japanese laws allow fathers to take up to a year of paid leave to care for children, just as new mothers can. Yet the participation rate among men in Japan is low – just over 6 per cent, up from about 3 per cent in 2016 – and most of those men take less than two weeks off when a child is born.

With a declining birthrate and the economic necessity that more women work, Japan desperately needs more fathers to help out at home.

In Japan's parliament, the Diet, some legislators have proposed that the government make paternity leave mandatory. A former education minister, Hirokazu Matsuno, said in June that if men took paternity leave it would help more women work and potentially help increase the country's birthrate.

Prime Minister Shinzo Abe said he understood that more men wanted to take paternity leave. "But there is a certain atmosphere and environment in companies or society which make them feel it is difficult to take it," he said.

One of the two men suing their employers in Japan is an American who was a managing director for global sales in the Tokyo office of Mitsubishi UFJ Morgan Stanley Securities, an investment bank. The man, Glen Wood, alleges he was demoted and eventually fired after he took paternity leave. Next month, the first public hearing is expected in his lawsuit.

"This is about harassing people for trying to spend time with their family and forcing men to marry their companies instead of their family," Wood said.

"We need to assure that there's work-life balance so that children can spend time with their parent and parents can spend time with their children. That has to be recognised as a universal human right."

Mitsubishi UFJ Morgan Stanley has denied paternity harassment and said it intended to fight the allegations in court, according to The Associated Press.

In the hearing on Thursday, an employee of Asics, the sportswear giant, said he had been subjected to harassment and demotion before and after he took paternity leave. The plaintiff, who requested anonymity because he fears he and his family will be attacked on social media, is seeking damages of close to $US85,000 ($123,000).

According to his legal complaint, the man, who is 38, started his career at Asics in 2011, working in the sports promotion department, often as a liaison between athletes and the company. His first demotion – to menial tasks such as scheduling fleet cars for maintenance – came after he alleged that one of his bosses drove drunk and that a colleague slapped him at a work party, according to his complaint.

His second demotion came when he returned to work after his first paternity leave of just over a year following the 2015 birth of his first son, his complaint alleges. The company assigned him to a warehouse loading and unloading goods, the complaint says. Eventually, he was transferred to the human resources department and his salary was cut.

The plaintiff's complaint says that after he finished a paternity leave with his second son – one that also lasted more than a year – he returned to his previous job in the human resources department, where he had been assigned to translate documents into English, despite having no expertise in translation.

On social media, critics who have heard about the case in the Japanese press have attacked the plaintiff for taking so much time off.

"Don't think that you'll have a place to belong when you, a man, take childcare leave for two years lol lol," wrote one commentator on Twitter. "You should be thankful that they're even allowing you to do warehouse work. You didn't work at all for two years, so don't try to think you're just the same as the other employees."

Asics issued a statement saying many of the allegations are "against the facts" and that it regretted not reaching a settlement with its employee.

Asics said it planned to continue improving its "work environment and support systems so that employees can actively participate in work even in the period of their pregnancy, childbearing or childcare".

The company said that in 2018, 100 per cent of its female employees who were eligible for maternity leave took it, while 7.8 per cent of the men did. It said it was actively working to increase the number of men taking parental leave to the level that the government had set as a national goal – 13 per cent by 2020.

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Analysts said it was not surprising that the few men who did take paternity leave were subject to pressure from their employers and colleagues.

Japanese work culture is very demanding, with employees putting in long hours late into the evenings and at weekends. Breaking long-standing patterns is difficult for workers, who not only fear they may be punished by their employers but may also believe that the success of their companies depends on their personal sacrifices.

"Men still believe that as long as they work seriously and hard in the company, then society and the family will function very well," said Naoto Sasayama, a lawyer who is representing the Asics employee. "That is the illusion they are still trying to cling to."

Many Japanese employees have a strong sense that they will impose too heavily on their colleagues if they take a leave.

"If a male full-time employee who is involved in decision-making or a core part of a project wants to take paternity leave for months, some people may question that," said Makoto Yoshida, a professor of industrial labour sociology at Ritsumeikan University in Kyoto.

Wood's lawyer, Yoshitatsu Imaizumi, said he hoped the recent paternity harassment cases might pave the way for other fathers, and make it easier for other men to exercise their right to care for their children.

"In Japan, there are a lot of good laws on the books," said Imaizumi. "But the problem is that few labourers exercise their rights under the law. We need some courageous people to exercise their rights and stand up in society."